Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model

We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of the site coordinates. When computing the growth process (last-passage times) at each site, the horizontal and vertical additive components of the rates are allowed to also vary respectively with the column and row number of that site. This setting includes several models of interest from the literature as special cases. Our main result provides simple explicit variational formulas for the a.s. first-order asymptotics of the growth process under a decay condition on the rates. Formulas of similar flavor were conjectured in arXiv:math/0004082, which we also establish. Subject to further mild conditions, we prove the existence of the limit shape and describe it explicitly. We observe that the boundary of the limit shape can develop flat segments adjacent to the axes and spikes along the axes. Furthermore, we record the formation of persistent macroscopic spikes and crevices in the cluster that are nonetheless not visible in the limit shape. As an application of the results for the growth process, we compute the flux function and limiting particle profile for the TASEP with the step initial condition and disorder in the jump rates of particles and holes. Our methodology is based on concentration bounds and estimating the boundary exit probabilities of the geodesics in the increment-stationary version of the model, with the only input from integrable probability being the distributional invariance of the last-passage times under permutations of columns and rows.

1. Introduction 1.1. Some background and the contribution of the present work. Stochastic growth far from equilibrium arises in models of diverse phenomena such as propagation of burning fronts, spread of infections, colonial growth of bacteria, liquid penetration into porous media and vehicular traffic flow [34,45]. In these models, a growth process describes the timeevolution of a randomly growing cluster that represents, for example, a tissue of infected cells. Mathematical study of growth processes dates at least back to Eden's model [23]. A fundamental type of result in this subject is that the cluster associated with a given growth process acquires a deterministic limit shape in a suitable scaling limit, analogously to the classical law of large numbers. Understanding the geometric properties of limit shapes has been one of the main research themes, for example, in percolation theory since at least the seminal work of D. Richardson [52]. See this brief introduction [19] and survey articles [3,18,48].
Of particular interest is to determine whether and in what manner local inhomogeneities in a growth model are manifested in the limit shape. This line of inquiry was pursued in both mathematics and physics literature with some early work in the 1990s, particularly on disordered exclusion and related growth processes [11,35,36,41,44,56,60,63,64]. It has now been rigorously observed in various settings that suitably introduced inhomogeneity into the parameters of a growth model can create new geometric features in the limit shape including flat segments [4,7,25,29,32,33,43,60,61], spikes [9], corners [29] and pyramids [1]. Such features are often an indication of a phase transition at the level of fluctuations of the growth process as observed, for example, in [5,7,10,32,33,43].
Significant research has been devoted particularly to the corner growth model (CGM), which enjoys rich connections to a variety of other models including last-passage percolation (LPP), random polymers, queueing systems, interacting particle systems, Young tableaux and determinantal point processes. The intense activity surrounding this model in the past three decades has been driven in part by rigorous results in connection with the Kardar-Parisi-Zhang universality [42]. Reviews of the CGM from many perspectives are available, for example, in [12,16,40,58]. Subsection 1.2 will briefly recall the model and the notion of limit shape therein.
The present paper revisits the exactly solvable, inhomogeneous CGM from [13,39] that has independent and exponentially distributed waiting times with possibly distinct rates given by an addivitely separable function of the site coordinates. Hence, the inhomogeneity can be represented in terms of real parameters (namely the additive components of the rates) attached to the columns and rows. The model arises naturally in several contexts, including from the totally asymmetric simple exclusion process (TASEP) with the step initial condition and particlewise and holewise disorder. As elaborated on in Subsection 1.4, we generalize the model slightly by computing the growth process at each site from a distinct collection of waiting times. With this enhancement, the model in particular unifies the following somewhat disparate settings from the literature.
(i) Random rates from some work in the 1990s on TASEP with the step initial condition and particlewise disorder [11,44,60] and more recently in [25,26]. (ii) Macroscopically inhomogeneous rates as in [14,29] in the special case that the speed function is additively separable. A discrete version of such a model (with geometrically distributed waiting times) appeared recently in [43]. (iii) Fixed defective rates on the south or west boundaries as in [6], on a thickened west boundary (a few columns) as in [5] and, more generally, on thickened south and west boundaries (a few columns and rows) as in [10]. (iv) Suitably rescaled defective rates in a few columns and rows considered in [10,13]. The discrete version of the model in [10] appeared later in [17]. (v) Defective rates near north or east boundaries used in [59].
Precise connections to the above models are made in Subsection 3.9.
When the rates are identical, a well-known result of H. Rost [54] identifies the limit shape as a certain explicit parabolic region (recalled in (1.6)). In this homogeneous case, the limit shape completely governs the growth of the cluster to the leading order in time. The purpose of this work is to study the macroscopic evolution of the cluster in presence of inhomogeneity.
We find that the inhomogeneity can influence the cluster qualitatively in two aspects, and the limit shape can fail to capture the full picture of growth at the macroscopic scale. We focus on the columns in the following discussion as analogous remarks hold also for the rows. First, when the smaller column parameters are sufficiently rare, the cluster evolves into an approximately flat shape near the vertical axis. This behavior creates a flat segment in the boundary of the limit shape adjacent to the vertical axis, and has been observed earlier in [25,60]. Second, the cluster can grow at distinct speeds across columns leading to persistent macroscopic variations in the height profile of the cluster. As a result, after a while the cluster visually resembles a structure with large crevices and spikes that do not disappear over time. Out of these features, the limit shape only remembers the maximal size of the spikes, and registers this information as a spike (line segment) along the vertical axis emanating from the vertical intercept of its boundary inside the quadrant. A systematic treatment of the formation of spikes and crevices in the CGM seems to be new.
In this paper, we obtain a complete and exact description for the macroscopic evolution of the cluster and elucidate the growth behavior outlined above. Our main result describes the first-order asymptotics of the growth process in terms of an explicit variational formula under a mild condition ensuring at least linear growth. With further reasonable assumptions, the formula leads to an exact description of the limit shape. In this model, the limit shape still controls the macroscopic growth asymptotically at sites increasingly away from both axes. The formula also yields the leading order growth of the cluster along a fixed set of columns or rows. This information is, in general, not encoded in the limit shape.
As an application of the results for the CGM, we also describe the macroscopic evolution of the particles in the associated disordered TASEP. In particular, we derive the flux function and limiting particle profile from the limit shape. Subsection 1.5 provides a more detailed account of our results.
It has come to our attention that a statement somewhat similar to our main result was conjectured by E. Rains in [51,Conjecture 5.2], which also contains analogous claims for various other integrable percolation models. Although peripheral to the present work, we reformulate and prove the part of the conjecture pertinent to our setting to highlight the connection.
A sequel [27] to the present paper will study geometric features such as Busemann limits, geodesics, and the competition interface in the inhomogeneous LPP model, and also apply these results to an inhomogeneous tandem of queues.
There is also some technical novelty in the treatment of the model. In the present setting, the existence of the limit shape does not follow from standard subadditive arguments and takes up a significant part of the paper to establish. We attain this through concentration bounds for the growth process, development of which utilizes explicit increment-stationary versions of the process. The arguments can likely be adapted to prove similar results for the geometric counterpart of the current model.
As proved in [13], the CGM studied here is connected to Schur measures [50] and thereby possesses a determinantal structure. In particular, the one-point distribution of the growth process can be written in terms of a Fredholm determinant with an explicit kernel. We take advantage of one feature that follows from this representation, namely, the distributional invariance of the growth process under permutations of columns and rows (stated in Lemma 4.4). Apart from this point, our methodology (described in Section 1.6) does not rely on integrable probability.
1.2. Limit shape in the CGM. The general two-dimensional CGM consists of a given collection of nonnegative real-valued random waiting times tωpi, jq : i, j P Z ą0 u and a corner growth process tGpi, jq : i, j P Z ą0 u defined through the recursion Gpi, jq " maxt1 tią1u Gpi´1, jq, 1 tją1u Gpi, j´1qu`ωpi, jq for i, j P Z ą0 . (1.1) This process represents a randomly growing cluster in the first quadrant of the plane, given at time t P R ě0 by Rptq " tpx, yq P R 2 ą0 : Gprxs, rysq ď tu. (1.2) In other words, the unit square pi´1, isˆpj´1, js is added to the cluster at time t " Gpi, jq for i, j P Z ą0 . The closure of (1.2) in R 2 ě0 is given by Rptq " tpx, yq P R 2 ě0 : Gprxs`1 tx"0u , rys`1 ty"0u q ď tu for t P R ě0 . (1. 3) The limit shape of the cluster is defined as the limiting set R " lim tÑ8 t´1Rptq, (1.4) with respect to the Hausdorff metric (on nonempty, closed, bounded subsets of R 2 ě0 , see Appendix A.4) provided that the limit exists 1 . The definition will be slighly modified in Subsection 3.5 via suitable truncation in the case of superlinear growth in time.
The first instance of the CGM in the literature had i.i.d. exponential waiting times and appeared in connection with a fundamental interacting particle system, TASEP, in a pioneering work [54] of H. Rost. Recall that the standard TASEP [62] is a continuous-time Markov process on particle configurations on Z that permit at most one particle per site (exclusion), and evolves as follows: Each particle independently attempts to jump at a common rate c ą 0 to the next site to its right. Per the exclusion rule, the jump is allowed only if the next site is vacant. The dynamics is unambiguously defined since simultaneous jump attempts a.s. never happen. To connect with the CGM, start the TASEP from the step initial condition meaning that the particles occupy the sites of Z ď0 at time zero. Label the particles with positive integers from right to left such that particle j is initially at site´j`1 for j P Z ą0 . Let Tpi, jq denote the time of the ith jump of particle j, and write ω 1 pi, jq " Tpi, jq´maxt1 tią1u Tpi´1, jq, 1 tją1u Tpi, j´1qu for i, j P Z ą0 . (1.5) By the strong Markov property, ω 1 pi, jq " Exppcq and are jointly independent for i, j P Z ą0 . Then, since the recursions in (1.1) and (1.5) are the same, the T-process is equal in distribution to the G-process defined with i.i.d. Exppcq-distributed waiting times.
A celebrated result in [54], based on the above correspondence with TASEP, identifies the limit shape of the CGM with i.i.d. Exppcq waiting times as the parabolic region given by R " tpx, yq P R 2 ě0 : ?
x`?y ď ? cu. (1.6) If the waiting times are i.i.d. and geometrically distributed, R is also explicitly known as a certain elliptic region [15,37,57]. More generally, for i.i.d. waiting times subject to mild conditions, the limit in (1.4) still exists and is a concave region with the boundary inside R 2 ą0 extending continuously to the axes [47]. Furthermore, the limit can be characterized in terms of variational formulas over certain infinite dimensional spaces [31]. However, these formulas presently do not yield detailed geometric information about the limit shape except in the above exactly solvable cases. For example, it is unclear precisely when the limit shape has flat segments in the boundary. If the waiting times attain their maxima frequently enough to create an infinite cluster of oriented percolation, the boundary of the limit shape becomes flat in a cone symmetric around the diagonal of the plane [31]. In the context of undirected 1 The definition of the cluster in some literature can differ slightly from (1.2). For example in [47], the growth process lives on Z 2 ě0 and the cluster at a given time is defined as a closed subset of R 2 ě0 with the floor function instead of the ceiling function. These variations do not have any impact on the limit shape.
first-passage percolation, this phenomenon goes back to the classic paper of R. Durrett and T. Liggett [22], and was subsequently studied in [2,46]. It is not known whether this is the only mechanism to produce flat segments with i.i.d. waiting times. Due to the limited knowledge in the general i.i.d. case, a natural starting point as a homogeneous setting for our study into the effects of inhomogeneity is the i.i.d. exponential model.

1.3.
Simulations of flat segments, spikes and crevices. Varying the rates of the exponential waiting times can create flat spots, spikes and crevices in the evolving shape of the cluster. Let us illustrate these features through some simulations of the CGM deferring their rigorous discussion to Subsections 3.6 and 3.7.
The simulations below share a common sample of independent Expp1q-distributed waiting times tωpi, jq : i, j P rN su where N " 4000. Each simulation constructs waiting times with specific rates λ m,n pi, jq ą 0 by setting ω m,n pi, jq " ωpi, jq λ m,n pi, jq " Exppλ m,n pi, jqq for m, n P rN s, i P rms, j P rns.
The value Gpm, nq of the growth process at each site pm, nq P rN s 2 is then computed through (1.1) with ω m,n pi, jq in place of ωpi, jq for i P rms, j P rns. Finally, the cluster Rptq is computed from (1.2) at time t " 1000. Figure 1.1 depicts a realization of Rptq together with the boundary of the limit shape approximation tR in four cases. For comparison, Figure 1.1a covers the homogeneous case where the rates are 1 and R is given by (1.6) with c " 1. In the remaining cases, R is the subset of R 2 ě0 given by R " tx ě y and ?
x`?y ď 1u Y tx ď y and 2px`yq ď 1u Y tp0, yq : 1{2 ď y ď 1u, (1.8) as can be seen Theorem 3.10. The justification for the asserted asymptotic sizes of the spikes and crevices below is in Subsection 3.7.
In Figure 1.1b, the rates along column i " 100 are altered to 0.5. Then the cluster becomes approximately flat above the diagonal to the right of column 100. This results in the flat segment between p0, 1q and p0.25, 0.25q in the boundary of R. To the strictly left of column 100, the cluster develops tall (order t) spikes relative to the boundary of tR. The spikes persist in the long time limit leaving a trace in the limit shape in the form of the vertical segment from p0, 0.5q to p0, 1q. The length of this segment indicates the asymptotic length of the maximal vertical spike in the cluster. In this example, all spikes are of the same length 0.5t asymptotically although this is not visible at time t " 1000.
The rates are further altered along column 50 to 0.75 in Figure 1.1c. As before, spikes appear stricly to the left of column 100. However, the spikes on the first 49 columns are now larger (order 0.5t) than the ones between columns 50 and 100 (order 0.25t). The cluster's true order of growth on columns with smaller spikes is not recorded in the limit shape.
The rates in Figure 1.1d are modified from the homogeneous case as follows: When computing the growth process in the first 99 columns, the rates on column 50 are taken as 0.25. For the remaining values of the G-process, the rates are set to 0.50 on column 100. Deep crevices now form between columns 50 and 100 where the height of the cluster is order 0.25t below the boundary of tR. The limit shape retains no information about these features.
1.4. Exponential CGM with inhomogeneous rates. In the basic version of our setting, ωpi, jq " Exppa i`bj q for i, j P Z ą0 for some real parameter sequences a " pa i q iPZ ą0 and   b " pb j q jPZ ą0 . To have positive rates the parameters are assumed to satisfy a i`bj ą 0 for i, j P Z ą0 . The earliest appearances of the above CGM were perhaps in [13,39]. The model arises via a limit transition from the CGM considered earlier in [38]. The latter has independent geometric waiting times with multiplicatively separable inhomogeneity in fail parameters and comes from a Schur measure [50]. As proved in [13][Theorem 1], the present model is closely linked to the complex Wishart ensemble (also known as Laguerre ensemble) in the sense that the square root of the largest singular value of a natural generalization of an mˆn sized realization of this ensemble has the same distribution as Gpm, nq for m, n P Z ą0 . This correspondence was observed earlier in [5, Proposition 6.1] when a or b is a constant sequence, and generalized later to the process level in [20].
The model can also be naturally motivated as a TASEP with the step initial condition, and particlewise and holewise disorder. The disorder in rates translates to the feature that the attempts for the ith jump of particle j occur at rate a i`bj for i, j P Z ą0 . For an alternative viewpoint, imagine the holes (empty sites) as another class of particles labeled with positive integers such that hole i is at site i for i P Z ą0 at time zero. Hole i moves by exchanging positions with the particle to its immediate left at rate a i and particle j moves by doing the same with the holes to its immediate right at rate b j for i, j P Z ą0 . Hence, when they encounter, hole i and particle j exchange positions at net rate a i`bj , and this exchange is precisely the ith jump of particle j for i, j P Z ą0 .
In this paper, we consider the following slightly more general setting that will permit us to simultaneously treat the models alluded to in items (i)-(vi) in Section 1.1. Fix two collections of real parameters a " ta m piq : m P Z ą0 and i P rmsu and b " tb n pjq : n P Z ą0 and j P rnsu. (1.9) Abbreviate a m " pa m piqq iPrms and b n " pb n pjqq jPrns for m, n P Z ą0 . Assume that a m piq`b n pjq ą 0 for m, n P Z ą0 and i P rms, j P rns (1.10) or, equivalently, min a m`m in b n ą 0 for m, n P Z ą0 . (1.11) For each m, n P Z ą0 , let ω a,b m,n pi, jq " Exppa m piq`b n pjqq and jointly independent for i P rms and j P rns, (1.12) and define G a,b pm, nq from these waiting times by (1.1). Then, for t P R ě0 , define the region R a,b t from the growth process tG a,b pm, nq : m, n P Z ą0 u by (1.2). A few points worth emphasizing: The G a,b -process itself does not necessarily satisfy the recursion (1.1) because the waiting times in (1.12) are allowed to vary with m, n. By the same token, G a,b -process need not be coordinatewise nondecreasing. Therefore, R a,b t may no longer be a connected subset of R 2 ą0 although we shall continue to call it a cluster. As corollaries we obtain results for the height process and cumulative particle current (flux process) of TASEP. In terms of the CGM these are defined respectively by H a,b pn, tq " maxtsuptm P Z ą0 : G a,b pm, nq ď tu, 0u (1.13) F a,b pm, tq " maxtsuptn P Z ą0 : G a,b pm`n´1, nq ď t, 0u (1.14) for m, n P Z ą0 and t P R ě0 . In the absence of m, n-dependence in (1.12) for m, n P Z ą0 , (1.13) gives the number of jumps executed by particle n by time t and also the height (namely, length) of the nth row of the cluster at time t, while (1.14) counts the number of particles that have jumped from site m´1 to site m by time t. See Subsection 2.4 for more details. These interpretations, although not valid in the full generality of (1.12), justify the names of the processes in (1.13)-(1.14).
1.5. Discussion of the main results. The main contributions of this paper are exact firstorder asymptotics of the growth process that lead to fairly explicit descriptions of the limit shape and the limiting flux function. Precise results are stated in Section 3. For the moment, we summarize some key points.
An explicit centering for the growth process (Theorem 3.2). The central result of the paper computes an explicit, deterministic approximation to the first order (a centering for short, see Definition 3.1 below for the precise meaning) for the growth process under a mild growth condition on the means of the waiting times. More specifically, assuming that min a m`m in b n does not decay too fast as m`n grows, G a,b pm, nq is a.s. approximately equal to provided that m`n is sufficiently large. Here, z serves as a convenient parameter indexing the increment-stationary versions of the growth process. This result is obtained by first developing summable concentration bounds for the growth process.
Resolution of a conjecture due to E. Rains (Theorem 3.4). A formula similar to (1.15) appeared in [51] within the continuous counterpart of Conjecture 5.2, which is not stated explicitly but can be discerned from the context. We state and prove the part of the conjecture related to the present model again by means of concentration bounds.
Shape function (Theorem 3.6). Assume further that the running minima min a n and min b n converge to some a, b P R Y t8u with a`b ą 0, respectively, and the empirical distributions associated with a n and b n converge vaguely to some subprobability measures α and β, respectively, on R as n Ñ 8. Then (1.15) leads to the simpler a.s. approximation for G a,b pm, nq when both m and n are sufficiently large. The centering in (1.16) is unique with the property that it extends to a positive-homogeneous and continuous function on R 2 ě0 . This extension is the shape function (see Definition 3.5) of the growth process. In many variants of the CGM from the literature, the shape function can be either represented as or derived from (1.16). Subsection 3.9 records numerous corollaries to this effect. Theorem 3.6 is a considerable strengthening of [25, Theorem 2.1] which derived the approximation (1.16) as m, n grow large along a fixed direction and in the special where the parameters in (1.9) are not m, n-dependent and are randomly chosen subject to a joint ergodicity condition. This condition enabled [25] to utilize subadditive ergodic theory to obtain the existence of the shape function and then compute it through convex analysis from the shape functions of the increment-stationary growth processes. An obstruction to implementing the above approach in the present setting is that the waiting times in (1.12) are not stationary with respect to lattice translations and, therefore, the existence of the shape function is no longer guaranteed by standard subadditive ergodic theory.
Growth near the axes (Theorem 3.7). Another consequence of (1.15) is that G a,b pm, nq is a.s. approximately # n ş R pb`min a m q´1βpdbq when n is large and m{n is small m ş R pa`min b n q´1αpdaq when m is large and n{m is small (1.17) provided that inf a`inf b ą 0 and the appropriate half of the vague convergence assumption above holds. In particular, (1.17) describes the asymptotics of the growth process along a fixed column or row. The result demonstrates the possibility of macroscopically uneven growth in the cluster, for example, across columns and reveals the underlying reason for this as the variations in the min a m sequence. Near the axes, (1.16) is approximately given by when n is large and m{n is small m ş R pa`bq´1αpdaq when m is large and n{m is small (1.18) in contrast with (1.17). Discrepancies in these approximations are manifested as macroscopic spikes and crevices in the cluster relative to the boundary of the limit shape near the axes as shown in Figure 1    2. An illustration of the cluster along column m P Z ą0 and after columns with much larger indices at time t (red) in the case min a m ‰ a. The lines y ş R pb`aq´1βpdbq " t (blue) and y ş R pb`min a m q´1βpdbq " t (purple) are shown. (a) A spike forms when min a m ą a (b) A crevice forms when min a m ă a.
Limit shape (Theorem 3.10). With the aid of (1.16)-(1.17) and assuming α, β ‰ 0, the limit shape (in the sense of (1.4)) can be identified as the union of the sublevel set " px, yq P R 2 ě0 : inf of the shape function, and the line segments min a m and B " sup nPZ ą0 min b n . See Figure 1.3 for an illustration. (When α " 0 or β " 0, the above set is unbounded but can still be viewed as the limit shape in a truncated sense, see Subsection 3.5). Computations behind the subsequent discussion are either omitted or postponed to Subsections 3.5-3.6. The statements pertinent to the vertical axis have obvious analogues for the horizontal axis. 3. An illustration of the boundary of the limit shape (blue) and the boundary of the region (1.22) (dashed gray). The strictly concave part and the (possibly empty) flat segments and spikes of the limit shape are indicated.
The boundary of the limit shape inside R 2 ą0 connects to the axes at the pointŝ Comparing with (1.20) shows that the limit shape has a vertical spike, namely, a vertical line segment above the second intercept in (1.21), if and only if A ą a. The latter is the precise condition for the occurrence of a vertical spike in the cluster. Thus, the limit shape retains some residual memory of the spikes in the cluster by encoding their maximal size (to the first-order asymptotics) as the lengths of its spikes. However, the crevices and non-maximal spikes of the cluster, despite being persistent macroscopic scale structures, are not visible in the limit shape.
The boundary of the limit shape can be explicitly parametrized. It is curved (strictly concave) inside the nonempty conic region given by 22) and is flat elsewhere. In particular, the boundary has a flat segment inside R 2 ą0 adjacent to the vertical axis if and only if ş R pa´aq´2αpdaq ă 8. This condition indicates that the small parameters in a m become sufficiently infrequent as m Ñ 8. It holds precisely when a ă inf supp α or ş R pa´inf supp αq´2αpdaq ă 8 since a ď inf supp α. The formation of flat segments can be understood geometrically in terms of the coalescence of geodesics in the associated LPP model, which we leave to the sequel [27].
Formulas (1.19)-(1.20) illuminate how the inhomogeneity introduced through the parameters a and b at the microscopic scale propogates to the limit shape. This happens by means of three partially independent mechanisms: the limiting empirical measures α, β, the limiting running minima a, b and the maximal running minima A, B. The dependence on the running minima implies that changing the means of the waiting times in a single column or row can alter the limit shape. This feature is reminiscent of the sensitivity of the flux function to a slow bond in TASEP [8].
The limiting height and flux functions (Theorem 3.11). The knowledge of the shape function also leads to explicit centerings for the height and flux processes. H a,b pn, tq and F a,b pm, tq are a.s. approximately equal to respectively, for sufficiently large m, n and all t. The formulas above are obtained from (1. 16) assuming further that the measures α, β are nonzero. We shall refer to (1.23) and (1.24) as the limiting height and flux functions, respectively.
Height of a fixed row (Theorem 3.12). In the case of spikes/crevices near the horizontal axis, (1.23) can be a poor approximation of the height of a fixed row. In fact, a correct centering for H a,b pn, tq is for fixed n and sufficiently large t. This result is derived from the behavior of the growth process near the axes.
1.6. Methodology. We study the growth process G a,b through couplings with its incrementstationary versions p G a,b,z indexed by the z-parameter in (1.15). The horizontal p G a,b,zincrements are independent along any row and exponentially distributed with explicit rates that are invariant under vertical translations, and an analogous statement holds for the vertical increments. This feature is sometimes referred to as the Burke property in reference to Burke's theorem from queueing theory since the increments correspond to interarrival/departure times of customers in an interpretation of the CGM as M/M/1 queues in tandem. The details can be found, for example, in [48,Section 7].
Due to the distributional structure of the increments, concentration bounds for sums of independent exponentials show that the p G a,b,z -process concentrates around the expression inside the infimum in (1.15) with overwhelming probability. Utilizing the coupling with z chosen as the unique minimizer ζ " ζ a,b pm, nq in (1.15), which exists due to strict convexity, we then derive similar concentration bounds for the G-process. The right tail bound comes easily since the G a,b -process is dominated by the p G a,b,z -process for each z. For the left tail bound, we first show that in the LPP representation of p G a,b,ζ the geodesic from the origin to pm, nq exits the boundary close to the origin with overwhelming probability assuming monotonicity of the parameters. In this case, a left tail bound for G a,b pm, nq can be extracted from that of p G a,b,z since the two quantities are close. On the other hand, the distributional invariance of the G a,b pm, nq under permutations of the parameters a m and b n implies that the bound continues to hold without the monotonicity condition. The bounds obtained in this manner are not sharp but suffice for the purposes of first-order asymptotics.
In the context of percolation and directed polymer models, the idea of coupling with increment-stationary processes to compute limit shapes dates back to [57].
An alternative path to the results proved in this work is to utilize the determinantal structure in the model. For example, formula (1.15) can be predicted from the correlation kernel. To obtain asymptotics in the strength of the present work, one would likely still turn to summable tail bounds for the growth process. Developing such bounds from the correlation kernel appears more involved than the more elementary arguments used here.

1.7.
Outline. The remainder of this text is organized as follows. Section 2 casts the growth process as an LPP process with inhomogeneous exponential weights. This section also constructs the TASEP with the step initial condition and disorder in particles and holes from the growth process. The main results are formulated precisely in Section 3. Concentration bounds for the growth process are developed in Section 4. The centerings (1.15) and (1.16) are derived in Sections 5 and 7, respectively. Section 8 obtains approximations to the growth process near the axes. Section 9 computes the limit shape. Section 10 computes the limiting flux and height functions for the disordered TASEP. Section 11 contains the proofs of various applications of the main results. Some standard and auxiliary facts are recorded in Appendix A.
1.8. Notation and conventions. Let Z, Q, R, and C denote the spaces of integers, rational, real and complex numbers, respectively. For a P R, define Z ěa " ti P Z : i ě au and make analogous definitions if the set is replaced with R or the subscript is replaced with ą a, ď a or ă a. Write H for the empty set. For n P Z ą0 , rns " t1, 2, . . . , nu with the convention that rns " ∅ for n P Z ď0 . For x P R, write rxs " inf Z ěx and x`" maxtx, 0u.
For a real sequence pc i q iPZ ą0 , write c min p,n " mintc i : i P rns rp´1su for n P Z ą0 and p P rns, and abbreviate c min 1,n " c min n for n P Z ą0 . For k P Z ě0 , denote with τ k the shift map yq for x, y, c ą 0. Being an increasing or decreasing function is understood in the strict sense.
For any space X and subset A Ă X, write 1 A for the indicator function of A that equals 1 on A and 0 on the complement X A. For any function f : A Ñ R Y t8,´8u, the product 1 A f denotes the function that equals f on A and 0 on X A. If X is a topological space, A denotes the closure of A in X.
The support of a Borel measure µ on R is the set supp µ " R U where U Ă R is the largest open set with µpU q " 0.
For λ P R ą0 , the exponential distribution with rate λ, denoted Exppλq, is the Borel measure on R with density x Þ Ñ 1 txą0u λe´λ x . Its mean and variance are λ´1 and λ´2, respectively. The statement X " Exppλq means that the random variable X is Exppλq-distributed.
For x P R, the Dirac measure δ x is the Borel probability measure on R such that δ x txu " 1.
A sequence of events pE n q nPZ ą0 in a probability space occurs with overwhelming probability if for any p P Z ą0 the probability of E n is at least 1´C p n´p for n P Z ą0 for some constant C p ą 0 dependent only on p.

LPP with inhomogeneous exponential weights
We study the corner growth process through its usual LPP representation. In this section, we reintroduce the model from the percolation perspective and discuss its special features due to the exponential weights (waiting times) that contribute to our analysis. We also discuss the disordered TASEP associated with the growth process and introduce several related quantities that will be of interest in the sequel.

2.1.
Last-passage times, geodesics and exit points. Let Ω " R Z 2 ą0 and p Ω " R Z 2 ě0 . Each ω " tωpi, jq : i, j P Z ą0 u P Ω represents a weight configuration on the bulk sites Z 2 ą0 while each p ω " tp ωpi, jq : i, j P Z ě0 u P p Ω represents a weight configuration on the bulk sites plus the boundary sites Z ě0ˆt 0u Y t0uˆZ ě0 .
A finite sequence π " pπ i q iPrps in Z 2 is an up-right path if π i´πi´1 P tp1, 0q, p0, 1qu for 1 ă i ď p. For k, l, m, n P Z, write Π m,n k,l for the set of all up-right paths with π 1 " pk, lq and π p " pm, nq.
Define the last-passage times on Ω by We work with k ď m and l ď n in the sequel, in which case Π m,n k,l is nonempty and the maxima above are finite. Any maximizer π P Π m,n k,l in (2.1) or (2.2) is called a geodesic. Being finite and nonempty, Π m,n k,l contains at least one geodesic. We abbreviate Gpm, nq " G 1,1 pm, nq (consistently with (1.1)) and p Gpm, nq " p G 0,0 pm, nq. For m, n P Z ą0 , k, l P Z ě0 with k ď m and l ď n, define the horizontal and vertical exit points by respectively. p H k pm, nq is the maximal i P tk, . . . , mu such that pi, 0q P π for some geodesic π P Π m,n k,0 , and then pi, 0q is the site where π exits the horizontal boundary Z ě0ˆt 0u. Likewise, p V l pm, nq is the maximal j P tl, . . . , nu such that there exists a geodesic in Π m,n 0,l that exits the vertical boundary t0uˆZ ě0 at site p0, jq. We abbreviate p Hpm, nq " p H 0 pm, nq and p Vpm, nq " p V 0 pm, nq.

2.2.
Bulk LPP process. Let P denote the Borel probability measure on Ω under which tωpi, jq : i, j P Z ą0 u are independent and ωpi, jq " Expp1q for i, j P Z ą0 .

(2.5)
Write E for the corresponding expectation. For m, n P Z ą0 , let tω m,n pi, jq : pi, jq P rmsˆrnsu be a collection of independent Expp1q-distributed random variables on the probability space pΩ, Pq. No assumption is made about the joint distribution of ω m,n pi, jq and ω m 1 ,n 1 pi 1 , j 1 q if pm, nq ‰ pm 1 , n 1 q.
Introduce inhomogeneity (disorder) through real parameters in (1.9) assumed to satisfy (1.10). A weight (waiting time) process with property (1.12) can be defined on pΩ, Pq by setting for m, n P Z ą0 and i P rms, j P rns.
m,n denote the distribution of the weights tω a,b m,n pi, jq : i P rms, j P rnsu under P for m, n P Z ą0 . In other words, P a,b m,n is the Borel probability measure on R rmsˆrns under which tωpi, jq : pi, jq P rmsˆrnsu are independent and ωpi, jq " Exppa m piq`b n pjqq for pi, jq P rmsˆrns.
Define the bulk LPP process via (2.1) using the weights in (2.6). Namely, the value of the process at site pm, nq is given by m,n pi, jq for m, n P Z ą0 and i P rms, j P rns.
This is a particular construction of the corner growth process discussed in Subsection 1.4.
Consider the special case ω m,n pi, jq " ωpi, jq, a m piq " a and b n pjq " b for m, n P Z ą0 and i, j P rmsˆrns (2.9) for some a, b ą 0 with c " a`b ą 0. This recovers the LPP process with i.i.d. Exppcq weights. Let us call the corresponding CGM/LPP model homogeneous. In this case, we will often replace the superscript a, b with c, for example, writing P c for the distribution of the bulk weights.

Stationary last-passage increments. The horizontal and vertical p
G-increments are defined by respectively, for m, n P Z ě0 . From the definitions, p Ipm, 0q " p ωpm, 0q and p Jp0, nq " p ωp0, nq for m, n P Z ą0 .
Let I a,b m,n " p´min a m , min b n q for m, n P Z ě0 with the convention min a 0 " min b 0 " 8. For m, n P Z ě0 and z P I a,b m,n , let p P a,b,z m,n denote the Borel probability measure on R prmsYt0uqˆprnsYt0uq under which (2.12) tp ωpi, jq : i P rms Y t0u, j P rns Y t0uu are independent, p ωp0, 0q a.s.
Under p P a,b,z m,n the bulk weights tp ωpi, jq : i P rms, j P rnsu have distribution P a,b m,n described in (2.7) and p G-increments are stationary in the sense that (2.13) t p Ipi, nq : i P rmsu are independent with p Ipi, nq " Exppa m piq`zq and t p Jpm, jq : j P rnsu are independent with p Jpm, jq " Exppb n pjq´zq.
A stronger version of this property is in [6, Lemma 4.2] for constant parameters. The extension to the general case is sketched in [25].
We study the bulk LPP process mainly through the coupling ωpi, jq " p ωpi, jq for i, j P Z ą0 . Then G k,l pm, nq " p G k,l pm, nq for m, n, k, l P Z ą0 , and for m, n P Z ą0 follows from definitions (2.3)-(2.4). Utilizing the stationarity and the independence structure of p G-increments, we establish sufficient control over the exit points and then gain access to the bulk LPP process via (2.14).
2.4. TASEP with particlewise and holewise disorder. Assume now that ωpi, jq ě 0 for i, j P Z ą0 . Define the height of an interface over site n P Z ą0 at time t P R ě0 by Hpn, tq " maxtsuptm P Z ą0 : Gpm, nq ď tu, 0u. (2.15) Since the weights are nonnegative, G is coordinatewise nondecreasing. Hence, Hpn, tq is nonincreasing in n and nondecreasing in t. Note that Hpn, tq also measures the width of the cluster in (1.2) at level n and time t.
The height variables also represent evolving configurations of particles on Z as follows: The position of particle n P Z ą0 at time t P R ě0 is given by σpn, tq " Hpn, tq´n`1.
Since σpn, tq is decreasing in n and nonincreasing in t, the particles move right over time retaining their order. In particular, each site is occupied by at most one particle at any time. Assume further that ωpi, jq ą 0 for i, j P Z ą0 . Then each particle jumps one step at a time and the particles start from the step initial condition i.e. σpn, 0q "´n`1 for n P Z ą0 .
Since the particles are initially at negative sites, it follows that Fpi, tq counts the number of particles that have jumped from i´1 to i by time t.
Define the height process tH a,b pn, tq : n P Z ą0 , t P R ě0 u, the particle process tσ a,b pn, tq : n P Z ą0 , t P R ě0 u and the flux process tF a,b pm, tq : m P Z ą0 , t P R ě0 u through (2.15), (2.16) and (2.17), respectively, using the bulk LPP process G a,b in place of G. We will provide first-order approximations to these processes.
The disordered TASEP arises in the special case a m piq " a i and b n pjq " b j for m, n P Z ą0 , i P rms, j P rns.

Main results
We state our main results in this section. Throughout, fix two collections of real parameters a " ta m piq : m P Z ą0 , i P rmsu and b " tb n pjq : n P Z ą0 , j P rnsu subject to condition (1.10).
3.1. An explicit centering for the LPP process. Definition 3.1. We call a (deterministic) function F : Z 2 ą0 Ñ R a centering for the G a,bprocess if for any ą 0, P-a.s., there exists a random L P Z ą0 such that The definition does not determine F uniquely since, for any function f : the function F`f also satisfies (3.1). The results of this subsection provide an explicit centering under a mild condition on the parameters. Condition (1.10) implies that the interval I a,b m,n " p´min a m , min b n q (with the convention min a 0 " min b 0 " 8) is nonempty for m, n P Z ě0 . When m, n P Z ą0 , the length |I a,b m,n | " min a m`m in b n of this interval equals the reciprocal of the maximal mean of the weights in the rectangle rmsˆrns: Erω a,b m,n pi, jqs for m, n P Z ą0 .
and z P C pt´a m piq : i P rmsu Y tb n pjq : j P rnsuq. When z P I a,b m,n , (3.2) gives the mean of p Gpm, nq under p P a,b,z m,n as can be seen from (2.10), (2.11) and (2.13). Next define Our first result bounds the difference G a,b pm, nq´M a,b pm, nq with error terms that are of smaller order than m`n provided that |I a,b m,n | does not decay too quickly. Note that the bounds require only one of m and n to be sufficiently large.
Theorem 3.2. Let p ą 0. Then, P-a.s., there exists a random L P Z ą0 such that m,n |´1pm`nq 9{10`p for m, n P Z ą0 with m`n ě L.
In particular, (3.3) is a centering for the G a,b -process under a mild condition. Corollary 3.3. Assume that, for some c ą 0 and η ą 0, Let ą 0. Then, P-a.s., there exists (random) L P Z ą0 such that |G a,b pm, nq´M a,b pm, nq| ď pm`nq for m, n P Z ě0 with m`n ě L.
m,n | is allowed to decay too fast. For example, let a n piq " b n piq " 2´i for n P Z ą0 and i P rns. Then M a,b pn, nq ď M a,b,0 pn, nq " 2 ř n i"1 2 i ď 2 n`2 , while Gpn, nq ě ωpn, nq ě 100¨2 n happens with at least probability e´2 00 for each n.
Remark 3.3.2. One would expect that the analogous estimates hold in the LPP with independent, Z ě0 -valued, geometrically distributed weights. The weight at site pi, jq has fail parameter a i b j for i, j P Z ą0 for some R ą0 -valued sequences pa i q iPZ ą0 and pb j q jPZ ą0 subject to a i b j ă 1 for i, j P Z ą0 . This model was introduced in [38]. For the precise geometric analogue of the present setting, fix positive real parameters a and b as in (1.9) but now subject to a m piqb n pjq ă 1 for m, n P Z ą0 , and i P rms, j P rns. For each m, n P Z ą0 , consider independent weights on pΩ, Pq such that m,n pi, jq ě ku " pa m piqb n pjqq k for k P Z ě0 , i P rms and j P rns. and z P C pta m piq : i P rmsu Y tb n pjq : j P rnsuq. This can likely be obtained adapting the proof of Theorem 3.2. In a random parameter setting, a similar centering appeared in [24]. Formula (3.7) can be obtained from the mean of the explicit increment-stationary version of the G a,b -process.

3.2.
A conjecture due to E. Rains. The next result can be viewed as a variant of Corollary 3.3 on account of the similarity of the centering (3.3) to the limit (3.10) below. The statement is a reformulation of a conjecture due to E. Rains [51].
where the equality is a definition. Also define On pΩ, Pq, consider the weights ω n pi, jq " ωpi, jq a ri{ns`brj{ns for i, j, n P Z ą0 . For each n P Z ą0 , define the last-passage times tG n pi, jq : i, j P Z ą0 u via (2.1) from the weights tω n pi, jq : i, j P Z ą0 u. Then, P-a.s., Remark 3.4.1. Since the conjecture addressed above is somewhat dispersed within the text of [51], we explain how to locate it. The statement is a special case of the continuous analogue of Conjecture 5.2. To obtain it, replace mpα, p`, p´q in (5.5) with m c pz; ρ`, u; ρ´q from (5.29) and set u " 0. The parameters ρ˘" pρȋ q and a play the role of a, b and z in our setting. Assumptions (2.14)-(2.15) there correspond to our (3.8)-(3.9). The infimum in (5.5) is now to be taken over z P p´inf a, inf bq. After these changes, the right-hand side of (5.5) becomes M a,b p8, 8q. In direct analogy with (5.7), the weights are chosen as in (3.11). Finally, the quantity λ 1 in (5.5) is precisely sup i,jPZ ą0 G n pi, jq, which can be inferred from the discussion preceding [51, Theorem 2.4].

Shape function.
Definition 3.5. We call a deterministic function γ : R 2 ą0 Ñ R ě0 the shape function of the G a,b -process if γ is coordinatewise nondecreasing, positive-homogeneous (see Subsection 1.8) and its restriction to Z 2 ą0 is a centering for G a,b in the sense of Definition 3.1. Definition (3.5) is consistent with the notion of shape function from earlier literature, see Remark 3.6.6 below. It follows from the definition and Lemmas A.6-A.7 that the shape function, if it exists, is necessarily unique and continuous. (The uniqueness also comes from (3.28) below). This subsection provides an explicit formula for the shape function under some natural sufficient conditions for its existence.
Let α and β be finite, nonnegative Borel measures on R. Introduce the functions The integrals above are well-defined, and A α and B β are holomorphic functions. For each As recorded in the last case of Corollary 3.16 below, the shape function (defined as in Definition (3.5) but with Z ě0 in place of Z ą0 ) of the increment-stationary LPP process can be given in the form (3.14). This fact will not be used in the sequel, however.
Assume that inf supp α`inf supp β ą 0. Then the intervals p´a, 8q and p´8, bq are contained in the domains of (3.12) and (3.13), respectively. In particular, each z P p´a, bq is a legitimate parameter in (3.14). Therefore, the following definition is sensible: As will be made precise with Lemma 7.2 below, one can view (3.18) as the continuous analogue of (3.3) obtained after letting m, n Ñ 8. From (3.14), (3.16) and (3.18) it follows that the function γ α,β,a,b is nonnegative, concave, finite, coordinatewise nondecreasing and positivehomogeneous.
The next result shows that the shape function of the G a,b -process exists and can be represented in the form (3.18) under mild conditions. In the statement, denote the empirical distributions associated with the parameters a and b, respectively. For the definition of vague convergence, see Appendix A.2.
for some subprobability measures α, β on R and a, b P R Y t8u such that (3.16) is satisfied. Then for any ą 0, P-a.s., there exists a random L P Z ą0 such that |G a,b pm, nq´γ α,β,a,b pm, nq| ď pm`nq for m, n P Z ěL . Remark 3.6.2. The condition m, n ě L can be weakened to m`n ě L if and only if A α pmin b n q " A α pbq and B β p´min a m q " B β p´aq for m, n P Z ą0 . (3.22) This is recorded in Corollary 3.9 below. Condition (3.22) is equivalent to min a m " a for m P Z ą0 when β ‰ 0 and min b n " b for n P Z ą0 when α ‰ 0 (3.23) by the strict monotonicity of A α and B β .
Remark 3.6.3. When either α or β is the zero measure, γ becomes a one-variable function and can be readily computed from (3.18) by monotonicity.
Hence, the shape function is identically zero in three cases: If none of the conditions in (3.25) holds then the shape function is nonzero on R 2 ą0 since γ α,β,a,b px, yq ě xA α pbq`yB β p´aq ě maxtxA α pbq, yB β p´aqu for x, y ą 0. (3.26) Remark 3.6.4. Assuming α, β ‰ 0, there exists a unique minimizer ζ ζ ζ α,β,a,b px, yq P r´a, bs in (3.18) for each px, yq P R 2 ą0 , see Section 7. It can be seen from (3.18) that ζ ζ ζ α,β,a,b is a continuous function of px, yq P R 2 ą0 and a decreasing function of the slope y{x. Furthermore, γ α,β,a,b is continuously differentiable with gradient given by The details, being similar to the proof of [25,Corollary 2.3], are omitted.
Remark 3.6.5. The hypotheses of the theorem can be motivated from the following statement that holds under the weaker assumption (3.8): Let pm i q iPZ ą0 and pn j q jPZ ą0 be increasing sequences in Z ą0 . By the (sequential) compactness of r´8, 8s and the space of subprobability measures (see Lemma A.4), there exist increasing sequences pk i q iPZ and pl j q jPZ ą0 in Z ą0 such that for some subprobability measures α, β on R and a, b P R Y t8u with (3.16). Repeating the argument in Section 7 with m " m k i and n " n l j yields that for any ą 0, P-a.s., there exists a random L P Z ą0 such that Remark 3.6.6. Theorem 3.6 together with positive-homogeneity and continuity of the shape function implies that there exists a single P-a.s. event on which lim kÑ8 k´1G a,b pm x k , n y k q " γ α,β,a,b px, yq (3. 28) holds for all choices of x, y ą 0 and all sequences tm x k , n y k u k P Z ą0 Ă Z ą0 such that m x k {k Ñ x and n y k {k Ñ y as k Ñ 8. In literature, (3.28) with m x k " rkxs and n y k " rkys is often taken as the definition of the shape function [48,58].
Remark 3.6.7. Unlike the usual sequence of arguments, we do not first establish the a.s. limit in (3.28) to obtain Theorem 3.6. Instead, we derive Theorem 3.6 from Corollary 3.3 by approximating the centering (3.3) by the shape function.

3.4.
Growth near the axes. The next theorem provides uniform approximations to the growth process at sites where one coordinate is large and the other is relatively small. In particular, the result describes the asymptotics along a fixed column or row. there exist a deterministic δ ą 0 and a random K P Z ą0 such that |G a,b pm, nq´mA α pmin b n q| ď m for m P Z ěK and n P S with n ď δm.
(b) Assume that the second limit in (3.20) holds, and inf b`inf mPS min a m ą 0. Then, P-a.s., there exist a deterministic δ ą 0 and a random K P Z ą0 such that |G a,b pm, nq´nB β p´min a m q| ď n for n P Z ěK and m P S with m ď δn.
With the assumptions of Theorem 3.6, the approximations in Theorem 3.7 can be replaced with simpler one-dimensional linear functions at sites where both coordinates are large but one is small relative to the other.  Then, P-a.s., there exists a random L P Z ą0 such that G a,b pm, nq ď γ α,β,a,b pm, nq` pm`nq for m, n P Z ą0 with m`n ě L. Then, P-a.s., there exists a random L P Z ą0 such that G a,b pm, nq ě γ α,β,a,b pm, nq´ pm`nq for m, n P Z ą0 with m`n ě L.
(c) Fix n P Z ą0 . Assume that α ‰ 0 and min b n ‰ b. Then, P-a.s., there exists a random K P Z ą0 such that G a,b pm, nq´γ α,β,a,b pm, nq A α pmin b n q´A α pbq ě p1´ qm for m P Z ěK .
(d) Fix m P Z ą0 . Assume that β ‰ 0 and min a m ‰ a. Then, P-a.s., there exists a random K P Z ą0 such that G a,b pm, nq´γ α,β,a,b pm, nq B β p´min a m q´B β p´aq ě p1´ qn for n P Z ěK .
Part (a) of the following result identifies the limit shape in terms of (3.32)-(3.33). Parts (b) and (c) describe precisely when the rescaled cluster is a.s. eventually squeezed between given scaling perturbations of (3.32). The statements involve truncations that restrict to a bounded set when the limit shape is unbounded. The result should be compared with limit shape statements in case of i.i.d. weights, [47, Theorem 5.1(i)], where the limit shape is the closure of (3.32) in R 2 ě0 . " px, yq P R 2 ě0 : x ď C 1 tα"0uYtb"8u and y ď C 1 tβ"0uYta"8u * (3. 34) with the convention 1{0 " 8. (a) P-a.s., If (3.29) does not hold then there exist 0 ą 0 and C 0 ą 0 such that if C ě C 0 and ď 0 then, P-a.s., there exists a random T 0 such that If (3.30) does not hold then there exist 0 ą 0 and C 0 ą 0 such that if C ě C 0 and ď 0 then, P-a.s., there exists a random T 0 such that S X pt´1R a,b t p1` qR α,β,a,b q ‰ H for t ě T 0 . These limits imply that the closure of R α,β,a,b in R 2 ě0 is R α,β,a,b " R α,β,a,b Y I α,β,a,b . When R α,β,a,b is bounded, R α,β,a,b coincides with the limit shape in part (c) if and only if (3.30) holds.
3.6. Flat segments. To describe the finer structure of the limit shape, define the regions To avoid trivialities, assume that α and β are both nonzero. In particular, a, b ă 8. The regions H α,β,a,b , V α,β,a,b and S α,β,a,b are pairwise disjoint. Note that (The other two integrals in (3.37)-(3.38) are finite by (3.16)). Also, S α,β,a,b ‰ H since where the inequality is strict by (3.16).
The justification for the following assertions can be seen from (7.1) below and Remark 3.6.4. The function γ α,β,a,b is affine on the regions H α,β,a,b and V α,β,a,b , and is given by (3.40) Hence, the boundaries (inside R 2 ą0 ) of the regions R α,β,a,b X V α,β,a,b and R α,β,a,b X H α,β,a,b are flat segments. On the other hand, the function γ α,β,a,b is strictly concave on S α,β,a,b and the boundary of the region R α,β,a,b X S α,β,a,b is curved (non-flat). The entire boundary of R α,β,a,b is the image of a continuously differentiable curve and does not have any corners.
While the curved part is nonempty with the hypotheses of Theorem 3.6, it is also possible to generate completely flat limit shapes as in Corollary 3.19 below.
The boundary of the limit shape admits an exact parametrization. Indeed, the curve Φpzq " pBB β pzq,´BA α pzqq A α pzqBB β pzq´B β pzqBA α pzq for z P p´a, bq (3.41) parametrizes the curved part since γ α,β,a,b pΦpzqq " 1 (because ζ ζ ζ α,β,a,b pBB β pzq,´BA α pzqq " z, see (7.1) below) and Φpzq R V α,β,a,b Y H α,β,a,b for z P p´a, bq. The flat boundaries inside R 2 ą0 are the line segments from p0, B β p´aq´1q to Φp´aq, and from pA α pbq´1, 0q to Φpbq. Finally, the boundary along the axes are the line segments from the origin to p0, B β p´Aq´1q and pA α pBq´1, 0q. 3.7. Spikes and crevices. In the absence of condition (3.23), large (in macroscopic scale) spikes/crevices form near the axes that are not visible in the limit shape. To demonstrate this, consider the case β ‰ 0 and min a m ‰ a for some m P Z ą0 , namely, that the first part of (3.23) fails. The other case is analogous. The precise claim is that, for any ą 0 with 2 ă |B β p´min a m q´1´B β p´aq´1|{2, P-a.s., there exists a random T ą 0 such that for t ě T . The line segments in (3.42) and (3.43) can be visualized as a vertical spike or crevice, respectively, in the rescaled cluster near the vertical axis.
To verify (3.42) for example, pick 0 ă δ ă B β p´aq such that By Theorem 3.7, P-a.s., there exists a random K P Z ą0 such that G a,b pm, nq ď npv`δq´1 for n P Z ěK . Therefore, K ď n ď tpv`δq implies that pm, nq P R a,b t . Consequently, tt´1muˆrt´1K, vs Ă t´1R a,b t for t ě T " maxtKv´1, δ´1u.
On the other hand, by (3.36), the shape function γ α,β,a,b pt´1m, uq tÑ8 Ñ uB β p´aq ą 1. Hence, 3.8. Centerings for the height and flux processes. The limit shape for the bulk LPP process leads to the following explicit centerings for the height and flux processes, respectively, discussed in Subsection 2.4. Assuming (3.16), (3.17) and that α, β ‰ 0, introduce the limiting height and flux functions by respectively.
The partial derivatives can be computed as where h " h α,β,a,b py, tq and ζ ζ ζ " ζ ζ ζ α,β,a,b . Hence, the height function satisfies the Hamilton-Jacobi equation where the speed function v is implicitly defined on the interval I " pB β {A α qtp´a, bqu by vˆB β pzq Similarly, the flux function can be alternatively given by f α,β,a,b px, tq " # 0 if t ď xA α pbq the unique y P p0, t{γ α,β,a,b p1, 1qs such that γ α,β,a,b px`y, yq " t otherwise.
The partial derivatives of f " f α,β,a,b px, tq read ą0 with t ą xA α pbq. The functions ρ "´B x f and j " B t f represent macroscopic density and current of the particles, respectively. A brief computation verifies that the density satisfies the conservation law ρvˆ1´ρ ρ˙* " 0 for px, tq P R 2 ą0 with t ą xA α pbq. (3.47) For TASEP with particlewise random disorder, variational representations of the height and density functions, and PDEs (3.46) and (3.47) appeared in [60, (16), (25)]. For discrete-time TASEP with step initial condition and particlewise macroscopic disorder, a formula for the height function is included in [43, (2.10)] as a special case, see Remark 3.20.3 below.
Remark 3.11.2. In the homogeneous case (2.9), various quantities in Remark 3.11.1 can be computed explicitly as h c py, tq " p ?
The next result approximates the location of particle with a fixed label after a long time.
Theorem 3.12. Assume the first limits in (3.20), (3.21) and that α ‰ 0. Fix n P Z ą0 and assume that a`min b n ą 0. Then, P-a.s., there exists a (random) T P Z ą0 such thaťˇˇˇH a,b pn, tq´t A α pmin b n qˇˇˇˇď t andˇˇˇˇσ a,b pn, tq´t A α pmin b n q`nˇˇˇˇf or t ě T.

3.9.
Applications of the shape theorem. We record special cases and applications of Theorem 3.6 and connect them with earlier literature. Recall Definition (3.5) for the shape function.
Fixed rates. First, consider the case where the parameters a and b come from two fixed sequences, i.e. no m, n-dependence in the terms.
Corollary 3.13. Let a " pa i q iPZ ą0 and b " pb j q jPZ ą0 be real sequences subject to (3.8) and (3.20). Then the shape function exists and is given by γ α,β,inf a,inf b .
Periodic rates. A further special case is the one with periodic rates.
Corollary 3.14. Let k, l P Z ą0 . Let a " pa i q iPZ ą0 and b " pb j q jPZ ą0 be real sequences subject to a min k`b min l ą 0, and with periods k and l, respectively, i.e. a i " a i`k and b j " b j`l for i, j P Z ą0 . Then the shape function exists and is given by Remark 3.14.1. For the homogeneous case (2.9), the preceding infimum can be computed explicitly to recover Rost's shape function [54] given by γ c px, yq " 1 c p ?
x`?yq 2 for x, y ą 0. (3.48) Random rates. The following application to the random parameter setting improves [25, Theorem 2.1] by strengthening directional limits (of the kind in (3.28)) to a shape theorem and by eliminating the joint ergodicity assumptions therein.
Corollary 3.15. Let a " pa i q iPZ ą0 and b " pb j q jPZ ą0 be individually ergodic sequences with marginal distributions a i " α and b j " β that satisfy (3.15). Then, for almost every realization of the parameters a and b, the shape function exists and is given by γ α,β,inf supp α,inf supp β . Boundary rates on one or two sides. Another application of Theorem 3.6 reveals the shape functions of the LPP processes with one-sided and two-sided boundaries. To define these processes, introduce the Borel probability measure p P on p Ω under which tp ωpi, jq : i, j P Z ě0 u are independent and p ωpi, jq " Expp1q for i, j P Z ě0 . for m, n, i, j P Z ě0 with i ď m and j ď n, w ą´min a m and z ă min b n . Abbreviate w, z in the superscript as z when z " w. For m, n P Z ě0 , the joint distribution of tp ω a,b,z m,n pi, jq : i P rms Y t0u, j P rns Y t0uu is precisely p P a,b,z m,n defined at (2.12). Now define three LPP processes t p G a,b,w 1,0 pm, nq, p G a,b,z 0,1 pm, nq and p G a,b,w,z pm, nq : m, n P Z ě0 u (3.51) for each w ą´min a m and z ă min b n via formula (2.2) using the weights in (3.50). The first two processes see the boundary weights only one one axis while the last process sees boundaries on both axes. For the last case in the theorem below, recall definition (3.14). . Let w, z P p´a, bq and ą 0. Then p P-a.s., there exists a (random) L P Z ą0 such that, for m, n P Z ą0 with m, n ě L, | p G a,b,w 1,0 pm, nq´γ α,β,a,w pm, n`1q| ď pm`nq | p G a,b,z 0,1 pm, nq´γ α,β,´z,b pm`1, nq| ď pm`nq | p G a,b,w,z pm, nq´maxtγ α,β,a,w pm, n`1q, γ α,β,´z,b pm`1, nqu| ď pm`nq | p G a,b,z pm, nq´Γ α,β z pm, nq| ď pm`nq.
Remark 3.16.1. In case of homogeneous bulk weights as in (2.9), LPP processes with onesided boundaries were considered in [5] (also see Remark 3.17.1 below) while the version with two-sided boundary appeared in [6,10]. Note that the more general model with thickened boundaries as in [10, (41)] is also included in (3.50).
Remark 3.16.2. If (3.16), (3.20) and (3.21) hold then the parameters r a " ta m pm`1´iq : m P Z ą0 , i P rmsu and r b " tb n pn`1´jq : n P Z ą0 , j P rnsu also satisfy these conditions with the same α, β, a and b. Therefore, the conclusion of Corollary 3.16 remain true with r a, r b in place of a, b. The processes in (3.51) computed with parameters r a, r b can viewed as LPP processes with boundary weights on north and/or east sides as in [59,  Defective rates along a few rows and columns. The next corollary records the shape function in the perturbations of the homogeneous model obtained by modifying the parameters on a fixed set of defective rows and columns. The law of large numbers and fluctuations of this variant of LPP (with defective columns) was studied in [5]. The model is interesting because, wiith careful tuning of the parameters, it exhibits a transition between Tracy-Widom and Gaussian type fluctuations, now known as Baik-BenArous-Péché (BBP) transition. Here, we obtain the LLN part of their result without resorting to the integrable probability techniques.
Corollary 3.17. Fix k, l P Z ě0 and sequences pa i q iPrks and pb j q jPrls in R ą´1 2 that satisfy a min k`b min l ą 0. For m, n P Z ą0 , i P rms, and j P rns define rate parameters a m piq " a i 1 tiďku`1 2 1 tiąku and b n pjq " b j 1 tjďlu`1 2 1 tjąlu .
Then the shape function exists and is given by x`?yq p ?
Remark 3.17.1. Set k " 0 and b j " b´1{2 for j P rrs and b j ą b´1{2 for j P rls rrs for some b ą 0 and r P rls Y t0u. Corollary 3.17 then yields x`?yq p ?
The last formula implies [5, Corollary 1.1] Convergent rates. The next corollary shows that if the parameters pa m piqq iPrms and pb n pjqq jPrns converge uniformly as m, n Ñ 8, then the shape function is the same as in the model with limiting parameters. Then the shape function exists and is given by γ α,β,a,b where a " infpa i q iPZ ą0 and b " infpb j q jPZ ą0 . Remark 3.18.1. The corollary covers the following type of model, the discrete version of which was considered in [17, Corollary 2.11]: Let k, l P Z ě0 , and x i , y j P R for i P rks and j P rls. Let η ą 0 and a, b P R with c " a`b ą 0. Let p m P rm´ks and q n P rn´ls for m, n P Z ą0 . Assume that a m piq " a`1 ti´pmPrksu x i m η for m P Z ą0 and i P rms b n pjq " b`1 tj´qnPrlsu y j n η for n P Z ą0 and j P rns.
Then the shape function is given by (3.48).
The shape function can still be computed if (3.8) is weakened to conditions (3.53) and (3.54) below. Corollary 3.19. Let pa i q iPZ ą0 and pb j q jPZ ą0 be real sequences such that (3.20) and (3.52) hold. For p P Z ě0 , write a p " infpa i q iąp and b p " infpb j q jąp . Assume further that minta k`b0 , a 0`bl u " inf pi,jqPZ 2 ą0 prksˆrlsq a i`bj ą 0 (3.53) lim LÑ8 sup m,něL tmin iPrks a m piq`min jPrls b n pjqu´1 m`n " 0 (3.54) for some k, l P Z ě0 . Then the shape function exists and is given by Remark 3.19.1. If a 0`b0 ą 0 then the conclusion also follows from Corollary 3.18. The interesting case is a 0`b0 " 0, in which case (3.53) implies that a i`bj " 0 for some i P rks and j P rls. Then Γ α,β a 0 " Γ α,β b 0 is finite because (3.20) and (3.53) imply that´a 0 " b 0 P p´a k , b l q Ă p´inf supp α, inf supp βq. The shape function is linear in this case.
Remark 3.19.2. It can be deduced from the corollary that the shape function in the LPP model studied in [13] is (3.48). To see this, let η ą 0, a, b P R ą0 , k, l P Z ě0 , and pick x m piq P R ą´b , y n pjq P R ą´a with x m piq`y n pjq ą 0 and |x m piq|, |y n pjq| ď C for m, n P Z ą0 , i P rks, j P rls for some constant C ą 0. Consider the rates a m piq " 1 tiąku a`x m piq m η 1 tiďku for m P Z ą0 and i P rks b n pjq " 1 tjąlu b`y n pjq n η 1 tjďlu for n P Z ą0 and j P rls.
Condition (3.8) fails for these rates but the hypotheses of Corollary 3.19 hold. Hence, the shape function is given by γpx, yq " x a`y b for x, y P R ą0 . Now fix x, y ą 0 and choose a " ? xp ?
Macroscopically inhomogeneous rates. Given a speed function λ : R 2 ą0 Ñ R ą0 , a scaling parameter N P Z ą0 and the weights in (2.5), define Consider the LPP process tG λ N pi, jq : i, j P Z ą0 u computed from these weights via (2.1) with the initial point at site p1, 1q. This variant of LPP was introduced in [53] with a continuous speed function and subsequently studied in more generality in [14], [29] and [30]. The weights in (3.57) can be emulated in our setup when the speed function is additively separable. The next lemma describes the shape function in this special case under further mild conditions. Define the parameters a and b by a m piq " Api{mq and b n pjq " Bpj{nq for m, n P Z ą0 , i P rms, j P rns. Remark 3.20.1. If A and B are upper semicontinuous then (3.21) holds with a " inf A and b " inf B. Interestingly, the speed function is assumed to be lower semicontinuous in [14] and [29].
Remark 3.20.2. We make the connection of the corollary to [14, Theorem 1] more precise. Let A, B : R ě0 Ñ R be continuous and subject to (3.58), and consider the speed function given by λpx, yq " Apxq`Bpyq for x, y P R ě0 . Furthermore, assume that Apx 0 q " inf A and Bpy 0 q " inf B for some x 0 , y 0 P R ě0 . For any x, y P Z 2 ą0 with x ě x 0 and y ě y 0 , the corollary applied with the functions t Þ Ñ Aptxq and t Þ Ñ Bptyq for t P r0, 1s gives The right-hand side defines the shape function U : R 2 ě0 Ñ R 2 ą0 (notation from [14, (5)]) for the model in (3.57). The directional limits above can be upgraded to a shape theorem via standard arguments using monotonicity and positive-homogeneity. An omitted computation verifies that U is precisely the solution of the boundary value problem in [14,Theorem 1] and, equivalently, satisfies [14, (17)]. . This formula appears in [25, Theorem 2.1] in the case of random parameters. Now, for each N P R ą0 , consider independent, Z ą0 -valued weights tω N pi, jq : i, j P Z ą0 u such that Ptω N pi, jq ě ku " pApi{N qBpj{N qq k´1 for i, j, k P Z ą0 , The term x`y in (3.63) comes because the geometric weights in (3.62) are supported on Z ě1 unlike (3.6). Now specialize to A " 1, introduce r B " 1{B´1, replace z with z´1 and rearrange terms in (3.63) to obtain γpx, yq " inf This is the shape function for a special case of the model in [43,Section 2.4] (with a i " r Bpi{N q, ν i "´a i , and after interchanging the roles of columns/rows). The limiting height function can be obtained from the shape function in analogy with (3.44) as which is in agreement with [43, (2.10)].
Growing rates. When the G a,b -process grows sublinearly, the shape function is trivial and does not capture the order of growth. This is the case, for example, if a and b are unbounded increasing sequences. However, the precise first-order asymptotics can still be deduced from Theorem 3.2 in some cases. The next corollary illustrates this with a model taken from [39].

Concentration bounds for the last-passage times
As a main step towards the proof of Theorem 3.2, we begin to derive concentration bounds for Gpm, nq around M a,b pm, nq under P a,b m,n for each site pm, nq P Z 2 ą0 . Writing B z for the z-derivative note that, for m, n P Z ě0 , If m or n is nonzero then the function z Þ Ñ M a,b,z pm, nq is strictly convex on I a,b m,n due to the strict positivity of (4.2). Also, if m, n ą 0 then M a,b,z pm, nq Ñ 8 as z approaches the boundary values t´min a m , min b n u within I a,b m,n . Hence, there exists a unique minimizer ζ " ζ a,b pm, nq in (3.3). This is the unique z P I a,b m,n that satisfies the implicit equation   The deviations of Gpm, nq from the centering M a,b pm, nq are naturally expressed below in terms of this variance.
For brevity, introduce the functions A a z pmq " M a,b,z pm, 0q " m ÿ i"1 1 a m piq`z for m P Z ě0 and z P C t´a m piq : i P rmsu, (4.5) for n P Z ě0 and z P C tb n pjq : j P rnsu (4. 6) with the convention that A a z p0q " B b z p0q " 0 for z P C. Because the concentration bounds below for site pm, nq P Z 2 ą0 depend on the parameters a and b only through a m and b n , it causes no loss in generality to assume in this section that a m piq " a i and b n pjq " b j for m, n P Z ą0 and i P rms, j P rns for some real sequences pa i q iPZ ą0 and pb j q jPZ ą0 .
We first record a concentration bound for p Gpm, nq around M a,b pm, nq under p P a,b,ζ m,n .
Lemma 4.1. Let m, n P Z ą0 and ζ " ζ a,b pm, nq. Let s ą 0 and p P Z ą0 . Then there exists a constant C p ą 0 (depending only on p) such that Proof. Abbreviate C " C a,b pm, nq. By the triangle inequality, a union bound and (2.13), the probability in the statement is at most

Now the result readily follows from definition (4.4) and Lemma A.2.
An immediate consequence is the next right tail bound. Proof. Write ζ " ζ a,b pm, nq. Since P a,b m,n is a projection of p P a,b,ζ m,n and Gpm, nq ď p Gpm, nq a.s. under p P a,b,ζ m,n , the result follows from Lemma 4.1.
We now turn to developing a corresponding left tail bound. To this end, first note the following right tail bound for the last-passage times defined on paths constrained to enter the bulk at a specific boundary site. Lemma 4.3. Let m, n P Z ą0 and ζ " ζ a,b pm, nq. Let k P rms, l P rns, s ą 0 and p P Z ě0 . Then there exists a constant C p ą 0 (depending only on p) such that Proof. By Lemma A.2 and since ř k i"1 pa i`ζ q´2 ď C a,b pm, nq, p P a,b,ζ m,n t p Gpk, 0q´A a ζ pkq ě s`C a,b pm, nq˘1 {2 u ď C p s´p.
Hence, by Lemma 4.2, The first of the claimed bounds now follows from the triangle inequality and a union bound. The proof of the second bound is analogous.
Some of the statements below require an ordering condition on the parameters a and b. This does not limit the scope of the tail bounds of G, however, because the distribution of G is invariant under permutations of the parameters as stated in the next lemma. Proof. One can readily observe this from the formula [13, (12)] for the distribution of G.
Next comes a comparison of M a,b with essentially (up to a single boundary term) the centering for the LPP values considered in Lemma 4.3. Denote the distance from the minimizer ζ to the boundary of I a,b m,n by ∆ a,b pm, nq " minta min m`ζ , b min n´ζ u. Lemma 4.5. Let m, n P Z ą0 , ζ " ζ a,b pm, nq, C " C a,b pm, nq and ∆ " ∆ a,b pm, nq. There exists an absolute constant c ą 0 such that the following statements hold.
(a) Let k P rms. If pa i q iPrms is nondecreasing then A a ζ pk´1q`M τ k´1 a,b pm´k`1, nq´M a,b pm, nq ď´c∆Cˆk´1 m˙2 .
(b) Let l P rns. If pb j q jPrns is nondecreasing then Proof. We prove only (a) since the proof of (b) is similar. For z P I a,b m,n , note the identities M a,b,z pm, nq´M a,b pm, nq " The last equality is obtained by adding m ÿ i"1 1 pa i`ζ q 2´n ÿ j"1 1 pb j´ζ q 2 " 0 to the previous line.
Next note that, for any z P p´a min k,m , b min n q, Set z " ζ´c pk´1q∆ m for some absolute constant c P p0, 1{2s to be selected below. This is a legitimate value for z since k ď m and ∆ ď a min m`ζ . Then, using the monotonicity of pa i q iPrms and bounding term by term, one obtains that The second inequality above uses a i`z ě 1 2 pa i`ζ q, which follows from c ď 1{2 in the choice of z. The subsequent steps use (4.4). Let c ă 1{3 and rename c´3c 2 as c.
With the aid of the preceding estimates, one can derive the upper bounds below for the exit probabilities. Lemma 4.6. Let m, n P Z ą0 , ζ " ζ a,b pm, nq, C " C a,b pm, nq and ∆ " ∆ a,b pm, nq. Let s ą 0 and p P Z ą0 . Then there exist an absolute constant s 0 ą 0 and a constant C p ą 0 (depending only on p) such that the following bounds hold subject to the indicated further assumptions.
(a) Let k P Z. Assume that pa i q iPrms is nondecreasing, s ě s 0 and k ě 1`s m ∆ 1{2 C 1{4 . Then p P a,b,ζ m,n t p Hpm, nq ě ku ď C p ms´p.
(b) Let l P Z. Assume that pb j q jPrns is nondecreasing, s ě s 0 and l ě 1`s n ∆ 1{2 C 1{4 . Then p P a,b,ζ m,n t p Vpm, nq ě lu ď C p ns´p.
Proof. We prove only (a) since the proof of (b) is analogous. One may assume that k ď m since the probability is zero otherwise. Set s 0 " 2c´1 {2 where c ą 0 is the absolute constant in Lemma 4.5(a). Then, by the lemma, Then a union bound combined with the tail bounds in Lemmas 4.1 and 4.3 yields p P a,b,ζ m,n t p Hpm, nq " ku " p P a,b,ζ m,n t p Gpm, nq " p Gpk, 0q`G k,1 pm, nqu ď p P a,b,ζ m,n t p Gpm, nq ď M a,b pm, nq´c s 2 4 ? Cù ? Cu

Now applying the last inequality with another union bound results in
which implies the claim in (a).
The next lemma is a provisional left tail bound for the last-passage times.
Lemma 4.7. Let m, n P Z ą0 , ζ " ζ a,b pm, nq, C " C a,b pm, nq and ∆ " ∆ a,b pm, nq. Let s ą 0 and p P Z ą0 . Then there exist an absolute constant s 0 ą 0 and a constant C p ą 0 (depending only on p) such that P a,b m,n tGpm, nq ď M a,b pm, nq´s∆´1 {4 C 3{8 pm`nq 1{2 u ď C p pm`nqs´p for s ě s 0 . Proof. Let k, l P Z ą0 and x P R. Note from the definitions (2.3)-(2.4) that, on the event p Hpm, nq ď k and p Vpm, nq ď l, the inequality Gpm, nq ě p Gpm, nq´p Gpk, 0q´p Gp0, lq holds. Using this with the union bound leads to P a,b m,n tGpm, nq ď xu " p P a,b,ζ m,n tGpm, nq ď xu " p P a,b,ζ m,n t p Hpm, nq ě k`1u`p P a,b,ζ m,n t p Vpm, nq ě l`1u (4.9)`p P a,b,ζ m,n t p Gpm, nq´p Gpk, 0q´p Gp0, lq ď xu. By virtue of Lemma 4.4, the sequences pa i q iPrms and pb j q jPrns are both nondecreasing without loss of generality. Also since s 2 ě s 0 , in the case s 2 ∆´1 {2 C´1 {4 ď 1, Lemma 4.6 gives (4.9) ď C p pm`nqs´p for some constant C p ą 0. The last bound also holds trivially in the case s 2 ∆´1 {2 C´1 {4 ą 1 because then (4.9) " 0 since k " m and l " n. Set x " M a,b pm, nq´y where y " cs∆´1 {4 C 3{8 pm`nq 1{2 and c ą 0 is an absolute constant to be determined below. Another union bound yields (4.10) ď p P a,b,ζ m,n t p Gpm, nq ď M a,b pm, nq´y{2u`p P a,b,ζ m,n t p Gpk, 0q ě y{4ù p P a,b,ζ m,n t p Gp0, lq ě y{4u.
It follows from definitions (4.4) and (4.7) that C ď mintm, nu∆´2. Using this bound with Lemma 4.1, one obtains that By the Cauchy-Schwarz inequality and and the choices of k and y, provided that c ě 8. Therefore, by Lemma A.2 and since ř k i"1 pa i`ζ q´2 ď C (as noted in (4.11)), The last step uses C ď m∆´2 once more and drops the factor pm`nq 3p{8 . In the same vein, Putting the preceding bounds together results in (4.10) ď C p s´p. Hence, P a,b m,n t p Gpm, nq ď xu ď (4.9)`(4.10) ď C p pm`nqs´p.
This implies the claim upon replacing s with s{c throughout.
We next derive a nontrivial left tail bound that does not depend on the location of the minimizer. One ingredient in our argument is the following set of elementary estimates on the minimizer with shifted parameters. Lemma 4.8. Let m, n P Z ą0 and ζ " ζ a,b pm, nq.
Proof. We only verify (a) since (b) is entirely analogous. A moment of inspecting (4.3) reveals that r ζ ď ζ. Hence, the second and last inequalities in the following derivation. The first inequality is by the assumption a 1 ď a min 2,m , and the equality comes again from (4.3). 2 pa 1`ζ q 2´1 pa min 2,m`r ζq 2 ě 1 pa 1`ζ q 2`1 pa min 2,m`ζ q 2´1 pa min 2,m`r ζq 2 Hence, a min 2,m`r ζ ě a 1`ζ ? 2 as claimed. Using r ζ ď ζ and (4.3) also yields 1 pa 1`ζ q 2`1 pb min n´r ζq 2´1 pb min n´ζ q 2 ě 1 pa 1`ζ q 2`n ÿ j"1 1 pb j´r ζq 2´n Lemma 4.9. Let m, n P Z ą0 and C " C a,b pm, nq. Let p P Z ą0 . Then there exist absolute constants s 0 , s 1 ą 0 and a constant C p ą 0 (depending only on p) such that Introduce a threshold 0 ă δ ď pa min m`b min n q{2 to be determined later. Let K " mintk P rms : ζ τ k´1 a,b pm´k`1, nq ď b min n´δ u (4.12) L " mintl P rns : ζ a,τ l´1 b pm, n´l`1q ě´a min m`δ u.  3). Also, since the intersection p´a min m ,´a min m`δ q X pb min n´δ , b min n q is empty, the inequalities K ą 1 and L ą 1 cannot both hold. Appealing to the row-column symmetry, let us assume that K " 1 for concreteness.
Write ∆ " ∆ a,b pm, nq. One can read off from definition (4.3) that the shifted minimizer ζ τ k´1 a,τ l´1 b pm´k`1, n´l`1q is nonincreasing in k P rms and nondecreasing in l P rns. Hence, the inequality below.
Now conclude from Lemma 4.7 and the bounds in (4.15) and (4.17) that one also has ?
Cu ď C q v´q (4.20) for some constant C q ą 0. Here, interpret Gp1, 0q " 0 that arises in the case L " 1.
With the preceding choices, y " 6sC 1{2 pm`nq 2{5 and the bound in (4.24) reads C p pm`nqs´5 p{4`C q s´qpm`nq´2 q{5 . Enlarging p by a factor 5{4 and choosing q ě 5p{4 makes the first term more dominant than the second and results in the bound (4.25) P a,b m,n tGpm, nq ď M a,b pm, nq´6sC 1{2 pm`nq 2{5 u ď C p pm`nqs´p. Redefining s to be s{6 and adjusting the constant completes the proof.

Proof of Theorem 3.2
Proof of Theorem 3.2. Let 0 ă q ă p. A Borel-Cantelli argument and Lemma 4.2 imply that, P-a.s., there exists a random M P Z ą0 such that Since ζ is at least 1 2 |I a,b m,n | away from one of the endpoints of I a,b m,n " p´min a m , min b n q, one concludes from definition (4.4) the trivial bound C a,b pm, nq ď 4 maxtm, nu|I a,b m,n |´2. Then, by the Cauchy-Schwarz inequality, where ζ " ζ a,b pm, nq. Hence, Let s 0 , s 1 ą 0 denote the absolute constants in Lemma 4.9. By (5.6) and since 0 ă q ă p, s 0 ď pm`nq q ď s 1 |I a,b m,n |C 1{2 pm`nq´2 {5 whenever m`n ě N 0 (5.7) for some sufficiently large constant N 0 P Z ą0 . Also, by (5.2), Let u ą 0. By (5.8) and an application of Lemma 4.9 with s " pm`nq q , one obtains that PtG a,b pm, nq ď M a,b pm, nq´2|I a,b m,n |´1pm`nq 9{10`q u ď PtG a,b pm, nq ď M a,b pm, nq´pm`nq q pC 1{2 pm`nq 2{5 qu ď C pm`nq qu whenever m`n ě N 0 and for some constant C ą 0 dependent only on u. The last bound is summable over Z 2 ą0 provided that u is sufficiently large. Hence, by the Borel-Cantelli argument, P-a.s., there exists a random N ě N 0 such that G a,b pm, nq ě M a,b pm, nq´2|I a,b m,n |´1pm`nq 9{10`q whenever m`n ě N. This implies the claimed lower bound since q ă p.

Proof of Theorem 3.4
Proof of Theorem 3.4. Fix n P Z ą0 for the moment. Since the weights in (3.11) are a.s. nonnegative, the G n -process is a.s. coordinatewise nondecreasing. Therefore, P-a.s., where the last equality defines G n .
Consider real parameters r a n and r b n subject to (1.10) (with a " r a n and b " r b n ) that also satisfies r a n ln piq " a ri{ns and r b n ln piq " b ri{ns for l P Z ą0 and i P rlns.
Then ω n pi, jq " ω r a n , r b n ln,ln pi, jq for i, j P rlns where the right-hand side is given by (2.6) (with ω m,n pi, jq " ωpi, jq). Hence, G n pln, lnq " G r a n , r b n pln, lnq for l P Z ą0 . Write z 0 " pinf a´inf bq{2 and ζ l " ζ a,b pl, lq for l P Z ą0 . Recalling definitions (4.4) and (4.7), note also that ∆ a,b pl, lq´2 " maxtpa min l`ζl q´2, pb min l´ζl q´2u ď C a,b pl, lq " Bound (6.6) follows upon considering the cases ζ l ď z 0 and ζ l ě z 0 separately. The last inequality uses that z 0 is the midpoint of p´inf a, inf bq. Denote the quantity in (6.7) by c 0 ą 0. By assumption (3.9), c 0 ă 8. Now, because ∆ a,b pl, lq ě c´1 {2 0 for l P Z ą0 , there exist δ ą 0 and L 0 P Z ą0 such that ζ l P I " p´inf a`δ, inf b´δq for l ě L 0 . Then The first term equals M a,b pl, lq for l ě L 0 while both series vanish as l Ñ 8 in (6.8).
In particular, with η ă 1{2, lim sup nÑ8 n´1G n ď M a,b p8, 8q. (6.11) Note from the bounds culminating in (6.7) that maxt∆ a,b pl, lq´2, C a,b pl, lqu ď c 0 for l P Z ą0 , which justifies the first step in the next derivation.

Proof of Theorem 3.6
The proof is based on Corollary 3.3 and an approximation of the centering M a,b by the shape function.
Let α and β be finite, nonnegative Borel measures on R. Fix x, y ą 0. The derivatives of (3.14) are given by B n z Γ α,β z px, yq " xp´1q n n! ż R αpdaq pa`zq n`1`y n! ż R βpdbq pb´zq n`1 for z P C p´supp α Y supp βq on account of (A.5). Assume that α and β are nonzero, that (3.15) holds, and suppose a, b P R satisfy (3.16) and (3.17). Then B 2 z Γ α,β z px, yq ą 0 for z P p´a, bq. Hence, the function z Þ Ñ Γ α,β z px, yq is strictly convex on p´a, bq. Consequently, there exists a unique z-value, denoted with ζ ζ ζ α,β,a,b px, yq, in the closed interval r´a, bs such that the infimum in (3.18) is given by γ α,β,a,b px, yq " Γ α,β z px, yq. Examining the sign of the first derivative reveals that ζ ζ ζ α,β,a,b px, yq " Proof. Note that a, b ă 8 since α, β ‰ 0. Pick δ ą 0 such that 2δ ă a`b. Then the interval r´a`δ, b´δs is nonempty. By (3.21), there exists L P Z ą0 such that | min a m´a | ă δ and | min b n´b | ă δ for m, n ě L.
Proof. Abbreviate ζ " ζ a,b pm, nq and ζ ζ ζ " ζ ζ ζ α,β,a,b pm, nq. Assume further that a, b ă 8. As in the preceding proof, pick δ ą 0 and L P Z ą0 such that 2δ ă a`b and r´a`δ, b´δs Ă I a,b m,n for m, n ě L. Let z P p´a`δ, b´δq. By (3.21), after increasing L if necessary, (7.4) a m piq`z ě δ and b n pjq´z ě δ for m, n ě L and i P rms, j P rns.
The first inequality above is by definition (3.3). The next two inequalities are due to (7.10)- (7.11). For the second-last inequality, first use a ě inf supp α ě a and δ ď a`b, and then recall a`b ě inf a`inf b. This completes the case β " 0. The case α " 0 is handled similarly. The preceding paragraph also includes the cases a " 8, b ă 8 and a ă 8, b " 8. Therefore, the only remaining case is a " b " 8 which implies that γ α,β,a,b " 0. Pick any z 0 P p´inf a, inf bq. Since α, β " 0 now, by Lemma A.5, M a,b pm, nq ď M a,b,z 0 pm, nq ď pm`nq for m, n ě L after increasing L if necessary. This completes the proof.
Proof of Theorem 3.6. This is immediate from Theorem 3.2 and Lemma 7.2.
8. Proofs of Theorem 3.7 and Corollaries 3.8 and 3.9 Proof of Theorem 3.7. Due to the symmetry, only (a) is proved below. Write p " inf nPS min b n . Introduce a small parameter η ą 0 such that 2η ă inf a`p. Since |I a,b m,n | " min a m`m in b n ě inf a`p ą η for m P Z ą0 and n P S, it follows from Theorem 3.2 that, P-a.s., there exists a random L P Z ą0 such that |G a,b pm, nq´M a,b pm, nq| ď pm`nq for m P Z ą0 and n P S with m`n ě L. The range of z in (8.2) can be extended to rp´η, 8q since the terms pa m piq`zq´1 for i P rms and the integrand pa`zq´1 are bounded from above by pinf a`zq´1 which is less than {4 for z ě p`4{ .
Choose K ě L, take m ě K and n P S below. Write ζ " ζ a,b pm, nq. By definition (3.3), (8.2) and since maxtζ, pu ď min b n , one has the lower bound The first two inequalities above come again from (3.3) and (8.2). The last inequality comes from 2η ă inf a`min b n ď inf supp α`min b n where we appealed to the assumption that the first limit in (3.20) holds. With η ď and n ď ηm, one then has M a,b pm, nq ď m ż R αpdaq a`min b n`C m, (8.4) where C " 2`2 ş R pa`min b n q´2αpdaq. Choose δ " mint1, ηu. Combining the bounds from (8.1), (8.3) and (8.4) with the triangle inequality results iňˇˇˇG a,b pm, nq´m ż R αpdaq a`min b nˇď pm`nq`C m ď mp1`δ`Cq ď p2`Cqm for m P Z ěK and n P S with n ď δm. The result follows upon replacing with p2`Cq´1 throughout.
Proof of Corollary 3.8. Assumptions (3.16) and (3.21) imply that inf a`inf b ą 0. Since (3.20) is also assumed, both parts of Theorem 3.7 hold with S " Z ą0 . The conclusion then follows from (3.21) and the continuity of A α and B β .
For the third step above, pb´b`ηq´1 ď η´1 for b ě inf supp β due to the second inequality in (3.17). Hence, the first claimed bound holds with δ " η 2 . The second bound follows similarly.
Proof of Corollary 3.9. By Theorem 3.6, P-a.s., there exists a random L P Z ą0 such that |G a,b pm, nq´γ α,β,a,b pm, nq| ď pm`nq for m, n P Z ěL .
Note that the assumptions of Theorem 3.7 hold with S " Z ą0 . Therefore, P-a.s., there exist a random K P Z ą0 and a deterministic δ ą 0 such that |G a,b pm, nq´A α pmin b n q| ď m 2 for m P Z ěK and n P Z ą0 with n ď δm |G a,b pm, nq´B β p´min a m q| ď n 2 for n P Z ěK and m P Z ą0 with m ď δn.
Pick δ ă 1 and set N " p1`δ´1q maxtK, Lu. Claim: If (3.29) holds then G a,b pm, nq ď γ α,β,a,b pm, nq` pm`nq for m, n P Z ą0 with m`n ě N. (8.6) This is clear from the choice of L if m, n ě L. If n ă L then m ě K and n ď δm, and it follows from the triangle inequality, A α pmin b n q ď A α pbq and the choices of K and δ that G a,b pm, nq´γ α,β,a,b pm, nq " pG a,b pm, nq´mA α pbqq`pγ α,β,a,b pm, nq´mA α pbqq ď pG a,b pm, nq´mA α pmin b n qq`pγ α,β,a,b pm, nq´mA α pbqq The case m ă L is similar. This gives (a), and (b) follows similarly.
To prove (c), assume that A α pmin b n q ą A α pbq since the other case is similar. Then, for m P Z ěK and n P Z ą0 with n ď mδ, one has G a,b pm, nq´γ α,β,a,b pm, nq " mpA α pmin b n q´A α pbqq´pγ α,β,a,b pm, nq´mA α pbqq pG a,b pm, nq´mA α pmin b n qq ě mpA α pmin b n q´A α pbqq´ m.
The proof of (d) is similar.
Lemma 9.2. Assume (3.16), (3.20) and (3.21). Let S denote the set from (3.34), and A and B be given by (3.35). Then there exists a constant C 0 ą 0 such that the following hold .
Turn to the remaining case px, yq P t´1R a,b t . Let η ą 0 to be chosen below. By Theorems 3.6 and 3.7, P-a.s., there exist random K, L P Z ą0 with K ě L such that G a,b pm, nq ě γ α,β,a,b pm, nq`ηpm`nq for m, n P Z ąL (9.1) G a,b pm, nq ě mA α pmin b n q`ηm for m P Z ěK and n P rLs.
After choosing η sufficiently small and K sufficiently large, the last term is at most t. If δtx ă K then x ď Kδ´1t´1 ď Kδ´1T´1 ď T ď t upon taking T sufficiently large.
Proof of Theorem 3.10. Corresponding to the given ą 0, P-a.s., there exist L P Z ą0 as in Lemma 9.3 and K, M, N P Z ą0 , T P R ą0 as in Lemma 9.4. Take t ě T below. Let C 0 ą 0 denote the constant from Lemma 9.2. To prove (a), it suffices to show that for sufficiently large T . The arguments in (9.11) are closed subsets of R 2 ě0 . The second argument is nonempty and bounded by Lemmas 9.1(b) and 9.2(a). The same is true for the first argument with large enough T by Lemma 9.2(b) and the first containment in Lemma 9.3. Hence, the left-hand side in (9.11) makes sense per definition of the Hausdorff metric in Subsection A.4.
Then px 1 , y 1 q P S X t´1R a,b t by Lemma 9.3 if tx, ty ě K with K ě p1´ qL and by Lemma 9.4 otherwise. Furthermore, |x´x 1 |, |y´y 1 | ď C 0 by taking T large enough. Hence, (9.11) is proved.
With ď 0 " 1 2 pB β p´aq´1´B β p´min a m q´1q, one has v " 1´2 B β p´aq ą 1 B β p´min a m q .
It then follows from (3.43) that, P-a.s., there exists a random T 0 ą 0 such that pmt´1, vq R t´1R a,b t for t ě T 0 . Since lim tÑ8 γ α,β,a,b pmt´1, vq " 1´2 by (3.36), after increasing T 0 if necessary, one also has pmt´1, vq P p1´ qR α,β,a,b . Finally, with C ě B β p´aq´1 and taking T 0 ě mB β p´aq, one has pmt´1, vq P S. The case α ‰ 0 and min b n ă b for some n P Z ą0 is treated similarly. The proof of (b) is now complete. The proof of (c) is analogous.
10. Proofs of Theorems 3.11 and 3.12 Proof of Theorem 3.11. We prove only the estimate for the flux process. The estimate for the height process can be obtained in the same manner, and then the estimate for the particle locations is immediate. Introduce a constant c ą 0 to be chosen later. By Theorem 3.6, P-a.s., there exists L 0 P Z ą0 such that |G a,b pm, nq´γ α,β,a,b pm, nq| ď c pm`nq for m, n P Z ěL 0 . (10.1) Choose L P Z ą0 with L ą 2L 0 maxt1, ´1 u. Fix m ě L and t P R ě0 below. Consider j P Z ą0 with j ą f α,β,a,b pm, tq` pm`tq. Then, by definition of the flux process and rearranging terms, pm`jqA α pzq`jB β pzq ą t` pA α pzq`B β pzqqpt`jq for z P p´a, bq. In particular, t ă c 0 pm`jq for some constant c 0 ą 0. Now setting z " ζ ζ ζ α,β,a,b pm`j, jq in (10.2), using monotonicity of A α and B β and the bound on t, one obtains that γ α,β,a,b pm`j, jq " pm`jqA α pζq`jB β pζq ą t` pA α pbq`B β p´aqqpc 0 pm`jq`jq ą t`c 1 pm`2jq for some constant c 1 ą 0. Since j ě L 0 , choosing c ď c 1 yields G a,b pm`j, jq ě γ α,β,a,b pm`j, jq´c 1 pm`2jq ą t in view of (10.1). Then it follows from the choice of m that F a,b pm, tq ď f α,β,a,b pm, tq` pm`tq. For the lower bound, it suffices to consider the case f α,β,a,b pm, tq ě pm`tq since the flux process is nonnegative. Now consider j P Z ą0 with L 0 ď j ă f α,β,a,b pm, tq´ pm`tq{2. Such j exists since the right-hand side is greater than L 0 . By definition (3.44), there exists z P p´a, bq such that pm`jqA α pzq`jB β pzq ă t´ 2 pA α pzq`B β pzqqpt`jq. (10.4) In particular, t ě jpA α pzq`B β pzqq ě jpA α pbq`B β p´aqq. Then, by (10.4), γ α,β,a,b pm`j, jq ă t´c 2 pm`2jq (10.5) for some constant c 2 ą 0. Now choosing c ď c 2 yields can be made arbitrarily close to zero by taking m, n sufficiently large. Therefore, the shape Proof. For n P Z ą0 , write S n for the set of all k " pk 1 , . . . , k m q P Z m ě0 such that ř iPrms k i " n. Let S 1 n denote the set of all k P S n with k i ‰ 1 for i P rms. Define two maps f, g : S 1 2p Ñ S p as follows: For each k P S 1 2p , there is an even number 2l of indices i P rms for which k i is odd. Let i 1 ă¨¨¨ă i 2l denote these indices. Define f pkq P S p and gpkq P S p by setting if i " i s for some s P rls pk i´1 q{2 if i " i s for some s P r2ls rls k i {2 otherwise.
if i " i s for some s P rls pk i`1 q{2 if i " i s for some s P r2ls rls k i {2 otherwise.
The inequality 2 ď t`1{t applied with t " ś l s"1 x is ś 2l s"l`1 1{x is yields which justifies the first inequality below. Note also that k i P t2f pkq i , 2f pkq i`1 , 2f pkq i´1 u and, because k i ‰ 1, one has k i ą 0 if and only if f pkq i ą 0 for i P rms. Then, since each u " pu 1 , . . . , u m q P S p has at most p nonzero coordinates, the number of elements in the preimage f´1tuu is bounded by 3 p . The same for g´1tuu. Hence, the second inequality below. The final line inserts the multinomial coefficients and appeals to the multinomial theorem. An application of the preceding lemma yields the concentration inequality below.
Lemma A.2. Let X i " Exppλ i q be mutually independent exponential random variables. Then for each p P Z ą0 there exists a constant C p ą 0 (depending only on p) such that, for all s ą 0 and n P Z ą0 , P "ˇˇˇˇn ÿ Proof. With S " ř n i"1 X i the claimed inequality is Pt|S´ES| ě s ? VarSu ď C p s´p. (A.1) For i P rns and q P Z ě0 , a brief computation gives the qth central moment of X i as ErpX i´λ´1 i q q s " c q λ´q i where c q " q! q ÿ l"0 p´1q l l! . Since c 1 " 0, the condition u i ‰ 1 for i P rks can be slipped into the outer sum without breaking the equality. Then, by virtue of Lemma A.1 and since 0 ď c u i ď u i !, the right-hand side of (A.2) is at most A.2. Vague convergence. For convenience, we recall the definition of and some standard facts about the vague convergence of real-valued Borel measures on the real line. These can be found for example in [21] and [28].
Let MpRq and C 0 pRq, respectively, denote the spaces of real-valued Borel measures on R and real-valued, continuous functions on R vanishing at infinity. The vague topology on MpRq is the minimal topology such that the maps µ Þ Ñ ş R f dµ for f P C 0 pRq are continuous. With this definition, a sequence pµ n q nPZ ą0 in MpRq converges to µ P MpRq in the vague topology (vaguely) if and only if ş R f dµ n Ñ ş R f dµ as n Ñ 8 for any f P C 0 pRq. Lemma A.3. Let pµ n q nPZ ą0 be a sequence in MpRq such that µ n Ñ µ vaguely as n Ñ 8 for some µ P MpRq. Then there exists C ą 0 such that µ n pRq ď C for n P Z ą0 .
A.3. Cauchy transform. This subsection derives a uniform strengthening of the continuity of the Cauchy transform (also known as the Stieltjes or Cauchy-Stieltjes transform) with respect to the vague topology.
Write M`pRq for the space of R ě0 -valued Borel measures on R. Let µ P M`pRq and D " C supp µ. The Cauchy transform of µ is defined as the convolution C µ pzq " The integral is well-defined since µpRq ă 8, µpR supp µq " 0 and the distance ∆pzq " inf t P supp µ |z´t| ą 0 for z P D, the latter due to supp µ being closed. By direct computation, |C µ pzq´C µ pwq| "ˇˇˇˇpw´zq ż R µpdtq pz´tqpw´tqˇˇˇˇď |z´w|µpRq ∆pzq∆pwq for z, w P D, (A.4) and C µ is holomorphic on D with B n z C µ pzq " p´1q n n! ż R µpdtq pz´tq n`1 for z P D and n P Z ě0 . (A.5) Lemma A.5. Let µ P M`pRq and pµ n q nPZ ą0 be a sequence in M`pRq such that lim nÑ8 µ n " µ vaguely. Let S Ă C denote the closure of supp µY Ť nPZ ą0 supp µ n . Let K Ă C S be compact. Then, for any ą 0, there exists n 0 P Z ą0 such that |C µn pzq´C µ pzq| ă for all n ě n 0 and z P K.
Proof. Since the real and imaginary parts of the integrand in (A.3) belong to C 0 pRq, it is immediate from the definition of vague convergence that C µn pzq Ñ C µ pzq as n Ñ 8 pointwise for z P C S. To upgrade to uniform convergence, first pick a constant C ą 0 as in Lemma A.3 such that µ n pRq ď C for n P Z ą0 . Also, since K is compact, S is closed and K X S " H, there exists δ ą 0 such that |z´x| ě δ for z P K and x P S. Hence, it follows from (A.4) that |C µn pzq´C µn pwq| ď C δ 2 |z´w| for z, w P K and n P Z ą0 . In particular, the sequence of functions pC µn q nPZ ą0 is equicontinuous on K. This property together with the pointwise convergence on the compact set K implies that C µn pzq Ñ C µ pzq uniformly in z P K as n Ñ 8 [55,Chapter 7].
A.4. Hausdorff metric. We include the definition of the Hausdorff metric given for example in [49, p. 280]. Let pX, dq be a metric space. For any ą 0 and A Ă X, denote the -fattening of A by A " tx P X : dpx, yq ă for some y P Au. Let X denote the space of nonempty, bounded, closed subsets of X. The Hausdorff metric on X is defined by Lemma A.6. Let f : R 2 ą0 Ñ R ě0 be a positive-homogeneous and coordinatewise nondecreasing function. Then f has a continuous extension to R 2 ě0 . Proof. By positive-homogeneity and coordinatewise monotonicity, |f px, yq´f pr x, r yq| ď f pmaxtx, r xu, maxty, r yuq´f pmintx, r xu, minty, r yuq ďˆmax " maxtx, r xu mintx, r xu , maxty, r yu minty, r yu *´1˙f pmintx, r xu, minty, r yuq ďˆmax " maxtx, r xu mintx, r xu , maxty, r yu minty, r yu *´1˙f px, yq for x, y, r x, r y P R ą0 .
In particular, f is continuous on R 2 ą0 . Extend f to a function R 2 ě0 Ñ R ě0 by setting f px, 0q " inf for x, y P R ą0 , and f p0, 0q " 0. The extension is also positive-homogeneous and coordinatewise nondecreasing. Then f px, yq ď maxtx, yuf p1, 1q and, consequently, f is continuous at p0, 0q. f is also continuous on the horizontal axis by (A.7) and on account of the bound |f px, 0q´f pr x, r yq| ď |xf p1, 0q´r xf p1, r y{r xq| ď xpf p1, r y{r xq´f p1, 0qq`|x´r x|f p1, r y{r xq ď xpf p1, 2r y{xq´f p1, 0qq`|x´r x|f p1, 2r y{xq for x, r x P R ą0 with r x ě x{2 and r y P R ě0 . The continuity on the vertical axis follows similarly.
Lemma A.7. Let f, g : R 2 ą0 Ñ R ě0 be positive-homogeneous and coordinatewise nondecreasing functions such that, for any ą 0, there exists L P Z ą0 such that |f pm, nq´gpm, nq| ď pm`nq for m, n ě L. (A.8) Then f " g on R 2 ą0 . Proof. Let ą 0 and choose L P Z ą0 such that (A.8) holds. Let p, q P Q ą0 . Then, by positive-homogeneity, |f pp, qq´gpp, qq| " n´1|f pnp, nqq´gpnp, nqqq| ď pp`qq for n P Z ą0 such that np, nq P Z ěL . Since ą 0 is arbitrary, it follows that f " g on Q 2 ą0 . Also, f and g are continuous on R 2 ą0 by Lemma A.6. Hence, f " g on R 2 ą0 .