Non-existence of bi-infinite polymers*

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inversegamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.


Directed polymers
The directed polymer model is a stochastic model of a random path that interacts with a random environment. In its simplest formulation on an integer lattice Z d , positive random weights tY x u xPZ d are assigned to the lattice vertices and the quenched probability of a finite lattice path π is declared to be proportional to the product ś xPπ Y x . In the usual Boltzmann-Gibbs formulation we take Y x " e´β ωx so that the energy of a path is proportional to the potential ř xPπ ω x and the strength of the coupling between the path π and the environment ω is modulated by the inverse temperature parameter β.
The directedness of the model means that some spatial direction u P R d represents time and the admissible paths π are required to be u-directed. One typical example

Organization of the paper
Section 2 develops enough of the general polymer theory from [25] so that in Section 2.3 we can state the main result Theorem 2.8 on the nonexistence of bi-infinite inversegamma polymer Gibbs measures. Along the way we apply results from [25] to prove for general weights that infinite polymers have to be directed into the open quadrant, unless they are rigid straight lines (Theorem 2.6). This result will also contribute to the proof of the main Theorem 2.8. Section 3 gives a quick description of the ratio-stationary inverse-gamma polymer and derives one estimate -that under the annealed measure, with high probability, stationary polymers will leave far from the characteristic ξ on the order of OpN 2{3 q when perturbing the density ρpξq properly on the order of OpN 1{3 q.
The heart of the proof is in Section 4. A coarse-graining argument decomposes the southwest boundary of a large 2Nˆ2N square into blocks of size N 2{3 . Two separate estimates are developed.
(a) The first kind is for the probability that a polymer path from an N 2{3 -block denoted by I goes through the origin and reaches the diagonally opposite block p I of size N 19{24 . This probability is shown to decay by controlling it with random walks that come from ratio-stationary polymer processes (Lemma 4.4).
(b) The second estimate (Lemma 4.5) controls the paths from I through the origin that miss p I. Such paths are rare due to KPZ bounds according to which the typical path remains within a range of order N 2{3 around the straight line between its endpoints. Section 4 culminates in Theorem 4.6 that combines the estimates.
Section 5 combines Theorem 4.6 with the earlier Theorem 2.6 to complete the proof of Theorem 2.8. The estimates for paths that go through the origin are generalized to other crossing points on the y-axis by suitably shifting the environment.
Since the background polymer material will be at least partly familiar to some readers, we have collected these facts in the appendix. Appendix A covers polymers on Z 2 with general vertex weights and Appendix B specializes to inverse-gamma weights. Appendix C states a positive lower bound on the running maximum of a random walk with a small negative drift that we use in a proof. This result is quoted from the technical note [11] that we have published separately.

Notation and conventions
Subsets of reals and integers are denoted by subscripts, as in Z ą0 " t1, 2, 3, . . .u and Z 2 ą0 " pZ ą0 q 2 . a, b denotes the integer interval ra, bs X Z if a, b P R, and the integer rectangle pra 1 , b 1 sˆra 2 , b 2 sq X Z 2 if a, b P R 2 .
Their strict versions mean that the defining inequalities are strict: px 1 , x 2 q ă py 1 , y 2 q if x r ă y r for r P t1, 2u, and px 1 , x 2 q ă py 1 , y 2 q if x 1 ă y 1 and x 2 ą y 2 .
Sequences are denoted by x m:n " px i q n i"m and x m:8 " px i q 8 i"m for integers m ď n ă 8 and also generically by x‚. An admissible path x‚ in Z 2 satisfies x k´xk´1 P te 1 , e 2 u. Limit velocities of these paths lie in the simplex re 2 , e 1 s " tpu, 1´uq : u P r0, 1su, whose relative interior is the open line segment se 2 , e 1 r .
The notations E and P refer to the random weights (the environment) ω, and otherwise E µ denotes expectation under probability measure µ. The usual gamma function for ρ ą 0 is Γpρq " ş 8 0 x ρ´1 e´x dx, and the digamma and trigamma functions are ψ 0 " Γ 1 {Γ and ψ 1 " ψ 1 0 . X " Gapρq if the random variable X has the density function f pxq " Γpρq´1x ρ´1 e´x on R ą0 , and X " Ga´1pρq if X´1 " Gapρq.
We often omit t¨u, for example, we write N 2{3 e 1 P Z 2 .

Polymer Gibbs measures 2.1 Directed polymers
Let pY x q xPZ 2 be an assignment of strictly positive real weights on the vertices of Z 2 . For vertices o ď p in Z 2 let X o,p denote the set of admissible lattice paths x‚ " px i q 0ďiďn with n " |p´o| 1 that satisfy x 0 " o, x i´xi´1 P te 1 , e 2 u, x n " p. Define point-to-point polymer partition functions between vertices o ď p in Z 2 by We use the convention Z o,p " 0 if o ď p fails. The quenched polymer probability distribution on the set X o,p is defined by When the weights ω " pY x q are random variables on some probability space pΩ, A, Pq, the averaged or annealed polymer distribution P o,p on X o,p is defined by (2. 3) The notation Q ω o,p highlights the dependence of the quenched measure on the weights. It is also convenient to use the unnormalized quenched polymer measure, which is simply the sum of path weights: (2.4) A basic law of large numbers object of this model is the limiting free energy density. Assume now the following: the weights pY x q xPZ 2 are i.i.d. random variables and Er| log Y 0 | p s ă 8 for some p ą 2. log Z 0,x´Λ pxq |x| 1 " 0 P-almost surely.
(2.6) (See Section 2.3 in [25].) In general, further regularity of Λ is unknown. In certain exactly solvable cases, including the inverse-gamma polymer we study in this paper, the following properties are known: Λ is differentiable and strictly concave on the open interval se 2 , e 1 r . (2.7) Fix the base point o " 0 (the origin) and consider sending the endpoint p to infinity in the quenched measure Q 0,p . Fix a finite path x 0:n P X 0,y where 0 ď y ď p and n " y¨pe 1`e2 q. To understand what happens as |p| 1 Ñ 8 it is convenient to write Q 0,p as a Markov chain: Q 0,p tX 0:n " x 0:n u " with initial state X 0 " 0, transition probability π 0,p px, x`e i q " Z´1 x,p Z x`ei,p Y x for p ‰ x P 0, p , and absorbing state p. The formulation above reveals that when the limit of the ratio Z x`ei,p {Z x,p exists for each fixed x as p tends to infinity, then Q 0,p converges weakly to a Markov chain. When p recedes in some particular direction, this can be proved under local hypotheses on the regularity of Λ. See Theorem 3.8 of [25] for a general result and Theorem 7.1 in [22] for the inverse-gamma polymer.
The limiting Markov chains are examples of rooted semi-infinite polymer Gibbs measures, which we discuss in the next section.

Infinite Gibbs measures
In this section we adopt mostly the terminology and notation of [25]. To describe semi-infinite and bi-infinite polymer Gibbs measures, introduce the spaces of semi-infinite and bi-infinite polymer paths in Z 2 : X u is the space of paths rooted or based at the vertex u P Z 2 . The indexing of the paths is immaterial. However, it adds clarity to index unbounded paths so that x k¨p e 1`e2 q " k, as done in [25]. We follow this convention in the present section. So in the definition of X u above take m " u¨pe 1`e2 q. The projection random variables on all the path spaces are denoted by X i px m:n q " x i for all choices´8 ď m ď n ď 8 and i in the correct range.
Fix ω P Ω and m P Z. Define a family of stochastic kernels tκ ω k,l : l ě k ě mu on semi-infinite paths x m:8 through the integral of a bounded Borel function f : κ ω k,l f px m:8 q " ż f py m,8 q κ ω k,l px m:8 , dy m,8 q " ÿ y k:l PXx k ,x l f px m:k y k:l x l:8 q Q ω x k ,x l py k:l q. (2.9) In other words, the action of κ ω k,l amounts to replacing the segment x k:l of the path with a new path y k:l sampled from the quenched polymer distribution Q ω x k ,x l . The argument x m:k y k:l x l:8 inside f is the concatenation of the three path segments. There is no inconsistency because y k " x k and y l " x l Q ω x k ,x l -almost surely. The key point is that the measure κ ω k,l px m:8 ,¨q is a function of the subpaths px m:k , x l:8 q. Note that the same kernel κ ω k,l works on paths x m:8 for any m ď k and also on the space X of bi-infinite paths by replacing m with´8 in the expressions above. With these kernels one defines semi-infinite and bi-infinite polymer Gibbs measures. Let F I " σtX i : i P Iu denote the σ-algebra generated by the projection variables indexed by the subset I of indices. Definition 2.1. Fix ω P Ω and u P Z 2 and let m " u¨pe 1`e2 q. Then a Borel probability measure ν on X u is a semi-infinite polymer Gibbs measure rooted at u in environment ω if for all integers l ě k ě m and any bounded Borel function f on X u we have E ν rf |F m,k Y l,8 s " κ ω k,l f . This set of probability measures is denoted by DLR ω u .  [21]. The reason is that the path space X is not a Z-indexed product space and the stochastic kernel κ ω k,l px´8 :8 ,¨q " Q ω x k ,x l p¨q is not defined for all pairs of boundary points px k , x l q, but only when x k and x l can be connected by a nearest-neighbor path.
The issue addressed in our paper is the nonexistence of nontrivial bi-infinite Gibbs measures. For the sake of context, we state an existence theorem for semi-infinite Gibbs measures. Ω 0 such that PpΩ 0 q " 1 and for every ω P Ω 0 the following holds. For each u P Z 2 and interior direction ξ P se 2 , e 1 r there exists a Gibbs measure Π ω,ξ u P DLR ω u such that X n {n Ñ ξ almost surely under Π ω,ξ u . Futhermore, these measures can be chosen to satisfy this consistency property: if u¨pe 1`e2 q ď y¨pe 1`e2 q " n ď z¨pe 1`e2 q " r, then for any path x n:r P X y,z , Π ω,ξ u pX n:r " x n:r | X n " yq " Π ω,ξ y pX n:r " x n:r q.
Uniqueness of Gibbs measures is a more subtle topic, and we refer the reader to [25]. Since the Gibbs measure Π ω,ξ u satisfies the strong law of large numbers X n {n Ñ ξ, we can call it (strongly) ξ-directed. In general, a path x m:8 is ξ-directed if x n {n Ñ ξ as n Ñ 8.
We turn to bi-infinite Gibbs measures. First we observe that there are trivial bi-infinite Gibbs measures supported on straight line paths.

Definition 2.5.
A path x ‚ is a straight line if for a fixed i P t1, 2u, x n`1´xn " e i for all path indices n. If x ‚ is a bi-infinite straight line then µ " δ x ‚ is a bi-infinite Gibbs measure because the polymer distribution Q u,u`mei is supported on the straight line from u to u`me i . More generally, any probability measure supported on bi-infinite straight lines is a bi-infinite Gibbs measure.
The next natural question is whether there can be bi-infinite polymer paths that are not merely straight lines but still directed into e i . That this option can be ruled out is essentially contained in the results of [25]. We make this explicit in the next theorem. It says that under both semi-infinite and bi-infinite Gibbs measures, up to a zero probability event, e i -directedness even along a subsequence is possible only for straight line paths. Note that (2.11) covers both e i -and p´e i q-directedness.
Non-existence of bi-infinite polymers Theorem 2.6. Assume (2.5). There exists an event Ω 0 Ď Ω such that PpΩ 0 q " 1 and for every ω P Ω 0 the following statements hold for both i P t1, 2u: (a) For all u P Z 2 and ν P DLR ω u , with m " u¨pe 1`e2 q, ν lim nÑ8 n´1|X n¨e3´i | " 0 ( " νtX n " u`pn´mqe i for n ě mu. (2.10) is an e i -directed bi-infinite straight lineu. (2.11) Proof. Let the event Ω 0 of full P-probability be the intersection of the events specified in Lemma 3.4 and Theorem 3.5 of [25].
Part (a). We can assume that the left-hand side of (2.10) is positive because the event on the right is a subset of the one on the left. Since A " tlim nÑ8 n´1|X n¨e3´i | " 0u is a tail event, it follows that r ν " νp¨| Aq P DLR ω u . The path space X u is compact in the product topology because once the initial point u is fixed, each coordinate x i has a finite range. Hence r ν is a mixture of extreme members of DLR ω u . (This is an application of Choquet's theorem, discussed more thoroughly in Section 2.4 of [25].) This mixture can be restricted to extreme Gibbs measures that give the event A full probability.
By Lemma 3.4 and Theorem 3.5 of [25], an extreme member of DLR ω u that is not directed into the open interval se 2 , e 1 r must be a degenerate point measure Π ei u , which is the probability measure supported on the single straight line path pu`pn´mqe i q n:něm . We conclude that r ν " Π ei u .
Let us show how we deduce (2.10). Let B ei u " tX n " u`pn´mqe i for n ě mu be the event that from u onwards the path is an e i -directed line. Then by conditioning, νpB ei u q " νpB ei u X Aq " r νpB ei u qνpAq " νpAq.
Part (b). Consider first the case n Ñ 8. Let m P Z and x¨pe 1`e2 q " m. Suppose µpX m " xq ą 0. Then, by Lemma 2.4 in [25], µ x " µp¨| X m " xq P DLR ω x . Part (a) applied to µ x shows that µtX m " x, lim nÑ8 n´1|X n¨e3´i | " 0u " µtX n " x`pn´mqe i for n ě mu. (2.12) By summing over the pairwise disjoint events tX m " xu gives, for each fixed m P Z, µt lim nÑ8 n´1|X n¨e3´i | " 0u " µtX n " X m`p n´mqe i for n ě mu.
The events on the right decrease as m Ñ´8, and in the limit we get µt lim nÑ8 n´1|X n¨e3´i | " 0u " µtX n " X m`p n´mqe i for all n, m P Zu which is exactly the claim (2.11) the case n Ñ 8.
The case n Ñ´8 of (2.11) follows by reflection across the origin. Let ω " pY x q xPZ 2 and define reflected weights r ω " p r Y x q xPZ 2 by r Y x " Y´x. Given µ P Ð Ý Ñ DLR ω , define the reflected measure r µ by setting, for m ď n and x m:n P X xm,xn , r µpX m:n " x m:n q " µpX i " x´i for i "´n, . . . ,´mq. Then r µ P Ð Ý Ñ DLR r ω . Directedness towards´e i under µ is now directedness towards e i under r µ, and we get the conclusion by applying the already proved part to r µ. . Fix ξ P se 2 , e 1 r . Then there exists an event Ω bi,ξ Ď Ω such that PpΩ bi,ξ q " 1 and for every ω P Ω bi,ξ there exists no measure µ P Ð Ý Ñ DLR ω such that as n Ñ 8, X n {n Ñ ξ in probability under µ.
We assumed (2.7) above to avoid introducing technicalities not needed in the rest of the paper. The global regularity assumption (2.7) can be weakened to local hypotheses, as done in Theorem 3.13 in [25].
The results above illustrate how far one can presently go without stronger assumptions on the model. The hard question left open is whether bi-infinite Gibbs measures can exist in random directions in the open interval se 2 , e 1 r . To rule these out we restrict our treatment to the exactly solvable case of inverse-gamma distributed weights.
That only directed Gibbs measures would need to be considered in the sequel is a consequence of Corollary 3.6 of [25]. However, we do not need to assume this directedness a priori and we do not use Theorem 2.7. At the end we will appeal to Theorem 2.6 to rule out the extreme slopes. As stated above, Theorem 2.6 does not seem to involve the regularity of Λ. But in fact through appeal to Theorem 3.5 of [25], it does rely on the nontrivial (but provable) feature that Λ is not affine on any interval of the type sζ, e 1 s (and symmetrically on re 2 , ηr ). This is the positive temperature counterpart of Martin's shape asymptotic on the boundary [29] and can be deduced from that (Lemma B.1 in [25]).

Bi-infinite Gibbs measures in the inverse-gamma polymer
A random variable X has the inverse gamma distribution with parameter θ ą 0, abbreviated X " Ga´1pθq, if its reciprocal X´1 has the standard gamma distribution with parameter θ, abbreviated X´1 " Gapθq. Their density functions for x ą 0 are f X´1 pxq " 1 Γpθq x θ´1 e´x for the gamma distribution Gapθq and f X pxq " 1 Γpθq x´1´θe´x´1 for the inverse gamma distribution Ga´1pθq. (2.13) Here Γpθq " ş 8 0 s θ´1 e´s ds is the gamma function.
Our basic assumption is: The weights pY x q xPZ 2 are i.i.d. inverse-gamma distributed random variables on some probability space pΩ, A, Pq. (2.14) The main result is stated as follows.
Due to Theorem 2.6(b), to prove Theorem 2.8 we only need to rule out the possibility of bi-infinite polymer measures that are directed towards the open segments s´e 2 ,´e 1 r and se 2 , e 1 r . The detailed proof is given in Section 5, after the development of preliminary estimates. For the proof we take Y x to be a Ga´1p1q variable. We note that there is no loss of generality due to our choice of the parameter 1 as using a different scale amounts to multiplying the weights by a scalar due to the scaling properties of the Gamma distribution.
For the interested reader, we mention that the semi-infinite Gibbs measures of the inverse-gamma polymer are described in the forthcoming work [20]. Earlier results appeared in [22] where such measures were obtained as almost sure weak limits of quenched point-to-point and point-to-line polymer distributions.

Stationary inverse-gamma polymer
The proof of Theorem 2.8 relies on the fact that the inverse-gamma polymer possesses a stationary version with accessible distributional properties, first constructed in [33]. This section gives a brief description of the stationary polymer and proves an estimate. Further properties of the stationary polymer are developed in the appendixes.
Let pY x q xPZ 2 be i.i.d. Ga´1p1q weights. A stationary version of the inverse-gamma polymer is defined in a quadrant by choosing suitable boundary weights on the south and west boundaries of the quadrant. For a parameter 0 ă α ă 1 and a base vertex o, introduce independent boundary weights on the xand y-axes emanating from o: The above convention, that the horizontal edge weight I α has parameter 1´α while the vertical J α has α, is followed consistently and it determines various formulas in the sequel.
For vertices p ě o define the partition functions The quenched polymer distribution corresponding to (3.2) is given by Q α o,p px‚q " pZ α o,p q´1 ś |p´o|1 i"0 r Y xi for x‚ P X o,p , and the annealed measure is P α o,p px‚q " ErQ α o,p px‚qs.
It will be convenient to consider also backward polymer processes whose paths proceed in the southwest direction and the stationary version starts with boundary weights on the north and east. For vertices o ě p let p X o,p be the set of down-left paths starting at o and terminating at p. As sets of vertices and edges, paths in p X o,p are exactly the same as those in X p,o . The difference is that in p X o,p paths are indexed in the down-left direction.
For o ě p, backward partition functions are then defined with i.i.d. bulk weights as and in the stationary case as The independent boundary weights I α o´ie1 and J α o´je2 (i, j ě 1) have the distributions We define functions that capture the wandering of a path x‚ P X o,p . The (signed) exit point or exit time τ o,p " τ o,p px‚q marks the position where the path x‚ leaves the southwest boundary and moves into the bulk, with the convention that a negative value indicates a jump off the y-axis. More generally, for 3 vertices o ď v ă p, τ o,v,p " τ o,v,p px‚q marks the position where x‚ P X o,p enters the rectangle v`e 1`e2 , p , again with a negative sign if this entry happens on the west edge tv`e 1`j e 2 : 1 ď j ď pp´vq¨e 2 u.
Here is the precise definition: Exactly one of the two cases above happens for each path x‚ P X o,p . The exit point from the boundary is then defined by τ o,p " τ o,o,p . An analogous definition is made for the backward polymer. For o ě v ą p and x‚ P p The signed exit point from the northeast boundary is p The remainder of this section is devoted to an estimate needed in the body of the proof. First recall that the digamma function ψ 0 " Γ 1 {Γ is strictly concave and strictly increasing on p0, 8q, with ψ 0 p0`q "´8 and ψ 0 p8q " 8. Its derivative, the trigamma function ψ 1 " ψ 1 0 , is positive, strictly convex, and strictly decreasing, with ψ 1 p0`q " 8 and ψ 1 p8q " 0. These functions appear as means and variances: for η " Ga´1pαq, Erlog ηs "´ψ 0 pαq and Varplog ηq " ψ 1 pαq. (3.6) In the stationary polymer Z α o,p in (3.2), the boundary weights are stochastically larger than the bulk weights. Consequently the polymer path prefers to run along one of the boundaries, its choice determined by the direction pp´oq{|p´o| 1 P re 2 , e 1 s. For each parameter α P p0, 1q there is a particular characteristic direction ξpαq P se 2 , e 1 r at which the attraction of the two boundaries balances out. For ρ P r0, 1s this function is given by The extreme cases are interpreted as ξp0q " e 1 and ξp1q " e 2 . The inverse function ρ " ρpξq of a direction ξ " pξ 1 , ξ 2 q P re 2 , e 1 s is defined by ρpe 2 q " 1, ρpe 1 q " 0, and The function ρpξq is a strictly decreasing bijective mapping of ξ 1 P r0, 1s onto ρ P r0, 1s, or, equivalently, a strictly decreasing mapping of ξ in the down-right order. The significance of the characteristic direction for fluctuations is that τ o,p is of order |p´o| 2{3 1 if and only if p´o is directed towards ξpαq, and of order |p´o| 1 in all other directions.
These fluctuation questions were first investigated in [33].
We insert here a lemma on the regularity of the characteristic direction.
Proof. As the function ψ 1 is smooth on p0, 8q Recall that to prove Theorem 2.8, our intention is to rule out bi-infinite polymer measures whose forward direction is into the open first quadrant, and whose backward direction is into the open third quadrant. The main step towards this is that, as N becomes large, a polymer path from southwest to northeast across the square ´N, N 2 , with slope bounded away from 0 and 8, cannot cross the y-axis anywhere close to the origin.
To achieve this we control partition functions from the southwest boundary of the square ´N, N 2 to the interval J " ´N 2{3 e 2 , N 2{3 e 2 on the y-axis, and backward partition functions from the northeast boundary of the square ´N, N 2 to the interval p J " e 1`J shifted one unit off the y-axis. Let ε ą 0. We establish notation for the southwest portion of the boundary of the square ´N, N 2 that is bounded by the lines of slopes ε and ε´1. With W for west and S for south, let B N W " t´N uˆ ´N,´εN , B N S " ´N,´εN ˆt´N u, and then The parameter ε ą 0 stays fixed for most of the proof, and hence will be suppressed from much of the notation. We also let o i " p´N,´εN q and o f " p´εN,´N q. A lattice point o " po 1 , o 2 q P B N is associated with its (reversed) direction vector ξpoq " pξ 1 poq, 1´ξ 1 poqq P se 2 , e 1 r and parameter ρpoq P p0, 1q through the relations ξpoq "ˆo
If we define the extremal parameters (for a given ε ą 0) by For o P B N define perturbed parameters (with dependence on r, N suppressed from the notation): ρ ‹ poq " ρpoq´rN´1 3 and ρ ‹ poq " ρpoq`rN´1 3 . (3.13) The variable r can be a function of N and become large but always rpN qN´1 {3 Ñ 0 as N Ñ 8. Then for N ě N 0 pεq the perturbed parameters are bounded uniformly away from 0 and 1: We consider the stationary processes Z Proof. We prove (3.16) as (3.15) is similar. We turn the quenched probability into a form to which we can apply fluctuation bounds. The justifications of the steps below go as follows.
(i) The first inequality below is from (A.14).  (iv) Choose an integer so that the vector from o`dN 2{3 e 1´ e 2 to N 2{3 e 2 points in the characteristic direction ξpρ ‹ poqq. Apply (A.5) and stationarity.
We show that ě c 0 pεqrN 2{3 for a constant c 0 pεq. Let o "´pN a, N bq, with ε ď a, b ď 1. Lemma 3.1 gives the next identity. The O-term hides an ε-dependent constant that is uniform for all ρpoq because, as observed in (3.12), the assumption o P B N bounds ρpoq away from 0 and 1.
Recall from Lemma 3.1 that φpρpoqq ą 0 is uniformly bounded away from zero for o P B N . For a small enough constant cpεq and large enough constants C 0 pεq and N 0 pεq, if we have 1 ď d ď cpεqN 1{3 , C 0 pεqd ď r ď cpεqN 1{3 and N ě N 0 pεq, the above simplifies to ě c 0 pεqrN 2{3 .
We can derive the final bound.
The final inequality comes from Theorem B.6.

Estimates for paths across a large square
After the preliminary work above we turn to develop the estimates that prove the main theorem. Throughout, d " pd 1 , d 2 q P Z 2 ě1 denotes a pair of parameters that control the coarse graining on the southwest and northeast boundaries of the square ´N, On the rectangle o c , N e 2 we define coupled polymer processes. For each u P I o,d we have the bulk process Z u,‚ that uses Ga´1p1q weights Y . Two stationary comparison processes based at o c have parameters ρ ‹ po c q and ρ ‹ po c q defined as in (3.13). Their basepoint is taken as o c so that we get simultaneous control over all the processes based at vertices u P I o,d .
Couple the boundary weights on the south and west boundaries of the rectangle o c , N e 2 as described in Theorem B.4 in Appendix B.2. In particular, for k, ě 1 we have the inequalities For all these coupled processes we define ratios of the partition functions from the base point to the y-axis, for all u P I o,d and i P ´N 2{3 , N 2{3 : Recall that J " ´N * .  On the event A oc,d,y , for any m, n P ´N 2{3 , N 2{3 such that m ă n we have the inequali- Proof. Bound (4.4) comes by switching to complements in Lemma 3.2. We show the second inequality of (4.5). The first inequality follows similarly. Let u P I o,d . The first inequality in the calculation (4.6) below is justified as follows in two cases. Recall the notation (2.4) for restricted partition functions Z o,p pAq.
on the y-axis, and on the x-axis takes any Y p1q u`me1 ă Y u`me1 for 1 ď m ď´u¨e 1 . Then the second inequality of (A.6) followed by the second inequality of (A.10) gives Next observe that the condition τ u,‚ ă j´d 1 N 2 3 ă 0 renders the boundary weights on the x-axis u`pZ ą0 qe 1 irrelevant. Therefore we can replace Y p1q u`me1 with the stationary boundary weights I ρ‹pocq u`me1 without changing the restricted partition functions on the right-hand side. This gives the first equality below: The second equality comes by multiplying upstairs and downstairs with the boundary weights J ρ‹pocq oc` e2 for 1 ď ď j " pu´o c q¨e 2 .
(ii) On the other hand, if u " o c`k e 1 for some 0 ď k ď d 1 N 2{3 , then first by (A.9) and then by applying the argument of the previous paragraph to u " o c : Now for the derivation.  The boundary weights in (4.1) and in (4.7) above are taken independent of each other.
Ratio weights on the shifted y-axis are defined by The collection of ration weights in (4.2) is independent of the collection in (4.8) above because they are constructed from independent inputs.
We have this analogue of Lemma 4.1. p J " e 1`J " e 1´N     Abbreviate the parameters for the base points as ρ ‹ " ρ ‹ po c q, ρ ‹ " ρ ‹ po c q, λ ‹ " ρ ‹ pp o c q, and λ ‹ " ρ ‹ pp o c q. Non-existence of bi-infinite polymers For i P ´N 2{3 , N 2{3 , take the Z-ratios from (4.2) and (4.8) and define A two-sided multiplicative walk M pXq with steps tX j u is defined by (4.14) The ratios from (4.13) above define the walks M u,v " M pX u,v q , M 1 " M pY 1 q and M " M pY q.   implies the independence of tJ ρ‹ i u iď0 and tJ ρ ‹ i u iě1 , and the independence of t p J λ‹ i u iě1 and t p J λ ‹ i u iď0 . With boundary weights on the southwest, the independence of tJ ρ‹ i u iď0 and tJ ρ ‹ i u iě1 is a direct application of Theorem B.4(i) with the choice pλ, ρ, σq " pρ ‹ , ρ ‹ , 1q. After reflection of the entire setting of Theorem B.4 across its base point u, the boundary weights reside on the northwest, as required for t p J λ‹ i u iě1 and t p J λ ‹ i u iď0 , and the direction e 2 has been reversed to´e 2 . Hence the inequalities i ď 0 and i ě 1 in the independence statement must be switched around.
To summarize, the collections tJ ρ‹ i , p J λ ‹ i u iď0 and tJ ρ ‹ i , p J λ‹ i u iě1 are independent of each other, which implies the independence of tY 1 i u iď0 from tY i u iě1 . We show the case n P 1, N 2{3 of (4.17).
An analogous argument gives the case n P ´N 2{3 ,´1 .
Each path that crosses the y-axis leaves the axis along a unique edge e i " pie 2 , ie 2`e1 q. Decompose the set of paths between u P B N and v P p B N according to the edge taken: be the quenched probability of paths going through the edge e i . We come to the important step that bounds these edge probabilities in terms of the multiplicative walk introduced above in (4.15). Namely, for all n P ´N 2{3 , N 2{3 we claim that Here is the verification for n ě 1: The case n ď´1 goes similarly.
We are ready to derive the key estimates. The first one controls the quenched probability of paths between I o,d and p I (4.21) By the independence in (4.16), To apply the random walk bound from Appendix C, we convert the multiplicative walks into additive walks. For given steps ξ " tξ i u define the two-sided walk Spξq by S n pξq " Recall the parameters defined in (4.12). With reference to (4.13) and (4.15), define the additive walks S n ă logp2N qC r´3. (4.23) We use Theorem C.1 to bound Pp max 1ďnďN 2{3 S n ă logp2N qq. Since we can establish constants 0 ă ρ min ă ρ max ă 1 and N 0 pεq P Z ą0 such that ρ ‹ , λ ‹ P rρ min , ρ max s for all o P B N and N ě N 0 pεq. As |o´o c | ď 1 2 d 1 N 2{3 and |p o´p o c | ď 1 2 d 2 N 2{3 , the restriction of the slope to rε, ε´1s implies that there is a constant C " Cpεq such that Then, since ρpoq " ρp´oq " ρpp oq, We conclude that for N ě N 0 pεq, the mean step of S n satisfies where the (new) constant C " Cpεq works for all o P B N . In Theorem C.1 set x " plog N q 2 to conclude that for N ě N 0 pεq Similarly one can show that P suṕ The lemma follows by inserting these bounds and r " N 2{15 into (4.23).
The next lemma controls the quenched probability of paths from points u P I o,d that go through the edge e 0 from 0 to e 1 but miss the interval p I      Geometrically, starting from the north pole N e 2 and traversing the boundary of the square ´N, N 2 clockwise, we meet the points (those that exist) in this order: (Figure 4.2). The set p F p o,d can be decomposed into two disjoint sets  For all u P I o,d and v P p F 1 p o,d , the pairs pu, vq and ph 1 , q 1 q satisfy the relation pu, vq ď ph 1 , q 1 q defined in (A.11). By Lemma A.3 we can couple random paths π u,v " Q u,v and π h 1 ,q 1 " Q h 1 ,q 1 so that π u,v ď π h 1 ,q 1 in the path ordering defined in Appendix A.3, simultaneously for all u P I o,d and v P p F 1 p o,d . Then π u,v P X 0 u,v forces π h 1 ,q 1 P Xh 1 ,q 1 , and we conclude that Hence the probability on the left of (4.27) ď PtQ h 1 ,q 1 pXh 1 ,q 1 q ą δu.
The last probability will be shown to be small by appeal to a KPZ wandering exponent bound from [33] stated in Appendix B.3. To this end we check that the line segment rh 1 , q 1 s from h 1 to q 1 crosses the vertical axis far above the origin on the scale N 2{3 . For o P B N and p o "´o P p B N , decompose h j " o`l j and q j " p o`r j . These vectors l j " pl j 1 , l j 2 q and r j " pr j 1 , r j 2 q satisfy , and r j 1 r j 2 ď 0. Use first the definition of h j and then q j (4.30) The first term on the last line is of order Θpd 2 N 2{3 q because there is no cancellation in the numerator. It is positive if j " 1 and negative if j " 2. This term dominates because d 2 " N 1 8 " 1 " d 1 . Let y 1 e 2 P rh 1 , q 1 s, that is, y 1 is the distance from the origin to the point where the line segment rh 1 , q 1 s crosses the y-axis. We bound this quantity from below. In addition to The last line of (4.30) gives (4.31) The last inequality used pd 1 , d 2 q " p1, N 1{8 q and took N ě p16ε´2q 8 . The wandering exponent bound stated in Theorem B.5 gives P h 1 ,q 1 pXh 1 ,q 1 q ď Cpεqd´3 2 for a constant Cpεq that works for all o P B N and N ě N 0 pεq. By Markov's inequality PtQ h 1 ,q 1 pXh 1 ,q 1 q ą δu ď Cpεqδ´1d´3 2 " Cpεqδ´1N´3 {8 . (4.32) The proof of (4.27) is complete.
We combine the estimates from above to cover all vertices on B N and p B N .
Theorem 4.6. There exist constants Cpεq, N 0 pεq such that for δ P p0, 1q and N ě δ´1 _ N 0 pεq, Proof. As before, d " p1, N 1 8 q. We first claim that for any o P B N , Next we coarse grain the southwest boundary B N . Let

Proof of the main theorem
Proof of Theorem 2.8. By Theorem 2.6(b), for almost every ω every bi-infinite Gibbs measure µ satisfies " tX‚ is a bi-infinite straight lineu µ-almost surely where X‚ " X´8 :8 is the bi-infinite polymer path under the measure µ. This equality follows because Theorem 2.6(b) has these consequences for (5.1): the union on the left is disjoint, the event on the right is a subset of the union on the left, and their µ-probabilities are equal. The complement of the union on the left is the following event: the limit points of |n|´1X n lie in s´e 2 ,´e 1 r when n Ñ´8 and in se 2 , e 1 r when n Ñ 8. Thus to complete the proof we show the existence of an event Ω 1 such that PpΩ 1 q " 1 and for each ω P Ω 1 , no µ P Ð Ý Ñ DLR ω assigns positive probability to this last property of the limit points of |n|´1X n .
We put ε back into the notation. For ε ą 0 let D ε " tξ P se 2 , e 1 r : Say that a bi-infinite path x‚ is p´D ε qˆD ε -directed if the limit points of |n|´1x n lie in´D ε when n Ñ´8 and in D ε when n Ñ 8. Recall the definition of the edges e i " pie 2 , ie 2`e1 q and define these sets of bi-infinite paths: X ε,i " x‚ P X : x‚ is p´D ε qˆD ε -directed and x‚ goes through e i ( .
Assume this proved. Let ε k " 2´k. Then for ω P Ω 1 and µ P which is the required result.
It remains to define the event Ω 1 and verify (5.2). Recall the definition (4.19) of p u,v i .
Define translations T x on weight configurations ω " pY x q by pT x ωq y " Y x`y . Define By Theorem 4.6, ξ ε N Ñ 0 in probability as N Ñ 8, and hence PpΩ 1 q " PpΩ 2 ε q " 1. A p´D ε qˆD ε -directed bi-infinite path intersects both B N, ε and p B N, ε for all large enough N . (This is because D ε bounds the slopes by ε 1{2 which is larger than ε.) Thus if we let Let ε " 2´k for some k ě 1, ε 1 " ε{2, and abbreviate N 1 " N`rN 2{3 s. In the scale N 1 consider the translated square ie 2` ´N 1 , N 1 2 centered at ie 2 , with its boundary portions ie 2`B N1, ε 1 in the southwest and ie 2`p B N1, ε 1 in the northeast. This translated N 1 -square contains ´N, N 2 for all i P ´N 2{3 , N 2{3 .
There exists a finite constant N 0 pεq such that |i|`ε 1 N 1 ď εN for all i P ´N 2{3 , N 2{3 and N ě N 0 pεq. Then every path x‚ P X N,ε,i necessarily goes through both ie 2`B N1, ε 1 Figure 5.1: The inner NˆN square is centered at 0 while the outer N 1ˆN1 square is centered at ie 2 . The (thick, dark) boundary segments of the outer square cover the (thick, light) boundary segments of the inner square. Thus the path through ie 2 that crosses B N,ε and p B N,ε is forced to also cross ie 2`B N1, ε 1 and ie 2`p B N1, ε 1 . and ie 2`p B N1, ε 1 . In other words, x‚ is a member of the translate ie 2`X N1,ε 1 ,0 of the class of paths that go through the edge e 0 . This is illustrated in Figure 5.1. On the event X N,ε,i let, in the coordinatewise ordering, X B " inftX‚ X pie 2`B N1, ε 1 qu be the first vertex of the path X‚ in ie 2`B N1, ε 1 and X p B " suptX‚ X pie 2`p B N1, ε 1 qu the last vertex of the path in ie 2`p B N1, ε 1 . Note that for u P pie 2`B N1, ε 1 q and v P pie 2`p B N1, ε 1 q, the event tX B " u, X p B " vu depends on the entire path X‚ only through its edges outside ie 2` ´N 1 , N 1 2 . Suppose µpX N,ε,i q ą 0 for some µ P Ð Ý Ñ DLR ω . Below we apply the Gibbs property, recall the definition (4.18) of X 0 u,v as the set of paths from u to v that take the edge e 0 " p0, e 1 q, and write Q ω so that we can include explicitly translation of the weights ω.

A General properties of planar directed polymers
This appendix covers some consequences of the general polymer formalism. We begin again with the partition function with given weights Y x ą 0:

A.1 Ratio weights and nested polymers
Keeping the base point u fixed, define ratio weights for varying x: The ratio weights can be calculated inductively from boundary values I u`ke1 " Y u`ke1 and J u` e2 " Y u` e2 for k, ě 1, by iterating On the boundary of the quadrant v`Z 2 ě0 , put ratio weights of the partition functions with base point u: for r P t1, 2u and i ě 1.
The ratio weights dominate the original weights: Y puq v`ier ě Y v`ier , and equality holds iff v " u`me r for some m ě 0.
Define a partition function Z puq v,w that uses these boundary weights and ignores the first weight of the path: for k, ě 1 and w P v`Z 2 ą0 , ą0 the definition from above can be rewritten as follows: Thus for all u ď v ď w we have the identity Ratio variables satisfy with the analogous identity J u,x " J puq v,x .
Recall the definition (3.5) of τ u,v,w . Let Q puq v,w be the quenched path probability on X v,w that corresponds to the partition function Z puq v,w . Then we have the identity Q u,w pτ u,v,w " q " Q puq v,w pτ v,w " q for 0 ‰ P Z.
Here is the derivation for the case where the path from u to w goes above v. Let k ě 1.

A.2 Inequalities for point-to-point partition functions
We state several inequalities that follow from the next basic lemma. The inequalities in (A.6) below are proved together by induction on x and y, beginning with x " u`ke 1 and y " u` e 2 . The induction step is carried out by formulas (A.2).
for all k, ě 1 and x P u`Z 2 ą0 . Then we have the following inequalities for x ě u`e 1 and y ě u`e 2 : From the lemma we obtain the following pair of inequalities for z P u`Z 2 ą0 : . (A.7) The first inequality above follows from the first inequality of (A.6) by letting the weights tY p2q u`je2 u jě1 tend to zero, and the second one by letting the weights tY p1q u´e2`ie1 u iě1 tend to zero.
Lemma A.2. Let x, y, z P Z 2 be such that x ď y and x, y ď z´e 1´e2 . We then have Proof. (A.8) follows from repeated application of (A.7) along the steps e 1 and´e 2 from x to y. Inequality (A.9) follows similarly.
Since u`ke 1 ě u and u` e 2 ď u for k, ě 0, inequalities (A.8)-(A.9) imply also these for 1 ď k ă px´uq¨e 1 and 1 ď ă py´uq¨e 2 : ď Z u,x pτ u,x ě kq Z u,x´e1 pτ u,x´e1 ě kq and Z u,y Z u,y´e2 ď Z u,y pτ u,y ď´ q Z u,y´e2 pτ u,y´e2 ď´ q for u ď x, y. (A.10) To illustrate the explicit proof of the first one: Figure A.1: On the left the pairs px 1 , y 1 q and px 2 , y 2 q satisfy px 1 , y 1 q ď px 2 , y 2 q, while on the right this relation fails. Consistently with this, on the left the paths π 1 P X x 1 ,y 1 and π 2 P X x 2 ,y 2 satisfy π 1 ď π 2 but on the right this fails.

A.3 Ordering of path measures
The down-right partial order ď on R 2 and Z 2 was defined by px 1 , x 2 q ď py 1 , y 2 q if x 1 ď y 1 and x 2 ě y 2 . Extend this relation to pairs of vertices px 1 , y 1 q, px 2 , y 2 q P Z 2ˆZ2 as follows (illustrated in Figure A px 1 , y 1 q ď px 2 , y 2 q if x 1 ď y 1 , x 2 ď y 2 , x 1 ď x 2 and y 1 ď y 2 . (A.11) Extend this relation further to finite paths: π 1 P X x 1 ,y 1 and π 2 P X x 2 ,y 2 satisfy π 1 ď π 2 if the pairs of endpoints satisfy px 1 , y 1 q ď px 2 , y 2 q and whenever z 1 P π 1 , z 2 P π 2 , and z 1¨p e 1`e2 q " z 2¨p e 1`e2 q, we have z 1 ď z 2 . Pictorially, in a very clear sense, π 1 lies (weakly) above and to the left of π 2 . See again Figure A.1.
Let µ and ν be probability measures on the finite path spaces X x 1 ,y 1 and X x 2 ,y 2 , respectively. We write µ ď ν if there exist random paths X 1 P X x 1 ,y 1 and X 2 P X x 2 ,y 2 on a common probability space such that X 1 " µ, X 2 " ν, and X 1 ď X 2 . In other words, µ ď ν if ν stochastically dominates µ under the partial order ď on paths. The following shows that for fixed weights there exists a coupling of all the quenched polymer distributions tQ x,y u xďy on the lattice Z 2 so that Q x,y ď Q u,v whenever px, yq ď pu, vq. Lemma A.3. Let pY x q xPZ 2 be an assignment of strictly positive weights on the lattice Z 2 . Then there exists a coupling of up-right random paths tπ x,y u xďy such that π x,y P X x,y , π x,y has the quenched polymer distribution Q x,y , and π x,y ď π u,v whenever px, yq ď pu, vq.
Proof. Let tU z u zPZ 2 be an assignment of i.i.d. uniform random variables U z " Unifp0, 1q to the vertices of Z 2 , defined under some probability measure P. For each pair x ď z such that x ‰ z, define the down-left pointing random unit vector If z " x`ke i this gives V x pzq "´e i due to the convention Z u,v " 0 when u ď v fails. Hence any path that starts at some vertex y ě x distinct from x and follows the steps from each z to z`V x pzq terminates at x. Since the paths from distinct points that follow increments V x pzq for a given x eventually coalesce, a realization of tV x pzqu zěx: z‰x defines a spanning tree T x rooted at x on the nearest-neighbor graph on the quadrant x`Z 2 ě0 . For x ď y let π x,y P X x,y be the path that connects x and y in the tree T x . Then for any path x‚ P X x,y , (A.12) implies that Q x,y px‚q " Ppπ x,y " x‚q. In other words, through the random paths tπ x,y u xďy we have a coupling of the quenched polymer distributions tQ x,y u xďy . Let x ď u. By Lemma A.2 Hence tV x pzq "´e 2 u Ď tV u pzq "´e 2 u and tV u pzq "´e 1 u Ď tV x pzq "´e 1 u.
It follows from (A.13) that two paths satisfy π x,y ď π u,v whenever px, yq ď pu, vq. This is because if these paths share a vertex z, then their subsequent down-left steps satisfy z`V x pzq ď z`V u pzq.
Let o ď x. In the tree T o constructed above, the path from x down to o stays weakly to the left of the path from x`e 1 down to o. This gives the inequality below: A similar bound holds for t e2 " pτ o,x q´.

A.4 Polymers on the upper half-plane
The stationary inverse-gamma polymer process that is our tool for calculations will be constructed on a half-plane. This section defines the notational apparatus for this purpose, borrowed from the forthcoming work [20].
Define mappings of bi-infinite sequences: I " pI k q k PZ and Y " pY j q jPZ in R Z ą0 that are assumed to satisfy From these inputs, three outputs r I " p r I k q k PZ , J " pJ k q k PZ and r Y " p r Y k q k PZ , also elements of R Z ą0 , are constructed as follows.
Let Z " pZ k q k PZ be any function on Z that satisfies I k " Z k {Z k´1 . This defines Z up to a positive multiplicative constant. Define the sequence r Z " p r Z q PZ by r Z " Under assumption (A.15) the sum on the right-hand side of (A.16) is finite. To check this choose a particular Z by setting Z 0 " 1. (Any other admissible Z is a constant multiple of this one.) Then Z k " The sequences r I, J and r Y are well-defined positive real sequences, and they do not depend on the choice of the function Z as long as Z has ratios I k " Z k {Z k´1 . The three mappings are denoted by r I " DpI, Y q, J " SpI, Y q, and r Y " RpI, Y q. (A.20) Beginning from r Z k " Y k pZ k`r Z k´1 q we derive these equations: The last formula iterates as follows: for ă m, We record two inequalities. From (A.21), If we start with two coordinatewise ordered boundary weights I j ď I 1 j (for all j) and use the same bulk weights Y to compute vertical ratio weights J " SpI, Y q and J 1 " SpI 1 , Y q, the inequality is reversed: Further manipulation gives the next lemma. We omit the proof.
Lemma A.4. To calculate t r I k , J k , r Y k : k ď mu, we need only the input tI k , Y k : k ď mu.
The next lemma is nontrivial and we include a complete proof.
Lemma A.5. The identity D`DpA, Iq, Y˘" D`DpA, RpI, Y qq, DpI, Y q˘holds whenever the sequences I, A, Y are such that the operations are well-defined.
Proof. Choose pZ j q and pB j q so that Z j {Z j´1 " I j and B j {B j´1 " A j . Then the output of DpA, Iq is the ratio sequence p r Next, the output of DpDpA, Iq, Y q is the ratio sequence pH m {H m´1 q m of Y j¯.

B The inverse-gamma polymer
This section reviews the ratio-stationary inverse-gamma polymer introduced in [33] and then constructs the two-variable jointly ratio-stationary process, which is a special case of the multivariate construction from the forthcoming work [20].

B.1 Inverse-gamma weights
Recall the inverse gamma distribution from (2.13) and it's mean from (3.6).
Lemma B.1. Define the mapping pI, J, Y q Þ Ñ pI 1 , J 1 , Y 1 q on R 3 ą0 by (b) Let α, β ą 0. Suppose that I, J, Y are independent random variables with distributions I " Ga´1pαq, J " Ga´1pβq and Y " Ga´1pα`βq. Then the triple pI 1 , J 1 , Y 1 q has the same distribution as pI, J, Y q.
Proof. Part (b) follows by applying the beta-gamma algebra (see Exercise 6.50 on page 244 of [1]) to the reciprocals that satisfy Lemma B.2. Let 0 ă ρ ă σ. Let I " pI k q k PZ and Y " pY j q jPZ be mutually independent random variables such that I k " Ga´1pρq and Y j " Ga´1pσq. Use mappings (A.20) to define r I " DpI, Y q r Y " RpI, Y q and J " SpI, Y q.
(a) tV k u k PZ is a stationary, ergodic process. For each k P Z, the random variables t r I j u jďk , J k , t r Y j u jďk are mutually independent with marginal distributions r I j " Ga´1pρq, r Y j " Ga´1pσq and J k " Ga´1pσ´ρq.
(b) r I and r Y are independent sequences of i.i.d. variables.
Proof. We start by verifying (A.15) to guarantee that the processes r I, r Y and J are almost surely well-defined and finite. To this end we show that Rewrite the above as where we can choose δ ą 0 to satisfy Erlog Y i´l og I i s "´ψ 0 pσq`ψ 0 pρq ă´3δ ă 0 (B.4) because ψ 0 is strictly increasing. Hence almost surely for large enough j ă 0, Non-existence of bi-infinite polymers The estimate below shows that, for any δ ą 0, sup jď0 e jδ Y j is almost surely finite: Erplog Y j q 2 s j 2 δ 2 ă 8.
The almost sure convergence of the series (B.2) has been verified. We turn to the proof of the lemma.
Part (b) follows from part (a) by dropping the J k coordinate and letting k Ñ 8. Stationarity and ergodicity of tV k u follow from its construction as a mapping applied to the independent i.i.d. sequences I and Y .
The distributional claims in part (a) are proved by coupling p r I k , J k´1 , r Y k q k PZ with another sequence of processes (indexed by N below) whose distribution we know. Let Z be a fixed Ga´1pσ´ρq variable that is independent of pI, Y q.
For each N ě 0, construct a process p p where ΘpI, J, Y q " pI 1 , J 1 , Y 1 q is the involution (B.1) in Lemma B.1. We claim that for each k P Z, Applying (A.23) gives where we chose δ ą 0 as in (B.4). Hence the last exponential factor above vanishes almost surely as N Ñ 8. The equation shows that tJ k u is a finite stationary process, and consequently e´N δ J´N Ñ 0 in probability. (B.9) implies the first limit in probability in (B.7).
To get the second limit in (B.7), apply (B.6) and the first limit as N Ñ 8: (B.11) For the last limit in (B.7), Next, we prove the following claim for each N ě 1: for each m ě´N`1, the random variables p I Ń N`1 , . . . , p are mutually independent with marginal distributions p I N k " Ga´1pρq , p J N m " Ga´1pσ´ρq, and p Y N j " Ga´1pσq.

B.2 Two jointly ratio-stationary polymer processes
Pick 0 ă λ ă ρ ă σ and a base vertex u " pu 1 , u 2 q P Z 2 . We construct two coupled polymer processes Z λ u,‚ and Z ρ u,‚ on the nonnegative quadrant u`Z 2 ě0 such that the joint process tpZ λ u,y {Z λ u,x , Z ρ u,y {Z ρ u,x q : x, y P u`Z 2 ě0 u of ratios is stationary under translations px, yq Þ Ñ px`v, y`vq. Both processes use the same i.i.d. Ga´1pσq weights tY x u x P u`Z 2 ą0 in the bulk. They have boundary conditions on the positive xand y-axes emanating from the origin at u, coupled in a way described in the next theorem.
For α P tλ, ρu, we repeat here the definition of the process Z α u,‚ given earlier in (3.2). On the boundaries of the quadrant we have strictly positive boundary weights tI α u`ie1 , J α u`je2 : i, j P Z ą0 u. Put Z α u,u " 1 and on the boundaries for k, l ě 1. (B.14) In the bulk for x " px 1 , x 2 q P u`Z 2 ą0 , Z α u,‚ does not use a weight at the base point u. Z x,y above is the partition function (A.1) that uses the bulk weights Y . Define ratio variables for vertices x P u`Z 2 ą0 by The next theorem describes the jointly stationary process that is used in the proofs of Section 4. Since those arguments work with the J-ratio variables on the y-axis, in order to tailor this theorem to its application we construct the joint process on the right half-plane and then restrict that process to the first quadrant. Consequently the upper half-plane of Sections A.4 and B.1 has been turned into the right half-plane, and thereby horizontal has become vertical. An important part of the theorem is the independence of various collections of ratio variables. These are illustrated in Figure B.1.
Theorem B.4. Let 0 ă λ ă ρ ă σ and u P Z 2 . There exists a coupling of the boundary weights tI λ u`ie1 , I ρ u`ie1 , J λ u`je2 , J ρ u`je2 : i, j P Z ą0 u such that the joint process pZ λ u,‚ , Z ρ u,‚ q has the following properties.
(i) (Joint) The joint process of ratios is stationary: for each v P u`Z 2 ě0 , ( . (B.17) (On the right above the implicit denominators Z λ u,u " Z ρ u,u " 1 were omitted.) The following independence property holds along vertical lines: for each x P u`Z 2 ą0 , the variables tJ λ x`je2 : u 2´x2`1 ď j ď 0u and tJ ρ x`je2 : j ě 1u are mutually independent.
(ii) (Marginal) For both α P tλ, ρu and for each v " pv 1 , v 2 q P u`Z 2 ě0 , the ratio variables tI α v`ie1 , J α v`je2 : i, j P Z ą0 u are mutually independent with marginal distributions The same is true of the variables tI α v´ie1 , J α v´je2 : 0 ď i ă v 1´u1 , 0 ď j ă v 2´u2 u.
(iii) (Monotonicity) The boundary weights can be coupled with i.i.d. Ga´1pσq weights tη u`ie1 , η u`je2 : i, j ě 1u independent of the bulk weights Y so that these inequalities hold almost surely for all i, j ě 1: Proof. We construct a joint partition function process pL λ x , L ρ x q x P u`Zě0ˆZ on the discrete right half-plane u`Z ě0ˆZ with origin fixed at u. The restriction of this process to the quadrant u`Z 2 ě0 then furnishes the process pZ λ u,‚ , Z ρ u,‚ q whose properties are claimed in the theorem.
In the interior put i.i.d. Ga´1pσq weights Y " tY x : x 1 ą u 1 u as before. (We write some weight configurations with bold symbols to distinguish the notation of this proof from earlier notation.) For α P tλ, ρu let Y λ " tY λ j u jPZ and Y ρ " tY ρ j u jPZ be independent sequences of i.i.d. variables with marginal distributions Y α j " Ga´1pαq, independent of Y. From these we define the boundary weights J λ " tJ λ u`je2 u jPZ and J ρ " tJ ρ u`je2 u jPZ on the y-axis through u by the equation pJ ρ , J λ q " pY ρ , DpY λ , Y ρ qq. D is the partition function operator from (A.20). This gives a pair of coupled sequences pJ ρ , J λ q. Marginally tJ α u`je2 u jPZ are i.i.d. Ga´1pαq. For α P tλ, ρu define the partition function values on the y-axis centered at u by L α u " 1 and " J α u`je2 for j P Z.
Complete the definitions by putting, again for α P tλ, ρu and now for x P u`Z ą0ˆZ , As in (A.16), the series converges because the boundary variables J α are stochastically larger than the bulk weights. This follows from the distributional properties established below. The evolution in (B.19) satisfies a semigroup property from vertical line to line: For k ě 0, denote the sequences of J-ratios on the vertical line shifted by ke 1 by J α,k " tJ α,k j u jPZ " tJ α u`ke1`je2 u jPZ and the sequences of interior weights by Y k " tY k j u jPZ " tY u`ke1`je2 u jPZ . J α,0 is the original boundary sequence J α we began with. One verifies inductively that J α,k " DpJ α,k´1 , Y k q for each k ě 1 and α P tλ, ρu.
Apply Lemma B.3 with parameters pσ, α 1 , α 2 q " pσ, ρ, λq. Directly from the definition pJ ρ , J λ q " pY ρ , DpY λ , Y ρ qq follows that pJ ρ , J λ q has the distribution of pI 1 , I 2 q in pJ ρ,k , J λ,k q d " pJ ρ , J λ q for all k ě 0. Equivalently, the joint distribution of the ratios along a vertical line is the same for all v P u`Z ě0ˆZ . The semigroup property (B.20) gives for both α P tλ, ρu The interior weights tY z : x P Z ě0ˆZ * (B.23) has the same distribution for all base points v P u`Z ě0ˆZ . Lemma B.3(v) gives the property that, for any x P u`Z ě0ˆZ , the ratio variables tJ λ x`je2 : j ď 0u and tJ ρ x`je2 : j ě 1u are mutually independent. (B.24) We claim that for α P tλ, ρu and for any new base point v P u`Z ě0ˆZ , tI α v`ie1 , J α v`je2 : i, j P Z ą0 u are mutually independent with marginal distributions I α v`ie1 " Ga´1pσ´αq and J α v`je2 " Ga´1pαq.

(B.25)
Since the joint distribution is shift-invariant, we can take v " u. As observed above, J α is a sequence of i.i.d. Ga´1pαq random variables by Lemma B.3(i). Thus it suffices to prove the marginal statement about tI α u`ie1 : i ě 1u because these variables are a function of tJ α u`je2 , Y u`pi,jq : i ě 1, j ď 0u which are independent of tJ α u`je2 : j ě 1u. The claim for tI α u`ie1 : i ě 1u follows from proving inductively the following statement for each n ě 1: tI α u`ie1 , J α u`ne1`je2 : 1 ď i ď n, j ď 0u are mutually independent with marginal distributions I α u`ie1 " Ga´1pσ´αq and J α u`ne1`je2 " Ga´1pαq. (B.26) Begin with the case n " 1. From the inputs given by boundary weights tI j " J α u`je2 : j ď 0u and bulk weights tY j " Y u`e1`je2 : j ď 0u, equation (A.17) computes the ratio weights t r I j " J α u`e1`je2 : j ď 0u and equation (A.18) gives J 0 " I α u`e1 . (Note here the switch between "horizontal" and "vertical".) Part of Lemma B.3(ii) then gives exactly statement (B.26) for n " 1. (The dual bulk weights q Y j that also appear in Lemma B.3(ii) are not needed here.) Continue inductively. Assume that (B.26) holds for a given n. Then feed into the polymer operators boundary weights tI j " J α u`ne1`je2 : j ď 0u and bulk weights tY j " Y u`pn`1qe1`je2 : j ď 0u, all independent of tI α u`ie1 : 1 ď i ď nu. Compute the ratio weights t r I j " J α u`pn`1qe1`je2 : j ď 0u and J 0 " I α u`pn`1qe1 . Lemma B.3(ii) extends the validity of (B.26) to n`1. Claim (B.25) has been verified.
To prove the full Theorem B.4 on the quadrant u`Z 2 ě0 , take the coupled boundary weights tI α u`ie1 , J α u`je2 : i, j ě 1, α P tλ, ρuu as constructed above. The partition function process tZ α u,x : x P u`Z 2 ě0 u defined by (B.14)-(B.15) is then exactly the same as the restriction tL α x : x P u`Z 2 ě0 u of L α . To verify this rewrite (B.15) as follows for x in the bulk u`Z 2 ą0 : Invariance (B.17) comes from the invariance statement about (B.23). The statement in part (i) about independence comes from (B.24). The first statement of part (ii) of the theorem comes from (B.25) and the second statement from (B.26).

B.3 Wandering exponent
We quote from [33] bounds on the fluctuations of the inverse-gamma polymer path. The results below are proved in [33] with couplings and calculations with the ratiostationary polymer process, without recourse to the integrable probability features of the inverse-gamma polymer.
Let the bulk weights pY x q xPZ 2 be i.i.d. Ga´1p1q distributed. Recall the definition of the averaged path distribution P 0,v from (2.3). On large scales the P 0,v -distributed random path X‚ P X 0,v follows the straight line segment r0, vs between its endpoints. Typical deviations from the line segment obey the Kardar-Parisi-Zhang (KPZ) exponent 2{3. The result below gives a quantified upper bound. It is used in the proof of Lemma 4.5.

C Bound on the running maximum of a random walk
In this appendix we quote a random walk estimate from [11], used in the proof of with two independent gamma variables G α " Gapαq and G β " Gapβq on the right. Denote the mean step by µ α,β " EpX α,β 1 q " ψ 0 pαq´ψ 0 pβq. Fix a compact interval rρ min , ρ max s Ă p0, 8q. Fix a positive constant a 0 and let ts N u N ě1 be a sequence of nonnegative reals such that 0 ď s N ď a 0 plog N q´3. Define a set of admissible pairs S N " tpα, βq : α, β P rρ min , ρ max s,´s N ď α´β ď 0u.
The point of the theorem below is that for pα, βq P S N the walk tS α,β m u 1ďmďN has a small enough negative drift that we can establish a positive lower bound for its running maximum.
Theorem C.1. [11,Corollary 2.8] In the setting described above the bound below holds for all N ě N 0 , pα, βq P S N , and x ě plog N q 2 : The constants C and N 0 depend on a 0 , ρ min , and ρ max .