On the convergence of massive loop-erased random walks to massive SLE(2) curves

Following the strategy proposed by Makarov and Smirnov in arXiv:0909.5377 (see also arXiv:0712.1952 and arXiv:0903.1023 for theoretical physics arguments), we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up of arXiv:0909.5377 appeared since then, we believe that such a treatment might be of interest for the community. We do not require any regularity of the limiting planar domain $\Omega$ near its degenerate prime ends $a$ and $b$ except that $(\Omega^\delta,a^\delta,b^\delta)$ are assumed to be `close discrete approximations' to $(\Omega,a,b)$ near $a$ and $b$ in the sense of Karrila arXiv:1810.05608.


Introduction
The classical loop-erased random walk (LERW) in a discrete domain Ω δ ⊂ δZ 2 is a curve obtained from a simple random walk trajectory by erasing the loops in the chronological order. In the famous paper [20] the convergence (in law) of LERW to the SLE(2) curves (see [18,15,5] and references therein) was proved by Lawler, Schramm and Werner. Namely, let Ω δ be discrete approximations to a simply connected domain Ω such that 0 ∈ Ω. Then, LERW obtained from simple random walks on Ω δ started at 0 and stopped when hitting ∂Ω converge to the so-called radial SLE(2) process in Ω. This result was generalized by Zhan [33] for multiply connected domains Ω and also for the chordal setup when the random walks are started at (discrete approximations of) a boundary point a ∈ ∂Ω and are conditioned to exit Ω δ through another boundary point b ∈ ∂Ω. Later on, a different generalization appeared in [32]: instead of δZ 2 one can consider any sequence of graphs Γ δ such that the simple random walks on Γ δ converge to the Brownian motion. Since then, variants of the LERW model become standard examples of lattice systems for which one can justify the convergence of interfaces to SLE and the Conformal Field Theory (CFT) predictions for correlation functions, e.g. see [14].
In parallel with a great success of studying the (conjectural) conformally invariant limits of critical 2D lattice models achieved during the last two decades, a program to study their near-critical perturbations was advocated by Makarov and Smirnov in 2009 (see [23]), with massive LERW (mLERW) being one of the cases most amenable for the rigorous analysis. More precisely, given m > 0, the law of the mLERW in Ω δ is defined by applying the same loop erasing procedure as above to random walks with the killing rate (i.e. probability to die at each step) m 2 δ 2 . The following result is provided by [23, Theorem 2.1]: Theorem 1.1. Let (Ω δ ; a δ , b δ ) be discrete approximations to a bounded simply connected domain (Ω; a, b) with two marked boundary points (prime ends). For each m > 0 the scaling limit γ of mLERW on (Ω δ ; a δ , b δ ) exists and is given by chordal Schramm-Loewner Evolution (2.11) whose driving term ξ t satisfies the SDE Ωt (a t , z) P Ωt (a t , z) z=b , Ωt (a t , ·) and P Ωt (a t , ·) denote the massive and the classical Poisson kernels in Ω t := Ω \ γ[0, t], and the logarithmic derivative with respect to a t := γ(t) is taken in the Loewner chart g t : Ω t → H. Moreover, (1.1) has a unique weak solution whose law is absolutely continuous with respect to √ 2B t . In order words, these scaling limits are absolutely continuous with respect to the classical SLE (2).
To the best of our knowledge, no follow up of [23] appeared since then. The goal of this paper is to provide technical details required for the proof of Theorem 1.1 as we believe that this might be of interest for the community and as we intend to pursue a rigorous understanding of further steps in the Makarov-Smirnov program (in particular, those discussed in [23, Sections 2.3 and 2.5] as well as [23,Question 4.12] for κ = 3, the latter is known to be of great interest).
We now discuss the setup in which we prove Theorem 1.1.
• Ω δ are assumed to converge to Ω in the Carathéodory topology (see Section 2.2 for more details). We do not assume any regularity of Ω (or Ω δ ) near degenerate prime ends a, b except that a δ , b δ are supposed to be close discrete approximations of a, b in the sense of the recent paper of Karrila [12]. It is worth noting that in [33] it was assumed that the boundary of Ω is 'flat' near the target point b, a technical restriction which was removed in [29] in the general setup of [32]. Our approach to this technicality is based upon the tools provided by [6] (see Section 3.2 for details), similar uniform estimates were independently obtained by Karrila [13] basing upon the conformal crossing estimates developed for the random walk in [16]. • The mode of convergence of discrete random curves γ δ to continuous ones is provided by the framework of Kemppainen and Smirnov [16] (with a recent addition of Karrila [12] in what concerns the vicinities of the endpoints a and b), see Section 2.3 for details. Namely, the weak convergence of the law of mLERW to that defined by (1.1) holds with respect to each of the following topologies: uniform convergence of curves γ δ to γ up to a reparametrization, convergence of conformal images γ δ H := φ Ω δ (γ δ ) to γ H := φ Ω (γ) under the half-plane capacity parametrizaiton, convergence of the driving terms ξ δ t in the Loewner equations describing γ δ H to ξ t . One can also use the result of Lawler and Viklund [21] on the convergence of LERW in the so-called natural parametrization to prove the same for mLERW.
There are several known strategies to prove the convergence of discrete random curves to classical SLEs, most of them relying upon the convergence of discrete martingale observables M δ (Ω δ ;a δ ,b δ ) (z) to M (Ω;a,b) (z) as (Ω δ ; a δ , b δ ) → (Ω; a, b); see (2.5) for the definition of these observables in the LERW case. The approach used in the original papers [20,33] on the subject (see also [11] for similar considerations in the Ising model context) relies upon the Skorohod embedding theorem and an approximate version of the Lévy characterization of the Brownian motion. A different viewpoint was advocated by Smirnov in [28]: once the tightness framework of [16] is set up, one gets the martingale property of ξ t and its quadratic variation from coefficients of the asymptotic expansion of M (Ωt;at,b) (z) near the target point b, e.g. see [28,Section 4.4] or [7] for sample computations. (Note however that [33] and [11] rely upon asymptotics of M (Ωt;at,b) (z) near the source point a t , which are known to be more useful in the multiple SLE context.) In the massive setup, one does not have conformal invariance, which makes these asymptotics of M Ωt (a t , z) rather sensitive to the local geometry of Ω t near b or a t . Moreover, even if we assume that the boundary of Ω is flat near b, these asymptotics are written in terms of Bessel functions instead of powers of (z − b). In this paper we use a combination of the two strategies: we do rely upon the tightness framework of [16] but analyze the stochastic processes M (Ωt;at,b) (z) at fixed points z ∈ Ω t instead of discussing their asymptotics (cf. [10], where this scheme was implemented for the first time in the literature, or [11,Section 3.1]).
In conformally invariant setups, it is known (e.g., see [10]) that one can easily derive the fact that the process ξ t is a continuous semi-martingale directly from the fact that M (Ωt;at,b) (z) are continuous (local) martingales, using explicit representations of ξ t via M (Ωt;at,b) (z). We illustrate this idea in Section 2.4 when discussing the convergence of the classical LERW to SLE (2). Despite the lack of explicit formulas, similar arguments might be used in the massive setup though being more involved. Nevertheless, we prefer to follow a more conceptual approach suggested in [3,2] and [23], which relies upon the Girsanov theorem and the fact that the mLERW can (and, arguably, should) be viewed as the classical LERW weighted by an appropriate density caused by the killing rate; in this approach the fact that ξ t is a semi-martingale does not require any proof (see Section 2.6).
Certainly, the idea of weighting SLE curves by martingales dates back to the very first developments in the subject, e.g. see [9,26] or [31,17] for recent examples. Note however that there exists an important difference between the 'critical/critical' and 'massive/critical' contexts. In the setup of Theorem 1.1, the density of mSLE (2) with respect to the classical SLE(2) does not coincide with the ratio of partition functions P (m) The reason is that the total mass of massive Brownian loops attached to the tip a t is strictly smaller than the mass of the critical ones, which results in a (positive) drift of this ratio; see also [3,Section 4] for a discussion of this effect from the theoretical physics perspective. Nevertheless, the expression for the drift λ t in (1.1) has exactly the same structure as in 'critical/critical' setups, see Remarks 2.9 and 4.10 for additional comments.
The rest of the article is organized as follows. In Section 2 we collect preliminaries and discuss the absolute continuity of mLERW with respect to LERW and that of their scaling limits. In Section 3 we prove the convergence of discrete martingale observables as δ → 0. Section 4 is devoted to a priori estimates and computations in continuum. The proof of Theorem 1.1 is summarized at the end of the paper. Let Ω ⊂ C be a bounded simply connected domain with two marked degenerate prime ends a, b (equivalence classes of sequence of inner point converging to a point on the boundary, see [24,Chapter 2]). We approximate (Ω; a, b) by (open) simply connected polygonal domains ( Ω δ ; a δ , b δ ) whose boundary consists of edges of square grid δZ 2 and the boundary points a δ , b δ lie on δZ 2 ); the precise mode of convergence of these approximations as δ → 0 is discussed in the next section. Each of these discrete domains can be also viewed as a subgraph Ω δ of δZ 2 by declaring its vertices and edges to be that of δZ 2 contained in Ω δ . We set IntΩ δ := V (Ω δ ) and define The reason for this definition of the boundary of Ω δ is that the same vertex v may be connected to several points v int ∈ IntΩ δ . When talking about exiting events of random walks, all such edges (v int , v) correspond to different possibilities to exit Ω δ .) Usually, we slightly abuse the notation and treat ∂Ω δ as a set of v ∈ δZ 2 without indicating the outgoing edges (v int , v) if no confusion arises. We also use the notation Ω δ := Ω δ ∪ ∂Ω δ . Given 0 < δ < m −1 ≤ +∞, a discrete domain Ω δ ⊂ δZ 2 , and two interior or boundary vertices w δ , z δ , we define the partition function of massive random walks running from w δ to z δ in Ω δ as where S Ω δ (w δ ; z δ ) denotes the set of all lattice paths connecting w δ and z δ inside Ω δ , and #π δ is the number of interior edges of Ω δ in π δ . (In other words, we do not count the edges (w δ , w δ int ) and (z δ int , z δ ) in #π δ if w δ ∈ ∂Ω δ and/or z δ ∈ ∂Ω δ .) To simplify the notation, we drop the superscript (m) when speaking about random walks without killing (i.e., m = 0). Splitting a trajectory π δ ∈ S Ω δ (w δ ; z δ ) into two parts (from w δ to v δ and from v δ to z δ ) and summing over all #π δ +1 possible choices of v δ , one easily sees that Let γ δ be a sample of the (massive or massless) LERW path from a δ to b δ in Ω δ . We denote by Ω δ \ γ δ [0, n] the connected component of this graph containing b δ . Let a sequence of vertices o δ be fixed so that o δ → 0 as δ → 0. A classical argument (e.g., see [20,Remark 3.6]) implies that, for each v δ ∈ IntΩ δ , the function is a martingale with respect to the filtration F n := σ(γ δ [0, n]) generated by first n steps of γ δ , until v δ is hit by γ δ or disconnected from b δ . Note that the additional normalization factor Z Ω δ (o δ , b δ ) does not depend neither on γ δ nor on m and is introduced for further convenience. As in the notation for partition functions, we drop the superscript (m) in (2.5) when speaking about classical (m = 0) LERW.

Carathéodory convergence of Ω δ and reparametrization by capacity.
Recall that we assume that all domains under consideration are uniformly bounded (that is, are contained in some B(0, R) for a fixed R > 0) and that 0 is contained in all domains. Let φ Ω : Ω → H be a conformal uniformization of Ω onto the upper half-plane H such that φ Ω (a) = 0, φ Ω (b) = ∞, and Im φ Ω (0) = 1, (2.6) note that these conditions define φ Ω uniquely and that one has We assume that discrete approximations ( Ω δ ; a δ , b) converge to (Ω; a, b) in the Carathéodory sense, which means that (e.g., see [24, Chapter 1]) • each inner point z ∈ Ω belongs to Ω δ for small enough δ; • each boundary point ζ ∈ ∂Ω can be approximated by ζ δ ∈ ∂Ω δ as δ → 0. Further, we require that a and b are degenerate prime ends of Ω and that a δ (resp., b δ ) is a close approximation of a (resp., of b) as defined by Karrila [12]: • a δ → a as δ → 0 and, moreover, the following is fulfilled: • Given r > 0 small enough, let S r be the arc of ∂B(a, r) ∩ Ω disconnecting (in Ω) the prime end a from 0 and all other arcs of this set, in other words this is the last arc from a (possibly countable) collection ∂B(a δ , r) ∩ Ω to cross for a path running from 0 to a inside Ω. We require that, for each r > 0 small enough, a δ is connected to the midpoint of S r inside Ω δ ∩ B(a, r).
We fix a uniformization φ Ω δ : Ω δ → C similarly to (2.6) so that φ Ω δ (a δ ) = 0, φ Ω δ (b δ ) = ∞, and Im φ Ω δ (0) = 1, note that the Carathéodory convergence of Ω δ to Ω can be reformulated as φ Ω on compact subsets of H and Ω, respectively. (2.8) From now onwards we assume (without loss of generality) that the discrete approximations Ω δ are shifted slightly so that the target point b δ = b is always the same. Inside all domains Ω δ (and similarly inside Ω), one can define the inner distance to the prime end b and the r-vicinities of b as follows: ρ Ω δ (b, z) := inf{r > 0 : z and b are connected in Ω δ ∩ B C (b, r)}, (2.9) Note that ρ Ω (b, z) is a continuous function of z ∈ Ω. Moreover, since a path connecting z to b inside Ω ∩ B C (b, r) eventually belongs to Ω δ as we assume that b δ is a close approximation of the prime end b. Let γ δ H := φ Ω δ (γ δ ) be the conformal images of LERW trajectories γ δ , considered as continuous paths in the upper half-plane H. These continuous simple curves can be canonically parameterized by the so-called half-plane capacity of their initial segments. Namely, a uniformization map g t : H \γ δ H [0, t] → H normalized at infinity is required to have the asymptotics g t (z) = z + 2tz −1 + O(|z| −2 ) as |z| → ∞.
Given t > 0 we define a random variable n δ t to be the first integer such that the half-plane capacity of φ Ω δ (γ δ [0, n]) is greater or equal than t. Further, given a small enough r > 0 we define n δ t,r to be the minimum of n δ t and the first integer such that γ δ (n) ∈ B Ω δ (b, r). Clearly, both n δ t and n δ t,r are stopping times with respect to the filtration F n := σ(γ δ [0, n]). We set , Ω δ t,r := Ω δ \ γ δ [0, n δ t,r ], a δ t,r := γ δ (n δ t,r ). The following lemma guarantees that the change of the parametrization from integers n δ t to the half-plane capacity t does not create big jumps. The proof given below is based upon compactness arguments though one can use standard estimates (e.g., see [5, Proposition 6.5]) of capacity increments in the upper halfplane H instead. However, it is worth noting that one does not have an immediate a priori bound of diam(γ δ H [0, n δ t,r ]) in the situation when the curve γ δ approaches b along the boundary of Ω δ , which might require to introduce additional stopping times to handle this scenario explicitly.
Proof. The set of all simply connected domains Ω δ \ γ δ [0, n] under consideration is precompact in the Carathéodory topology (with respect to points near b). Suppose on the contrary that the one-step increments of the half-plane capacities By compactness, one can find a subsequence along which Ω δ \ γ δ [0, n δ ] converge in the Carathéodory sense (with respect to points near b). Clearly, Ω δ \ γ δ [0, n δ −1] converge to the same limit and hence one can find conformal homeomorphisms that become arbitrary close to the identity on each compact subset K ⊂ B Ω (b, r), note that one necessarily has K ⊂ B Ω δ (b, r) for small enough δ due to (2.10). Due to (2.8), this implies that the conformal maps become (as δ → 0) arbitrary close to the identity on compact subsets of the fixed vicinity φ Ω (B Ω (b, r)) of ∞ in the upper half-plane. This contradicts to the assumption that the half-plane capacities of φ Ω δ (γ δ [0, n δ −1]) and φ Ω δ (γ δ [0, n δ ]) differ by a constant amount as δ → 0.

2.3.
Chordal SLE(2) and topologies of convergence. We now discuss a few basic facts on the construction of SLE curves, the interested reader is referred to [18,15,5] for more details. Let γ H be a continuous non-self-crossing curve in the upper half-plane H := {z ∈ C : Imz > 0}, growing from 0 to ∞.
. Assume that γ H is parameterized by half-plane capacity so that the conformal map g t : H \ K t → H (normalized at ∞) has the asymptotics g t (z) = z + 2tz −1 + O(|z| −2 ) as |z| → ∞.
Then there exists a unique real-valued function ξ t , called the driving term, such that the following equation, called the Loewner evolution equation, is satisfied: where we use the shorthand notation ∂ t for the partial derivative in t. Vice versa, given a nice function ξ t one can reconstruct the growing family K t and, further (under some assumptions on ξ t ), the curve γ H by solving (2.11) with g 0 (z) = z.
Classical SLE H (2) curves in the upper half-plane correspond to random driving is a standard Brownian motion. It is known that • almost surely, SLE H (2) is a simple curve in the upper half plane H, see [25]; • almost surely, the Hausdorff dimension of SLE H (2) is equal to 5 4 , see [4]. Moreover, one can use the corresponding Minkowski content of the initial segments of SLE H (2) to introduce the so-called natural parametrization of these curves, see [19].
Generally, given a simply connected domain Ω with boundary points (prime ends) a, b ∈ Ω, chordal SLE Ω curves from a to b in Ω are defined as preimages of SLE H under a conformal uniformization φ Ω : Ω → H satisfying φ Ω (a) = 0 and φ Ω (b) = ∞. Note that this definition does no require to fix a normalization of φ Ω due to the scale invariance of the law of SLE H curves.
When speaking about the tightness of random curves in (Ω δ ; a δ , b δ ) we rely upon a powerful framework developed by Kemppainen and Smirnov in [16] as well as upon a recent work of Karrila [12] (in which the behaviour in vicinities of the endpoints a, b is discussed). Let ξ δ be a random driving term corresponding via (2.11) to the conformal images γ δ H := φ Ω δ (γ δ ) of LERWs in (Ω δ ; a δ , b). It is known since the work [1] of Aizenman and Burchard (see also [20]) that appropriate crossing estimates imply that (1) the family of random curves γ δ (except maybe in vicinities of endpoints) is tight in the topology induced by the metric min ψ1,ψ2 γ 1 • ψ 1 − γ 2 • ψ 2 ∞ , with minimum taken over all parameterizations ψ 1 , ψ 2 of two curves γ 1 , γ 2 .
The results of Kempainen and Smirnov (see [16, Theorem 1.5 and Corollary 1.7] as well as [16,Section 4.5] where the required crossing estimates are checked for the loop-erased random walks) give much more: Moreover, a weak convergence in one of the topologies (2)-(4) imply the convergence in two others. Furthermore, provided that ( Ω δ ; a δ , b) converge to (Ω; a, b) in the Carathéodory sense so that a δ and b δ = b are close approximations of degenerate prime ends a and b of Ω, the following holds: (5) if a sequence of random curves γ δ H converges weakly in the topologies (2)-(4) to a random curve γ H then γ δ also converges weakly to a random curve which, almost surely, is supported on the limiting domain Ω due to [16,Corollary 1.8], and has the same law as φ −1 Ω (γ H ) due to [12,Theorem 4.4].

Convergence of classical LERW to chordal SLE(2).
To keep the presentation self-contained, in this section we sketch (a variant of, cf. the strategy used in [10]) a proof of the classical result: convergence of the usual loop-erased random walks to SLE(2), in the setup of Theorem 1.1 discussed in the introduction. As discussed above, the family of LERW probability measures on (Ω δ ; a δ , b) is tight, provided that the curves γ δ are parameterized by the half-plane capacities of their conformal images φ Ω δ (γ δ ) in (H; 0, ∞). Since the space of continuous functions is metrizable and separable, by Skorokhod representation theorem we can suppose that for each weakly convergent subsequence of these measures we also have γ δ → γ almost surely.
Let τ r := inf{t > 0 : γ(t) ∈ B Ω (b, r)} and τ δ r be the similar stopping times (in the half-plane capacity parametrization) for the discrete curves γ δ . Given an (unknown) law of γ, it is easy to see that for almost all r > 0 one almost surely has τ δ r → τ r . Indeed, let ρ t := ρ Ω (b, γ t ). Since the curves γ δ converge to γ in the capacity parametrization, one has τ δ r → τ r unless the continuous process ρ t has a local minimum at level r, which can happen only for a countable set of r's. Therefore, λ({r > 0 : r is a local minimum of ρ t }) = 0 (almost) surely and hence P[ r is a local minimum of ρ t ] = 0 for almost all r > 0.
due to the Fubini theorem.
Let t > 0 and assume that r > 0 is chosen as discussed above so that, almost . The martingale property of the discrete observables (2.5) gives where f is a bounded continuous test function on the space of curves. We now pass to the limit (as δ → 0) in this identity using the following two facts: as δ → 0. We discuss such convergence results in Section 3 (see Proposition 3.14 for this concrete statement). • The martingale observables are uniformly (with respect to δ and all possible as δ → 0 due to Corollary 3.8 (which allows one to replace b by an inner point b εr lying close enough to b, cf. the proof of Proposition 3.5) and Corollary 3.3 (which provides the convergence of Green's functions).
Passing to the limit δ → 0 in (2.12) we are now able to conclude that, for each r > 0, the (continuous, uniformly bounded) process (2.14) We now claim that the process ξ t∧τr is a continuous local semi-martingale since it can be uniquely reconstructed as a deterministic function of the values of continuous martingales (2.13) evaluated at two distinct points v 1 , v 2 ∈ B Ω (b, 1 2 r) and differentiable processes g t (φ(v 1 )), g t (φ(v 2 )). Using the Loewner equation (2.11) and Itô's lemma, one gets the following formula: (here and below we use the sign d for the stochastic differential). As this process should be a martingale for each v ∈ B Ω (b, 1 2 r), the only possibility is that both processes ξ t∧τr and ξ, ξ t∧τr − 2d(t ∧ τ r ) are (local) martingales.
Since τ r → +∞ almost surely, one concludes that ξ t Remark 2.2. The martingale property (2.14) can be directly generalized to the massive setup. Namely, for each subsequential limit (in the same topologies as above) of massive LERW on (Ω δ ; a δ , b δ ) the following holds: where the massive Poisson kernels P (m) are given by (3.12). In order to prove (2.15) one mimics the arguments given above basing upon Ω\γ[0,t∧τr] (v) as δ → 0 provided by Proposition 3.16; • the uniform boundedness of massive observables (until time t ∧ τ r ), which follows from Corollary 2.7 and the uniform boundedness of massless ones. We identify the law of ξ t in the massive setup in Section 4.3 using (2.15) in the same spirit as discussed above in the classical situation; see (4.19),(4.20).
2.5. The density of mLERW with respect to the classical LERW. Given a discrete domain (Ω δ ; a δ , b δ ) and m < δ −1 , denote by the probabilities that a simple lattice path γ δ running from a δ to b δ inside Ω δ appears as a classical (m = 0) or a massive LERW trajectory, respectively.
Let Ω δ be a simply connected discrete domain, a δ , b δ be its boundary points,and v δ ∈ IntΩ δ . Then, the following estimate holds: with a universal (i.e., independent of Ω δ , a δ , b δ , and v δ ) constant.
Proof. E.g., see [6, Proposition 3.1] which claims that the left-hand side is uniformly comparable to the probability that the random walk trajectory started at a δ and conditioned to exit Ω δ at b δ intersects the ball B(v δ , 1 3 dist(v δ , ∂Ω δ )).

Proposition 2.4.
There exists a universal constant c 0 > 0 such that, for each discrete domain Ω δ ⊂ B(0, R), boundary points a δ , b δ ∈ ∂Ω δ and m ≤ 1 where the massive random walk partition function Z where the expectation is taken over simple random walks π δ ∈ S Ω δ (a δ , b δ ) started at a δ and conditioned to exit Ω δ at b δ , whereas Lemma 2.3 gives The desired uniform estimate (2.16) follows easily.
where the expectation is taken over the classical LERW measure P (Ω δ ;a δ ,b δ ) .
Proof. (i) By definition, where LE denotes the loop-erasure procedure applied to the simple random walk trajectory π δ . The estimate (2.16) gives the desired uniform upper bound.
(ii) By Jensen's inequality and since Z (m) where the first expectation is taken with respect to the LERW measure while the second is with respect to the simple random walk measure on the set S Ω δ (a δ , b δ ). The proof is completed in the same way as the proof of Proposition 2.4.
Below we also need the following extension of Lemma 2.3 and Proposition 2.4. Lemma 2.6. Let Ω δ be a discrete domain, z δ , w δ ∈ Ω δ and v δ ∈ IntΩ δ . Then,
, then both sides are comparable due to the Harnack principle, otherwise one has Z Ω δ (w δ , v δ ) ≤ const · Z Ω δ (z δ , v δ ) ≤ const · Z Ω δ (z δ , w δ )). In particular, this proves the desired estimate in the situation when z δ (or, similarly, w δ ) is within 1 3 To handle the case when both w δ and z δ are at least 1 3 d Ω δ (v δ ) apart from v δ , note that the left-hand side of (2.17) satisfies the maximum principle in both variables w δ and z δ ; is uniformly bounded due to Lemma 2.3 if both w δ , z δ ∈ ∂Ω δ ; and is also uniformly bounded if at least one of these two vertices is at distance 1 3 d Ω δ (v δ ) from v δ due to the argument given above.
Corollary 2.7. There exists a universal constant c 0 > 0 such that, for each discrete domain Ω δ ⊂ B(0, R), two vertices w δ , z δ ∈ Ω δ and m ≤ 1 2 δ −1 , one has Z (m) Proof. The proof mimics the proof of Proposition 2.4. Indeed, one has due to Lemma 2.6 and standard estimates of the discrete Green functions.
2.6. Absolute continuity of mSLE(2) with respect to SLE (2). As discussed in Section 2.3, the classical LERW probability measures P (Ω δ ;a δ ,b δ ) on curves in discrete approximations (Ω δ ; a δ , b) are tight. Moreover (see Section 2.4), the only possible weak limit of P (Ω δ ;a δ ,b δ ) , as δ → 0, is given by the SLE(2) measure on curves in (Ω; a, b), which we denote by P (Ω;a,b) . Due to Corollary 2.5(i), the densities of the massive LERW measures on curves in (Ω δ ; a δ , b δ ) with respect to the classical ones are uniformly bounded from above by exp(c 0 m 2 R 2 ). Therefore, the measures P (m) (Ω δ ;a δ ,b δ ) are also tight in the topologies discussed in Section 2.3. (Ω;a,b) and P (Ω;a,b) are mutually absolutely continuous. Proof. Denote C := exp(c 0 m 2 R 2 ). Both results can be easily deduced from Corollary 2.5 by passing to the limit δ → 0. As probability measures on metrizable spaces are always regular, each Borel set A can be approximated by a compact subset F ⊂ A. In its turn, F can be approximated by its open ε-neighborhood F ε that can be without loss of generality assumed to be a continuity set for both measures under consideration. The first claim easily follows since for such approximations of A, here and below we write Ω instead of (Ω; a, b) and Ω δ instead of (Ω δ ; a δ , b δ ) for shortness. Therefore, P (m) and approximate each A k by F ε k as explained above. Provided that ε > 0 is small enough (depending on the choice of F k ), the sets F ε k are still disjoint and hence The proof is completed by applying the uniform estimate E Ω δ [ log D (m) Ω δ ] ≥ − log C provided by Corollary 2.5(ii), passing to the limit δ → 0, and then passing to the limit in the choice of approximations F ε k of a given disjoint collection A k . We now discuss how the law of the driving term ξ t = √ 2B t of SLE(2) changes when the measure P (Ω;a,b) is replaced by P Let τ n → ∞ be stopping times that localize L (Ω;a,b) . Assume that, for an adapted process λ t , one has Due to (2.18), this implies that the martingale part of the process L t (which is a semi-martingale under P (m) (Ω;a,b) . Therefore, in order to find the law of ξ t it is enough to identify λ t in (2.20). It is worth noting that in the massive setup a standard identity, e.g., in the multiple SLE context. The reason is that the total mass of massive RW loops attached to the tip a δ t is strictly smaller then the mass of the critical ones. Because of that, the process N (m) t actually has a negative drift (which can be computed explicitly, see (4.21)) and one cannot easily deduce Theorem 1.1 relying only upon the analysis of this process; cf. Remark 4.10.

Convergence of martingale observables
3.1. Convergence of discrete harmonic functions. In this section we recall two useful results from [8]: convergence of the discrete Green functions Z Ω δ (u δ , v δ ) and of the discrete Poisson kernels Z Ω δ (a δ , u δ )/Z Ω δ (a δ , v δ ) as Ω δ → Ω, where u, v are inner points and a is a boundary point (more accurately, a prime end) of Ω.
Let Ω ⊂ C be a simply connected bounded domain and r > 0. We say that points u, v ∈ Ω are jointly r-inside Ω if they can be connected by a path L uv ⊂ Ω such that dist(L uv , ∂Ω) > r. In other words, u and v belong to the same connected component of the r-interior of Ω.
Recall that we denote by Ω δ the polygonal representation of a discrete domain Ω δ .
Proposition 3.2. Let 0 < r < R be fixed. There exists a function ε(δ) = ε(δ, r, R), defined for small enough δ ≤ δ 0 (r, R), such that ε(δ) → 0 as δ → 0 and that the following is fulfilled for all simply connected discrete domains Ω δ ⊂ B(0, R) and all pairs of points u δ , v δ lying jointly r-inside Ω δ and such that |u δ − v δ | ≥ r: Proof. This follows from (a more general in several aspects) uniform convergence result provided by [8,Corollary 3.11] and the convergence of the discrete full-plane Green function to − 1 2π log |u δ − v δ | for r ≤ |u δ − v δ | ≤ 2R and δ → 0, the latter being a standard fact of the discrete potential theory on the square grid.

Corollary 3.3.
Let Ω ⊂ B(0, R) be a simply connected planar domain and u, v ∈ Ω be two distinct points of Ω. Assume that discrete domains Ω δ ⊂ B(0, R) approximate Ω (in the Carathéodory topology with respect to u or v) as δ → 0. Then, Moreover, for each r > 0 this convergence is uniform provided that u and v are jointly r-inside Ω and |u − v| ≥ r.
Proof. Let L uv ⊂ Ω be a path connecting u and v inside Ω and r := 1 2 dist(L uv , ∂Ω). It follows from the Carathéodory convergence of Ω δ to Ω that u δ and v δ are jointly r-inside of Ω δ provided that δ is small enough. Since (the continuous) Green function is conformally invariant, G Ω δ (u δ , v δ ) → G Ω (u, v) as δ → 0 uniformly for such u and v and thus the claim trivially follows from (3.1).
Remark 3.4. In Section 3.3 we prove an analogue of (3.2) in the massive setup along the lines of [8] though do not discuss an analogue of (3.1). Note that in [8] the uniform estimate (3.1) is actually deduced from (3.2) by compactness arguments; cf. the proofs of Proposition 3.5 and Corollary 3.6 discussed below. Proposition 3.5. Let 0 < r < R be fixed. There exists a function ε(δ) = ε(δ, r, R), defined for small enough δ ≤ δ 0 (r, R), such that ε(δ) → 0 as δ → 0 and that the following is fulfilled for all simply connected discrete domains Ω δ ⊂ B(0, R), all boundary points a δ , and all inner points u δ , v δ ∈ Ω δ lying jointly r-inside Ω δ : where P Ω δ (a δ , ·) denotes the Poisson kernel in the polygonal representation Ω δ with mass at the point a δ ∈ ∂ Ω δ , note that its normalization is irrelevant for (3.3).
Proof. This result is provided (again, in a stronger form) by [8,Theorem 3.13]. For completeness of the exposition we sketch the key ingredients of this proof, which goes by contradiction. If the uniform estimate (3.3) was wrong, it would fail (for a fixed ε 0 > 0) along a sequence of configurations (Ω δ ; a δ , u δ , v δ ) with δ → 0. As the set of all simply connected domains Λ satisfying B(u, r) ⊂ Λ ⊂ B(0, R) is compact in the Carathéodory topology, we could pass to a subsequence and assume that ( Ω δ ; a δ , u δ , v δ ) → (Ω; a, u, v) as δ → 0 in the Carathéodory sense, with u and v being jointly r-inside Ω. The Poisson kernel P Λ (a, u)/P Λ (a, v) is conformally invariant and so is stable under this convergence. Thus, it is enough to prove that in order to obtain a contradiction, where u, v ∈ Ω and a is a prime end of Ω.
Let d > 0 be small enough and let a point a d be chosen so that the circle ∂B(a d , 1 2 d) separates the prime end a from u and v in Ω. Since (Ω δ ; a δ ) converges to (Ω; a), the circle ∂B(a d , d) then separates a δ from u δ and v δ in Ω δ , for all sufficiently small δ. Let L δ d ⊂ ∂B(a d , d) denote the arc separating u δ and v δ from a δ and all the other arcs forming the set ∂B(a d , d) ∩ Ω δ , in other words this is the first arc of ∂B(a d , d) ∩ Ω δ to cross for a path running from, say, u δ to a δ ; see [8,Fig. 4].
The key argument of the proof is the following uniform (for small enough δ) estimate: We refer the reader to [8, pp. 26-27] for the proof of this statement which is based on the fact that the discrete harmonic measure ω δ (v δ ; K δ 3d ; Ω δ d ) of each path K δ 3d started in Ω δ 3d and running to L δ d is uniformly bounded from below due to [8,Theorem 3.12] and [8,Lemma 3.14]; note that u δ is not assumed to be located in the r-interior of Ω δ in (3.5).
The proof can be now completed in a standard way. The (uniform in δ) weak-Beurling estimate (see Lemma 3.11) allows one to improve the uniform bound (3.5) near the boundary of Ω δ : Since uniformly bounded discrete harmonic functions are also equicontinuous (cf. Lemma 3.10), one can pass to a subsequence once again to get the (uniform on compact subsets) convergence Each subsequential limit h is a positive harmonic function in Ω normalized so that h(v) = 1 and satisfies, for each d > 0, the same estimate Thus, h has Dirichlet boundary conditions, except at the prime end a. These properties characterize the Poisson kernel h(u) = P Ω (a, u)/P Ω (a, v) uniquely.

Corollary 3.6.
Let Ω ⊂ B(0, R) be a simply connected planar domain, a ∈ ∂Ω be its prime end, and u, v ∈ Ω be two, not necessarily distinct, inner points. Assume that discrete domains Ω δ ⊂ B(0, R) with marked boundary points a δ ∈ ∂Ω δ approximate (Ω; a) in the Carathéodory topology with respect to u or v. Then, Moreover, for each r > 0 this convergence is uniform if u, v are jointly r-inside Ω.
Proof. For a fixed pair u, v of points of Ω, this result is given by (3.4) and is a key step of the proof of Proposition 3.5. The fact that the convergence is uniform provided that u and v are jointly r-inside Ω can be, for instance, deduced from (3.3) and the conformal invariance of the Poisson kernel. Indeed, the Carathéodory convergence of ( Ω δ ; a δ ) to (Ω; a) implies that P Ω δ (a, u)/P Ω δ (a, v) → P Ω (a, u)/P Ω (a, v) as δ → 0, uniformly for such u and v.

3.2.
Boundary behavior of discrete harmonic functions. Since we work in the chordal setup, in order to prove the convergence of the martingale observables (2.5) we need convergence results for (both classical and massive) Poisson kernels normalized at the boundary. To make the exposition self-contained and accessible to readers who are not familiar with the classical potential theory in 2D, we start this section with a remark on the boundary behavior of continuous harmonic functions defined in a vicinity B Ω (b, r) ⊂ Ω of the point b and satisfy the Dirichlet boundary conditions on ∂B Ω (b, r) ∩ ∂Ω. Given two such (positive) functions h 1 , h 2 : B Ω (b, r) → R + , we claim that their ratio h 1 /h 2 is always continuous at b and we slightly abuse the notation by writing Indeed, let φ : B Ω (b, r) → H be a conformal uniformization of B Ω (b, r) onto the upper half-plane H such that φ(b) = 0. Both functions h 1,2 • φ −1 are harmonic in H and thus must behave like c 1,2 Im z + O(|z| 2 ) as z → 0, which justifies the existence of the limit c 1 /c 2 in (3.7). Below we justify a similar effect in discrete, uniformly over all possible shapes of discrete domains Ω δ near b.
Lemma 3.7. There exists a universal constant k < 1 such that the following holds for each simply connected discrete domain Ω δ , a point b ∈ ∂Ω δ and r > 2δ: for each pair of positive discrete harmonic functions H 1 , H 2 : B Ω δ (b, r) → R + satisfying the Dirichlet boundary conditions on ∂B Ω δ (b, r) ∩ ∂Ω δ , one has Proof. For shortness, denote B r := B Ω δ (b, r), C r := ∂B Ω δ (b, r) \ ∂Ω, and let Given a discrete harmonic function H : B r → R satisfying the Dirichlet boundary conditions on ∂B r ∩ ∂Ω = ∂B r \ C r and a point u ∈ B r/2 , one can write where u ′ stands for the last visit of a random walk trajectory from u to x to B r/2 . Applying this identity four times (for both functions H 1 and H 2 as well as for both point u and v) and rearranging terms one sees that Therefore, in order to derive the desired estimate it is enough to prove that (uniformly in all the parameters involved) Without loss of generality, assume that the boundary points u ′ , v ′ , y, x of A r are listed in the counterclockwise order. Then, (3.8) is equivalent to the following uniform lower bound for the discrete cross-ratio of the quadrilateral (A r ; u ′ , v ′ , y, x): Due to [6,Proposition 4.5] and [6, Theorem 7.1], this estimate (with some universal constant k < 1) follows from the following uniform lower bound on the discrete extremal length (aka effective resistance) between the arcs [u ′ v ′ ] and [xy] in A r : which holds true since the discrete and the continuous extremal lengths are uniformly comparable to each other (e.g., see [6, Proposition 6.2]) and A r is a part of an annulus with the fixed aspect ratio.
Let Ω δ be a simply connected discrete domain, b ∈ ∂Ω δ , q ∈ N and r > 2 q δ. Let H 1 , H 2 : B Ω δ (b, r) → R + be positive discrete harmonic functions satisfying the Dirichlet boundary conditions on ∂B Ω δ (b, r) ∩ ∂Ω. Then, one has with the same universal constant k < 1 as in Lemma 3.7.
Proof. This estimate follows easily by iterating q times the result of Lemma 3.7, which gives

3.3.
Convergence of the massive Green function. In this section we prove an analogue of the uniform convergence (3.2) for massive Green functions Z (m) Ω δ (u δ , v δ ). To prove this result, Proposition 3.12, we need several preliminary facts. Lemma 3.9. Let (X n ) n∈N be a simple random walk with killing rate m 2 δ 2 on δZ 2 . For an annulus A = A(v 0 , r 1 , r 2 ), denote by E(A) the event that X n , started at v ∈ A ∩ δZ 2 , makes a non-trivial loop around v 0 before exiting A, that is, there exists 0 ≤ s < k < τ C\A such that X s = X k and X| [s,k] is not null-homotopic in A. There exists a universal constant such that one has The desired event can be easily constructed from a few events of a type that a random walk started at the center u of a rectangle [u− 1 4 r, u+ 1 4 r]×[u− 1 8 r, u+ 1 8 r] exists it through a prescribed side not dying along the way. As we require that the killing rate m 2 δ 2 is scaled accordingly to the mesh size and that r ≤ m −1 , standard estimates imply that the probability of each of these events is uniformly bounded from below by a universal constant, independent of δ and r.
Given m > 0, we say that a function H is massive discrete harmonic at a vertex v ∈ δZ 2 if Trivially, if H is positive, then it satisfies the maximum principle: H(v) cannot be bigger than all four values H(v 1 ) at v 1 ∼ v. Using Lemma 3.9 one can easily prove an a priori regularity of massive discrete harmonic functions on δZ 2 .
Lemma 3.10. There exists universal constants C, β > 0 such that the following holds: for each positive massive discrete harmonic function H defined in the disc B(v 0 , 2r) ∩ δZ 2 with r ≤ m −1 and for each v 1 , v 2 ∈ B(v 0 , r) ∩ δZ 2 one has Proof. Without loss of generality, assume that |v 2 − v 1 | ≤ 1 4 r. The maximum principle yields the existence of a path γ connecting v 2 to the boundary of B(v 0 , r) such that the values of H along γ are larger than H(v 2 ). Consider a family of concentric annuli Due to Lemma 3.9, for each k the probability that the random walk with killing rate m 2 δ 2 started from v 1 is killed or does not hit γ while crossing A k is uniformly bounded away from 1. At the same time, standard estimates imply that the probability that this random walk is killed before crossing all A k is uniformly bounded from above by const · m 2 r|v 2 − v 1 | ≤ const · |v 2 − v 1 |/r. Hence, the probability that this random walk hits γ before dying or exiting B(v 0 , 2r) is at least 1 − C(|v 2 − v 1 |/r) β . Therefore, with universal constants C, β > 0.
We also need the so-called weak-Beurling estimate which applies to both discrete massive harmonic and usual (m = 0) discrete harmonic functions. Lemma 3.11. Let Ω δ ⊂ δZ 2 be a simply connected discrete domain, c δ ∈ ∂Ω δ be a boundary point, and r ≤ m −1 . Let H be discrete massive harmonic function defined in the r-vicinity B Ω δ (c, r) of c in Ω δ and let H satisfy the Dirichlet boundary conditions on ∂B Ω δ (c, r) ∩ ∂Ω δ . There exist universal constants C, β > 0 such that one has ) and B Ω δ (c, r) are defined by (2.9).
Proof. The proof is similar to the proof of Lemma 3.10: the simple random walk with killing rate m 2 δ 2 started at v hits ∂Ω δ or dies before reaching ∂B Ω δ (c, r) \ ∂Ω δ with probability at least 1 − C · (ρ Ω δ (c, v)/r) β .
We are now ready to prove an analogue of Proposition 3.2 for massive Green functions. Given a simply connected domain Λ ⊂ C we denote by G  Let Ω ⊂ B(0, R) be a simply connected planar domain and u, v ∈ Ω be two distinct points of Ω. Assume that discrete domains Ω δ ⊂ B(0, R) approximate Ω (in the Carathéodory topology with respect to u or v). Then, (3.10) Moreover, for each r > 0 this convergence is uniform provided that u and v are jointly r-inside Ω and |u − v| ≥ r.
It remains to check that (−∆ + m 2 )h(u, ·) = δ u (·) in the sense of distributions. Let φ ∈ C ∞ 0 (Ω) be a smooth function such that suppφ ⊂ Ω and hence suppφ ⊂ Ω δ provided that δ is small enough. For v δ ∈ IntΩ δ , denote The function Z (m) Ω δ (u δ , ·) satisfies (3.9) everywhere in Ω δ except at the vertex u δ . Due to the discrete integration by parts, this implies the identity . The upper bound (3.11) implies that the sums over ρ-vicinities of u are uniformly (in δ) small as ρ → 0.
Hence, the convergence of Z (m) Ω δ (u δ , ·) to h(u, ·) away from u implies that Therefore, each subsequential limit h(u, ·) must coincide with G Ω (u, ·), which proves (3.10) for fixed u and v. The fact that the convergence is uniform follows from the equicontinuity of functions Z (m) Ω δ (u δ , v δ ) discussed above and the compactness of the set of pairs (u, v) under consideration.
Remark 3.13. It follows from the convergence (3.10) that, for u, v ∈ Ω ⊂ B(0, R), due to the similar uniform estimate in discrete provided by Corollary 2.7.

3.4.
Convergence of martingale observables. Recall that (Ω δ ; a δ , b) are discrete approximations on scale δ of (Ω; a, b) in the Carathéodory sense. It follows from the absolute continuity of massive LERW with respect to the massless one (see Section 2.6) that the family of mLERW probability measures in (Ω δ ; a δ , b) is tight, when parameterized by the half-plane capacities of their conformal images (under the mappings φ Ω δ ) in (H; 0, ∞). Using the Skorokhod representation theorem as in Section 2.4, we can always assume that, almost surely, where Ω δ t,r = Ω δ \ γ δ [0, n δ t,r ] and a δ t,r = γ δ (n δ t,r ). The goal of this section is to show that in this situation the martingale observables (2.5), evaluated in the 1 2 r-vicinity of b, also converge almost surely to their continuous analogues. In other words, Proposition 3.14 (for m = 0) and Proposition 3.16 (for m = 0) are deterministic statements, which we later apply for all possible limiting curves. For shortness, below we drop the second subscript r and simply say that t ≤ τ r instead.
We start by proving the convergence result for the classical (i.e., massless) LERW observable normalized at the boundary point b.
Proposition 3.14. In the setup described above, let t ≤ τ r and v ∈ B Ω (b, 1 2 r). Then where the Poisson kernel P Ωt (a t , ·) in the domain Ω t is normalized so that one has P Ωt (a t , z) ∼ P Ω (a, z) ∼ G Ω (0, z) as z → b, see (2.7) and Section 4.1.
Since we also know that the claim follows by first sending δ → 0 and then ε → 0.
We now move on to the convergence of the martingale observable in the massive setup. In order to formulate an analogue of Proposition 3.14 in this situation, we need to introduce the massive Poisson kernel Ωt (w, z)dA(w). (3.12) We refer the reader to Section 4.1 (more precisely, to Remark 4.3(i)), where the convergence of this integral is discussed; note that no regularity assumptions on Ω t are required for this fact.
Proposition 3.15. In the setup described above, let z ∈ Ω t (note that we do not need to assume that this point is close to b). Then, as δ → 0, one has in this part of (3.13) are uniformly (in δ) small as ρ → 0. Indeed, due to (3.5) we have a uniform (provided that δ is small enough) upper bound At the same time, since w δ is not ρ-jointly inside Ω δ t with z, there exists a ball of radius ρ which intersects the boundary of Ω δ t and separates these two points in Ω δ t . Therefore, the weak-Beurling estimate (see Lemma 3.11) and the uniform boundedness of the Green functions Z (m) Ω δ t (·, z δ ) outside of the 1 2 dist(z, ∂Ω)-vicinity of z allow us to conclude that Combining (3.14)-(3.17) together and sending first ρ → 0 and then ρ a → 0 we get exhaust Ω t . The proof is completed.
We now introduce the quantity which keeps track of the normalization of the massive observable at the point b.
The existence of this limit is discussed in Section 4.1, see (4.12). It is worth noting Ωt (a t , b) ≤ exp(c 0 m 2 R 2 ) due to the convergence (3.19) and Corollary 2.7. The next proposition is the main result of this section. Proposition 3.16. In the setup of Proposition 3.14 (i.e., t ≤ τ r and v ∈ B Ω (b, 1 2 r)), the following convergence holds true as δ → 0: where the quantities in the right-hand side are defined by (3.12) and (3.18).
Proof. We start by generalizing the result of Proposition 3.15 to z = b: We use the same argument as in the proof of Proposition 3.14. Given ε > 0 we pick a point b εr ∈ B Ω (b, εr) and note that, due to the identity (2.4) and Corollary 3.8 one has Note that the identity (4.4) is nothing but a continuous counterpart of the similar identity (2.4) for the partition functions of random walks discussed in Section 2.1.
Lemma 4.1. There exists an absolute constant C > 0 such that, for each simply connected domain Λ ⊂ C, its uniformization φ Λ : Λ → H, and z, w ∈ Λ, the following estimates are fulfilled: Proof. It is easy to see that both expressions are invariant under Möbius automorphisms of H preserving the point 0. Therefore, one can assume φ Λ (z) = i without loss of generality. In this situation, the required estimates (4.5) are nothing but the claim that both functions are bounded in the upper half-plane, which is clearly true since both of them are continuous in φ = φ Λ (w) ∈ H (including at the point i) and decay as |φ| → ∞.
Remark 4.2. For later purposes, it is useful to rewrite (4.5) as We now introduce massive counterparts of the functions (4.1) as follows: Λ (z) is well-defined since the only possible pathology in the integral is at w = z, where the integrand is bounded from above by a multiple of the Green function G Λ (w, z). Moreover, one easily sees that exp(−c 0 m 2 R 2 )P t (a t , z) ≤ P (m) t (a t , z) ≤ P t (a t , z) (4.10) due to Proposition 3.15 and similar uniform bounds provided by Corollary 2.7.
(ii) On the contrary, (4.7) guarantees that the function Q (m) Λ is well-defined only under the additional assumption Λ P Λ (w)dA(w) < +∞. Though this is not always true in general, it follows from Corollary 4.6(i) given below that this assumption holds for almost all (in t) domains Λ = Ω t generated by a Loewner evolution in Ω.  Proof. Note that the integral converges due to (4.6) and since P (m) Λ (w) ≤ P Λ (w). Moreover, one has Λ P (m) Λ (w)G Λ (w, z)dA(w) where the application of the Fubini theorem in the second equality is justified by the uniform estimate P Λ (w)G (m) Λ (w, w ′ )G Λ (w ′ , z) ≤ P Λ (z)(G Λ (w, w ′ ) + C)(G Λ (w ′ , z) + C) which follows from (4.6).
Assume now that b := φ −1 Λ (∞) is a degenerate prime end of Λ. The representation (4.11) together with the discussion given in Section 3.2 allows one to define the following quantity (note that here and below we abuse the notation in a way similar to Section 3.  Indeed, one can exchange the limit z → b and the integration over w ∈ Λ due to the uniform estimate (4.6), which provides a majorant P (m) Λ (w) G Λ (w, z) P Λ (z) ≤ P Λ (w)G Λ (w, z) P Λ (z) ≤ G Λ (w, z) + C, and the fact that max z∈BΛ(b,r) BΛ(b,2r) G Λ (w, z)dA(w) → 0 as r → 0, which follows from (4.3) and allows one to neglect the contributions of vicinities of the point b (where the Green function blows up and thus no uniform in z majorant is available).

4.2.
Hadamard's formula. We now move to the Loewner equation setup and assume that a decreasing family of subdomains Ω t ⊂ Ω is constructed according to (2.11) and that their uniformizations onto the upper half-plane are fixed as φ t := (g t − ξ t ) • φ Ω : Ω t → H so that, in particular, φ t (a t ) = 0 and φ t (b) = ∞. For shortness, from now onwards we replace the subscript Ω t by t, thus we write G t (w, z) instead of G Ωt (w, z), P t (z) instead of P Ωt (z) = P Ωt (a t , z), etc. The following lemma is classical.  (z, w) is differentiable in t (until the first moment when either z ∈ Ω t or w ∈ Ω t ) and ∂ t G t (w, z) = −2πP t (w)P t (z). (4.13) Proof. Let w H := φ Ω (w) and z H := φ Ω (z), note that one has G t (w, z) = − 1 2π log g t (w H ) − g t (z H ) g t (w H ) − g t (z H ) .
Since both g t (w H ) and g t (z H ) satisfy the Loewner equation (2.11), one easily obtains As pointed out in [23], it immediately follows from the Hadamard formula that the integrals Ωt P t (w)dA(w) converge for almost all t, see the next corollary. In our analysis we also need a stronger estimate which guarantees the convergence of integrals Ωt (P t (w)) 2 dA(w) for almost all t provided that γ t is an SLE(2) curve. 2π Ω Ω G 0 (w, z)dA(w)dA(z) < +∞.
We now derive a counterpart of Lemma 4.5 in the massive setup.