Stabilization of DLA in a wedge

We consider Diffusion Limited Aggregation (DLA) in a two-dimensional wedge. We prove that if the angle of the wedge is smaller than $\pi/4$, there is some $a>2$ such that almost surely, for all $R$ large enough, after time $R^a$ all new particles attached to the DLA will be at distance larger than $R$ from the origin. This means that DLA stabilizes in growing balls, thus allowing a definition of the infinite DLA in a wedge via a finite time process.

where −π/2 ≤ θ 1 < θ 2 ≤ π/2. Here we used the convention that (0, 0) belongs to all wedges and that arctan(y/0) equals π/2 for y > 0 and −π/2 for y < 0. In Appendix A we prove the existence of the harmonic measure in W θ 1 ,θ 2 which is needed for the definition of the DLA in the wedge.
For R > 0, let B R = {(x, y) ∈ Z 2 : x 2 + y 2 < R 2 } be the discrete Euclidean ball of radius R around the origin, and define W R θ 1 ,θ 2 = W θ 1 ,θ 2 ∩ B R . Throughout the paper, we consider W θ 1 ,θ 2 and W R θ 1 ,θ 2 for R > 0 as graphs, with vertices W θ 1 ,θ 2 and W R θ 1 ,θ 2 respectively and edges induced from the graph Z 2 . We denote by P x θ 1 ,θ 2 the law of a simple random walk (S n ) n≥0 in the graph W θ 1 ,θ 2 , starting from x and for B ⊂ W θ 1 ,θ 2 , denote by τ + B = inf{n ≥ 1 : S n ∈ B} the first return time of the random walk into the set B. Finally, we set P = P θ 1 ,θ 2 to be the law of the DLA (A n ) n≥0 in W θ 1 ,θ 2 (see Section 3 for the formal definition). For future use, for n ≥ 1 we denote by a n the particle added to the aggregate at time n, namely the unique vertex in W θ 1 ,θ 2 such that A n = A n−1 ∪ {a n }.
Our main result is the stabilization of the DLA in sufficiently sharp wedges.
Theorem 1. Assume −π/2 ≤ θ 1 < θ 2 ≤ π/2 satisfy θ 2 − θ 1 < π/4 and fix a > 2π+4(θ 2 −θ 1 ) π−4(θ 2 −θ 1 ) . Then P θ 1 ,θ 2 -almost surely, for every R > 0 sufficiently large, the random sets (A n ∩ B R ) n≥R a are all the same. In other words, for all R sufficiently large, none of the particles (a n ) n≥R a added to the system after time R a will attach to the aggregate inside W R θ 1 ,θ 2 . The main tool in proving Theorem 1 is a discrete Beurling estimate for random walk in a wedge, which enables us to control the harmonic measure of finite, connected subsets of W θ 1 ,θ 2 . Unlike in the work of H. Kesten [Kes87b], the proof of the discrete Beurling estimate here does not rely on Green function calculations.

Discussion and open problems
Since the introduction of DLA in 1983, rigorous understanding of the model was limited. The main exception being H. Kesten's upper bound on the growth rate [Kes87b], see also [BY17]. Lately, similar results were obtained for DLA in the upper half plane with Dirichlet boundary conditions [PZ17b,PZ17a]. The main technical difference between this paper and previous works is that the former does not use any Green function calculations. The main reason is the lack of control over the discrete Green function in the wedge that would allow hitting probability calculations (See [GP17] for the best known control in the case of general Neumann boundary conditions).
There are many interesting open questions regarding DLA. First natural questions are about the growth rate, the fractal dimension and the relation between the two (see [HMP86]). For these important questions our paper does not add to the discussion. Another natural question is about the number of arms in DLA growing in a wedge (or in Z 2 ). The physics literature does not provide clear conjectures or even definitions for the number of arms in a wedge. In [KOO + 98], D. A. Kessler, Z. Olami, J. Oz, I. Procaccia, E. Somfai and L. M. Sander claim evidence for a critical angle ν between 120 and 140 degrees which guarantees coexistence of two arms in a wedge of angle ν.
One immediate contribution of our result is to provide a method to sampling the DLA in W R θ 1 ,θ 2 , for every finite R > 0, via a finite time random process. By Theorem 1 there is some a > 0 such that for any R large enough almost surely the sets (A n ∩ B R ) n≥R a are all the same. As a result, for all R > 0 sufficiently large, we can define the DLA in W R θ 1 ,θ 2 to be A R a ∩ B R , which is a finite time random process. Returning to discuss the number of arms, since the sets (A R a ∩ B R ) are monotonic increasing in R, we can define Let ‫ג‬ be an infinite graph. The number of ends of ‫ג‬ is defined to be the supremum on the number of infinite, connected components of ‫ג‬ \ K, where we run over all finite K ⊂ ‫.ג‬ Hence, one can define the number of arms of the DLA as the number of ends of the graph ‫ג‬ = A ∞ . Due to the fact that A ∞ can be written as the limit of the sets A R a ∩ B R , we can erase a finite set K in finite times and only look on the dynamics after such times.
Remark 2.1. Computer simulations seem to suggest that θ 0 is smaller than π/4. See Figure 1. From our results one can deduce that if in a wedge of angle smaller than π/4, there are two connected components and one is behind the other by a polynomial order, then asymptotically the smaller component will cease to grow. However, without a lower bound on the growth rate this is not enough to prove the conjecture. That being said we would like to suggest one more ambitious conjecture for the DLA in a wedge. Define the growth rate of (A n ) n≥0 , denote gr((A n ) n≥0 ) by gr((A n ) n≥0 ) = sup β ≥ 1/2 : lim sup where diam stands for the diameter of the set in the Euclidean distance.

Formal definition of the DLA process in a wedge
This section is devoted to the formal definition of the DLA process in a wedge. In particular, we state a result regarding the existence of the harmonic measure from infinity in it, whose proof we postpone to Appendix A.
For x = (x 1 , x 2 ) ∈ Z 2 we denote by x 2 = x 2 1 + x 2 2 it Euclidean distance from the origin. Fix −π/2 ≤ θ 1 < θ 2 ≤ π/2. Recall that for x ∈ W θ 1 ,θ 2 we denote by P x θ 1 ,θ 2 the law of a simple random walk (S n ) n≥0 in W θ 1 ,θ 2 staring from x, and for A ⊂ W θ 1 ,θ 2 and y ∈ Z 2 , define H A (x, y) = P x θ 1 ,θ 2 (S τ A = y). Theorem 3. For every A ⊂ W θ 1 ,θ 2 and y ∈ Z 2 , the following limit, called the harmonic measure of A from infinity exists Unlike the analouge problem in the whole plain Z 2 , c.f. [LL10, Proposition 6.6.1], the existence of the limiting harmonic measure in a wedge is highly nontrivial and delicate. The main issue here is that, as of right now, there seems to be no discrete Green function approximations on the wedge (or part of it) that is precise enough to match our needs in applying that same approach. A detailed proof for the existence of the limit can be found in Appendix A.
Combining Theorem 2 and Theorem 3 we obtain Corollary 3.1. Fix −π/2 ≤ θ 1 < θ 2 ≤ π/2. For every ε > 0, there exists M ∈ N and C ∈ (0, ∞) such that for all r, L ∈ N satisfying r ≥ M and L/r ≥ M and every connected subset A ⊂ W θ 1 ,θ 2 , such that W r θ 1 ,θ 2 ⊂ A and A ∩ ∂W L θ 1 ,θ 2 = ∅. (2) Using the existence of the limit H ∞ ∂A (y), we can now formally define the DLA in the wedge, denoted (A n ) n≥0 , to be a sequence of random subsets of W θ 1 ,θ 2 such that where, given A n , the vertex a n+1 ∈ W θ 1 ,θ 2 is sampled according to H ∞ ∂An (·). Note that (A n ) n≥0 is an increasing family of subsets in W θ 1 ,θ 2 , and we denote its limit by

Proof of Theorem 1
Throughout the remainder of the paper we fix −π/2 ≤ θ 1 < θ 2 ≤ π/2. In this section we assume Corollary 3.1 and turn to prove Theorem 1. Let M > 0 and R ∈ N such that R > M . Denote by (L n ) n≥0 a sequence such that L 0 ≥ R and L n+1 /L n ≥ M for every n ≥ 1, and by σ i the sequence of successive first exit times of (A n ) n≥0 from B L i , namely, and turn to bound each of the terms on the right hand side.
Fix i ≥ 1. Note that |W θ 1 ,θ 2 and note that the random set B j is a connected set containing W R θ 1 ,θ 2 such that B j ∩ ∂W L i θ 1 ,θ 2 = ∅. Therefore, by Corollary 3.1, for every σ i ≤ j < σ i+1 , where for the first inequality we used the fact that a particle starting from a sufficiently far point must hit ∂W R θ 1 ,θ 2 before hitting B R . Consequently, the random variable (A τ i+1 \ A τ i ) ∩ B R is stochastically dominated by a sum of πL 2 i+1 independent Bernoulli random variables with success probability Fix some b > 1 and choose the sequence L i := M i R b . By Lemma 4.1, for every ε > 0 and M, R > 0 sufficiently large Since ε > 0 can be chosen to be arbitrary small, if θ 2 − θ 1 < π/4, then the sum on the right hand side is finite and equals . Furthermore, for every fixed b > 1, the expectation goes to zero as R → ∞ as soon as As b → ∞, this assumption coincides with the previous one, namely θ 2 − θ 1 < π/4. In fact, for θ 1 , θ 2 such that θ 2 − θ 1 < π/4, the power of R on the right hand side can be made arbitrarily small by increasing b, and in particular, if the power is strictly smaller than −1. Assuming (6) it follows that from the Markov inequality that and therefore, using the Borel Cantelli lemma, that P θ 1 ,θ 2 -almost surely, for all R > 0 sufficiently large, there are no particles hitting B R after time σ 1 = σ 1 (M R b ), namely, after the aggregate reaches distance M R b . Taking a > 2b and noting that for sufficiently large R we have σ 1 ≤ |B L 1 | ≤ πL 2 1 = M 2 R 2b < R a the result follows.

Discrete Beurling estimate in a wedge
The goal of this section is to provee Theorem 2. We start by describing the proof strategy.
(i) Applying the strong Markov property, and the fact that a random walk starting from radius R must hit radius L before r, we conclude that it suffices to consider random walks starting from radius L. (ii) Using time reversibility of the random walk, we rewrite the hitting probability as the ratio between the escape probability from radius r to radius L while avoiding A, and the probability starting from radius L to hit A before returning to the starting point. (iii) We bound the probability to hit A before returning to the starting point is bounded from below using the theory of electrical networks. (iv) Finally, we bound the escape probability from radius r to radius L from above with the help of the invariance principle and geometric observations on the set A for which the escape probability is maximal.
5.1. Proof of Theorem 2 -Reversibility and key lemmas. Recall that for a set B ⊂ W θ,θ 2 we denote by τ + B = inf{n ≥ 1 : S n ∈ B} the first return time to B and let τ B = inf{n ≥ 0 : S n ∈ B} be the first hitting time of B.
We start our proof by rewriting the hitting probabilities from far away appearing in (1) as escaping probabilities. First, note that for all y ∈ W r θ 1 ,θ 2 and x ∈ ∂W R θ 1 ,θ 2 the hitting probability P x θ 1 ,θ 2 (S τ ∂A = y), can only increase if we replace A with A ∩ W L θ 1 ,θ 2 . Thus, without loss of generality, we can assume that A ⊂ W L θ 1 ,θ 2 . For a set B ⊂ W θ 1 ,θ 2 , define B = B ∪ ∂B to be its clousre and note that the assumption For any R L and x ∈ ∂W R θ 1 ,θ 2 , observe that a random walk starting at x must hit ∂W L θ 1 ,θ 2 before hitting ∂A. Thus, by the strong Markov property, for any For any u ∈ ∂W L θ 1 ,θ 2 \ A, we now rewrite the hitting probability from u to y as the escaping probabiity from y to u. This is done using the reversibility property of simple random walk on graphs. The method of replacing the hitting probability with the escaping probability was used in H. Kesten [Kes87a] work on the DLA (see also [PZ17b] for the case of a domain with a boundary).
First, note that for each u ∈ ∂W L θ 1 ,θ 2 \ A and y ∈ ∂W r θ 1 ,θ 2 , Here we used the fact that A is a connected set and that u / ∈ A. As a result, the whole path must stay outside of the set A until it first hit y.
Let Γ A,n u,y be the collection of all paths γ = (x 0 , x 1 , x 2 , · · · , x n ) of length n in W θ 1 ,θ 2 such that x 0 = u, x n = y and x 1 , · · · , x n−1 / ∈ A. For γ = (x 0 , . . . , x n ) ∈ Γ A,n (u, y) denote by γ = (x n , . . . , x 0 ) ∈ Γ n,A (y, u) the path in the reverse direction. Then, the reversibility of simple random walk implies Combining (8) and (9), By the strong Markov property for the stopping time where in the last step we used the fact that the number of visits to u before hitting A when starting from x is a geometric random variables with parameter P u θ 1 ,θ 2 (τ + A < τ + u ).
Consequently, in order to prove Theorem 2, it suffices to show that there is a constant C < ∞ such that for every ε > 0, y ∈ ∂W r θ 1 ,θ 2 and every u ∈ ∂W L θ 1 ,θ 2 \ A, for sufficiently large L. The last inequality is an immediate corollary of the following two lemmas.
Lemma 5.1. There exists a constant c ∈ (0, ∞) independent of r and L such that uniformly for all A ⊂ W L θ 1 ,θ 2 as in Theorem 2 and all u ∈ ∂W L θ 1 ,θ 2 \ A. Lemma 5.2. For every ε > 0 there exists a constant C ∈ (0, ∞) such that for every r, L sufficiently large, every y ∈ ∂W r θ 1 ,θ 2 , every A ⊂ W L θ 1 ,θ 2 as in Theorem 2 and every u ∈ ∂W L θ 1 ,θ 2 \ A The proof of Lemma 5.1 is presented in Subsection 5.2 and the proof of Lemma 5.2 can be found in Subsection 5.3. 5.2. Proof of Lemma 5.1. We will use the theory of electrical networks in order to estimate the effective resistance from u to 0 in the graph W θ 1 ,θ 2 . Since 0 ∈ W r θ 1 ,θ 2 ⊂ A it follows that τ + A ≤ τ 0 , and therefore there was a mistake here with the definition of effective resistance, but the proof is still correct. .
5.3. Proof of Lemma 5.2 -Escaping probability to distance L. Having completed the proof of Lemma 5.1, we turn to the proof of Lemma 5.2. The proof of the latter contains several subclaims: (i) Finding the worst possile choice for the set A, (ii) explicit calculation for the continuous counterpart of the upper bound obtained in step (i) and (iii) using the invariance principle to compare the discrete and the continuous probabilities. We now turn to implement this strategy.
5.3.1. Proof of Lemma 5.2 part (i) -The worst choice for A. We start by showing that the worst choice for A, namely the set which maximizes the probability P y θ 1 ,θ 2 (τ + ∂W L θ 1 ,θ 2 < τ + A ) among all sets A ⊂ W L θ 1 ,θ 2 as in Theorem 2 is given by one of the lines along the wedge boundary.
Next, we turn to compare the hitting probability in A to the hitting probability in Γ u,r,L and Γ l,r,L . We separate the proof into three cases according to the position of y with respect to the path γ: (1) The point y satisfies arctan(y 2 /y 1 ) > arctan(x 0 2 /x 0 1 ). (2) The point y satisfies arctan(y 2 /y 1 ) < arctan(x 0 2 /x 0 1 ). (3) The point y satisfies y = x 0 . We start with case (1), since W θ 1 ,θ 2 is a connected, planar graph and γ is a connected path in it from ∂W r θ 1 ,θ 2 to ∂W L θ 1 ,θ 2 , every path from y to Γ l,r,L must hit A . In particular, paths starting in y that hit Γ l,r,L before hitting ∂W L θ 1 ,θ 2 must hit A before hitting ∂W L θ 1 ,θ 2 . This implies that P y θ 1 ,θ 2 (τ + Γ l,r,L ≤ τ + ) and therefore the required inequality. Similarly, in case (2), every path from y to Γ u,r,L must hit A and therefore paths hitting Γ u,r,L before ∂W L θ 1 ,θ 2 must also hit A before ∂W L θ 1 ,θ 2 . Hence P y θ 1 ,θ 2 (τ + . Finally, we turn to deal with case (3). After one step of the random walk we have ≤ τ A ) = 0 and for z / ∈ A we can repeat the argument in (1) and (2) we conclude that as required.
Let K be a constant to be chosen later on, for i ≥ 0, define M i = rK i and let N = N (r, L, K) be the largest integer such that M N ≤ L. Then and by strong Markov property In order to estimate each of the probabilities on the right hand side we first turn to evaluate their continuous counterpart.

5.3.3.
Proof of Lemma 5.2 part (iii) -from continuous to discrete and completion of the proof. In this subsection, we use Lemma 5.4 in order to find an upper bound on the discrete probability Lemma 5.5. Let K > 0. For every ε > 0, there exists a constant L 0 ∈ (0, ∞) such that for all L ≥ L 0 , and all z ∈ ∂W L θ 1 ,θ 2 The main ingredient in the proof of lemma 5.5 is the invariance principle for simple random walk in a wedge. Suppose that x ε ∈ D ε and x ε → x 0 ∈ D as ε → 0. Let {W ε t , t ≥ 0} be simple random walk on D ε with W ε 0 = x ε . Then (W ε t ) t≥0 converge weakly to reflected Brownian motion on D starting from x 0 as ε → 0.
Remark 5.1. In [BQ06], it was assumed throughout the paper that D is a bounded, connected, open set with analytic boundary. The analyticity assumption is made for technical reason, needed in proving their main result. However, it is noted in the paper that the lemma above is derived from Theorem 6.3 of [SV71], which holds for smooth regions as well.
Proof of Lemma 5.5. Suppose the lemma does not hold. Then, there exists ε 0 > 0 and an increasing sequence (L n ) n≥1 going to infinity together with a sequence of points (z n ) n≥1 such that z n ∈ ∂W Ln θ 1 ,θ 2 for all n ≥ 1 such that Noting that |z n /L n | → 1 as n → ∞, it follows that there exists a subsequence k n such that lim n→∞ z kn /L kn = z 0 ∈ {x ∈ W θ 1 ,θ 2 : x 2 = 1}. At this point we have all the ingredients needed in order to complete the proof of Theorem 2. Recalling Lemma 15, we observe that there exists a universal constant C ∈ (0, ∞) such that Taking ε = ε K 0 = K −π/2ϕ 0 in Lemma 5.5, we conclude that exists R 0 ∈ (0, ∞) such that for all r ≥ R 0 and all z ∈ ∂W r θ 1 ,θ 2 Next, recall that N K 0 was defined to be the largest integer n such that rK n 0 ≤ L, which implies that N K 0 = log K 0 (L/r) . Consequently, for all r ≥ R 0 , and L sufficiently large so that L r Thus, the proof of Lemma 5.2 is complete.
Appendix A. Existence of infinite harmonic measure In this section, we prove Theorem 3. The convergence is proved by showing that for any y ∈ W θ 1 ,θ 2 and any sequence (x n ) n≥1 in W θ 1 ,θ 2 such that lim n→∞ x n 2 = ∞, the sequence (H A (x n , y)) n≥1 is Cauchy.
Let (x n ) n≥1 be a sequence as above. Since A is finite one can find r > 0 such that A ⊂ W r θ 1 ,θ 2 and thus H A (x, y) = 0 for all y / ∈ W R θ 1 ,θ 2 and x ∈ W θ 1 ,θ 2 . Hence, we can restrict attention to y ∈ W r θ 1 ,θ 2 . Since lim n→∞ x n 2 = ∞, we can assume without loss of generality that x n 2 > r for all n ≥ 1 Note that for any m, n ∈ N such that x n 2 < x m 2 , a random walk starting in x m must hit ∂W xn 2 θ 1 ,θ 2 before hitting A. Thus it is enough to prove that for every y ∈ W r θ 1 ,θ 2 As mentioned in Section 3, there is no discrete Green function approximation on the wedge which is accurate enough to allow us to follow the proof outline of H. Kesten in Z 2 . Instead, we will prove the result using the following strategy (1) Show that the number of steps needed for a random walk, starting from ∂W R θ 1 ,θ 2 , to reach A is asymptotically bigger than R 2 .
(2) Show that the mixing time for a random walk started from x ∈ ∂W R θ 1 ,θ 2 is much smaller than the hitting time of A and therefore, using a coupling argument, that two random walks starting from x and x in ∂W R θ 1 ,θ 2 respectively, will coincide with high probability before hitting A.
Note that carrying out the strategy above requires careful choices of parameters in the proof. This is the content of the following subsections.
A.1. Coupling, two key propositions and the proof of Theorem 3. For R > 0 and x 1 , x 2 ∈ ∂W R θ 1 ,θ 2 , denote by P x 1 ,x 2 R a coupling of two continuous time Markov processes (B R 1 (t), B R 2 (t)) t≥0 each with state space (W θ 1 ,θ 2 /R) defined as follows: (a) B R 1 (t) is a continuous time, simple random walk on W θ 1 ,θ 2 /R, starting at x 1 /R, with fixed jump rate 2R 2 . (b) B R 2 (t) is a continuous time, simple random walk on W θ 1 ,θ 2 /R, starting at x 2 /R, with fixed jump rate 2R 2 . (c) (B R 1 (t)) t≥0 and (B R 2 (t)) t≥0 are coupled according to the maximum coupling, see For i ∈ {1, 2}, we define (S R i (n)) n≥1 to be the embedded, discrete time, simple random walk in (B R i (t)) t≥0 , and for s ≥ 0, denote by N R i (s), the number of jumps made by the Markov process (B R i (t)) t≥0 up to time s. It follows from the definitions above that B R i (s) = S R i (N R i (s)) for i ∈ {1, 2}, R > 0 and s ≥ 0. Denoting by τ i A = inf{t ≥ 0 : S R i (t) ∈ A/R} the hitting time of (S R i (t)) t≥0 in A/R, it follows from the definition of H A (·, ·) that for every y ∈ W r θ 1 ,θ 2 Define the stopping time and note that from the definition of the coupling and the stopping time B 1 (s) = B 2 (s) for all s ≥ T . For T 0 > 0 and R > 0, define the event

Then, by the Markov property
, where in the last inequality we used large deviation estimate for the random variable Combining all of the above, we conclude that for every T 0 > 0, R > 0, x 1 , x 2 ∈ W R θ 1 ,θ 2 and y ∈ W r θ 1 ,θ 2 Consequently, the proof of Theorem 3 is an immediate consequence of the following two propositions: Proposition A.1. For every finite set A ∈ W θ 1 ,θ 2 and every T ∈ (0, ∞) Proof of Theorem 3. Let A ⊂ W θ 1 ,θ 2 be a finite set. The discussion above combined with Proposition A.1 and Proposition A.2 implies that (23) holds. As a result, the limit lim x 2 →∞ H A (x, y) exists for every y ∈ W θ 1 ,θ 2 and thus the existence of the Harmonic measure follows.
A.2. Lower bound on the hitting time -Proof of Proposition A.1. We start with some results for the continuous analogue of reflected Brownian motion in the continuous wedge W θ 1 ,θ 2 , whose law when starting in u ∈ W θ 1 ,θ 2 we denote by P u θ 1 ,θ 2 . For L > 0 denote by σ L the hitting time of the reflected Brownian motion in ∂W L θ 1 ,θ 2 . Lemma A.1. For every ε > 0, C > 1 and u ∈ W θ 1 ,θ 2 such that u 2 = 1 Proof. The proof follows from the fact that log |x| is the Green function in R 2 and therefore, if (B(t)) t≥0 is a standard two-dimensional Brownian motion, then (log |B(t)|) t≥0 is a martingale. Indeed, note that the reflected Brownian motion in a smooth region is conformally invariant up to a time change, c.f. Theorem 9.3 of [LSW03]. By the conformal mapping theorem we can map the reflected we can map the wedge into C \ [0, ∞), which transforms the Brownian motion in W θ 1 ,θ 2 to a Brownian motion in C 2 reflected on the line {(x, 0) : x ∈ R}. Note that the original event {σ C −1 < σ C ε } is mapped under this transformation to the event {σ C −2π/(θ 2 −θ 1 ) < σ C 2πε/(θ 2 −θ 1 ) }. Next, observe that by the reflection principle, the reflection on the line {(x, 0) : x ∈ R} does not change the probability of the event {σ C −2π/(θ 2 −θ 1 ) < σ C 2πε/(θ 2 −θ 1 ) }.
Due to the fact that (log(|B(t)|)) t≥0 is a martingale, where (B(t)) t≥0 is a standard two-dimensional Brownian motion. It follows from the optional stopping theorem for the stopping time σ : which proves the result.
Lemma A.2. For every ε > 0, there is a constant C ε ∈ (0, ∞) such that for all u ∈ W θ 1 ,θ 2 satisfying u 2 = 1 and all T > 0 Proof. It suffices to prove the result for all sufficiently large T . Recall that the time change of the process under the conformal map is given, due to Ito's formula, by where f is the conformal map from the the upper half plane to W θ 1 ,θ 2 , given by with ϕ := (θ 2 − θ 1 )/π, and B(s) is the reflected Brownian motion in the upper half plane. For T > 0, define Starting from the first term on the right hand side of (27), note that u 1/ϕ 2 = 1 for every u = (u 1 , u 2 ) such that u 2 = 1. Also, observe that a reflected Brownian motion in the upper half plane can be constructed by replacing the y−coordinate of a standard 2-dimensional Brownian motion by its absolute value. Since, B(t) 2 ≥ T implies that one of the coordinates of B(t) is bigger than T /2 it follows that (28) where B 1 (t) is a one-dimensional Brownian motion. By reflection principle for one dimensional Brownian motion 2T for all |a| ≤ 1 and all T > 0 sufficiently large. Thus it remains to control the second term on the right hand side of (27), and show that for every ε > 0 there exists C ε ∈ (0, ∞) such that Recalling that f (x) = x ϕ , we have |f ( B(s))| 2 = ϕ 2 | B(s)| 2(ϕ−1) . Define δ ε = ϕε/(3+ ε). For any n ∈ N, using similar argument as in (28), we have (30) for all sufficiently large n. Consequently Thus it suffices to show that ζ(T 1+ε/2 ) > T on the event This indeed holds since for all n ≥ [T (1+ε/2)/ϕ /2] − 1 n n−1 ϕ 2 |B(s)| 2(ϕ−1) ds > ϕ 2 n (1+δε)(ϕ−1) ≥ ϕ 2 n −1+ϕ−δε = ϕ 2 n −1+ϕ/(1+ε/3) and therefore (32) With Lemma A.2 at hand, by choosing C = T 1+ 1 2ε one obtains: Lemma A.3. For every ε > 0, every T ∈ (0, ∞) and any u ∈ W θ 1 ,θ 2 satisfying u 2 = 1, Next, using the invariance principle and an argument similar to the one in the proof of Lemma 5.5, we prove analogue results to the ones in Lemmas A.1 -A.3, for simple random walk in W θ 1 ,θ 2 .
For ε > 0, R ≥ 1 and C > 0, let be a simple random walks in W θ 1 ,θ 2 , starting from x R ∈ ∂W R θ 1 ,θ 2 , stopped at the first hitting time of ∂W R/2C θ 1 ,θ 2 or ∂W 2C ε R θ 1 ,θ 2 . Due to the invariance principle, for every sequence of points (x R ) R≥1 for which the limit x ∞ := lim R→∞ x R /R exists, the linear interpolation of (S R,C,ε n ) n≥0 converges weakly to reflected Brownian motion in the wedge, starting from x ∞ ∈ W θ 1 ,θ 2 (satisfying x ∞ 2 = 1) until the stopping time σ.
Lemma A.4. For every ε > 0, T ∈ (0, ∞) and C > 1, there exists C ε ∈ (0, ∞) such that the following holds Proof. As alluded above, we use a similar argument to the one in the proof of Lemma 5.5. Suppose (34) does not hold. Then there is a sequence (R n ) n≥1 going to infinity and a sequence (x n ) n≥ such that for every n ≥ 1, x n ∈ ∂W Rn θ 1 ,θ 2 and P xn θ 1 , Since x n /R n is a bounded sequence in R 2 , there exists a subsequence (n k ) k≥1 such that lim k→∞ x n k /R n k = x ∞ ∈ W θ 1 ,θ 2 , satisfying x ∞ 2 = 1. Thus by the invariance principle and Lemma Lemma A.1, which contradicts the assumption. Repeating the argument with Lemma A.2 and Lemma A.3 replacing Lemma A.1, yields (35) and (36) respectively.
Proof of Proposition A.1. Since A is a fixed finite set, for every C > 1 and R sufficiently large (depending only on C and A) we have A ⊂ W R/C θ 1 ,θ 2 . Hence, for every x ∈ ∂W R θ 1 ,θ 2 and R sufficiently large τ C ε R ∧ τ R/C ≤ τ R/C ≤ τ A , P x θ 1 ,θ 2 − a.s and therrefore, for every T > 0, C > 1, ε > 0 and x ∈ ∂W R θ 1 ,θ 2 Taking the maximum over x ∈ ∂W R θ 1 ,θ 2 , then the limit R → ∞, then the limit C → ∞ and finally the limit ε → 0, the result follows from Lemma A.4.
A.3. Proof of Proposition A.2. Since P x 1 ,x 2 R couples the two random walks via a maximum coupling for Markov chains, it follows that For M 0 ∈ (0, ∞), one can split the sum over y into two parts, those satisfying y 2 ≥ M 0 and the ones satisfying y 2 < M . We turn to estimate each of the terms separately. For the first sum, note that a similar argument the one in (25) yields Fixing some ε > 0 and defining M 0 = T 1/2+ε 0 , it follows from Lemma A.4 (see (35)) that the sum is bounded by provided R is sufficiently large. Next, we turn to estimate the second term, namely (37) y∈W θ 1 ,θ 2 /R y 2 <M 0 The strategy for bounding the last sum is to use known bounds on the mixing time and total variation distance for random walks on finite graphs, obtained by intersecting scaled version of Z 2 with some bounded and sufficiently regular domains in R 2 . Note however, that the continuous time, simple random walk in (37) is defined on the cone W θ 1 ,θ 2 /R, which is not bounded. Thus our first step is to show that the last sum can be well approximated by a corresponding sum for a continuous time, simple random walk in W M 0 R θ 1 ,θ 2 /R, with M 0 chosen to be T 1/2+ε 0 .