Consider the Ising model at low temperatures and positive external field λ on an N×N box with Dobrushin boundary conditions that are plus on the north, east and west boundaries and minus on the south boundary. If λ=0, the interface separating the plus and minus phases is diffusive, having O( N) height fluctuations, and the model is fully wetted. Under an order one field, the interface fluctuations are O(1), and the interface is only partially wetted, being pinned to its southern boundary. We study the critical prewetting regime of λN↓0, where the height fluctuations are expected to scale as λ−1/3 and the rescaled interface is predicted to converge to the Ferrari–Spohn diffusion. Velenik (Probab. Theory Related Fields 129 (2004) 83–112) identified the order of the area under the interface up to logarithmic corrections. Since then, more refined features of such interfaces have only been identified in simpler models of random walks under area tilts. In this paper we resolve several conjectures of Velenik regarding the refined features of the Ising interface in the critical prewetting regime. Our main result is a sharp bound on the one-point height fluctuation, proving e−Θ(x3/2) upper tails reminiscent of the Tracy–Widom distribution, capturing a tradeoff between the locally Brownian oscillations and the global field effect. We further prove a concentration estimate for the number of points above which the interface attains a large height. These are used to deduce various geometric properties of the interface, including the order and tails of the area it confines and the polylogarithmic prefactor governing its maximum height fluctuation. Our arguments combine classical inputs from the random-line representation of the Ising interface with novel local resampling and coupling schemes.
考虑低温下的Ising模型和N×N盒子上的正外场λ，Dobrushin边界条件在北、东、西边界上为正，在南边界上为负。如果λ=0，分离正相和负相的界面是扩散的，具有O（N）高度波动，并且模型是完全润湿的。在一阶场下，界面波动为O（1），界面仅部分润湿，被钉扎在其南部边界。我们研究了λN的临界预润湿状态↓0，其中高度波动预计为λ−1/3，并且重新缩放的界面预计收敛于Ferrari–Spohn扩散。Velenik（Probab.Theory Related Fields 129（2004）83-112）确定了界面下区域的顺序，直至对数校正。从那时起，这种界面的更精细的特征只在区域倾斜下随机行走的更简单模型中被识别出来。在本文中，我们解决了Velenik关于临界预润湿状态下Ising界面精细特征的几个猜想。我们的主要结果是一点高度波动的尖锐边界，证明了e-θ（x3/2）上尾让人想起Tracy–Widom分布，捕捉到了局部布朗振荡和全局场效应之间的折衷。我们进一步证明了界面达到较大高度的点的数量的浓度估计。这些用于推导界面的各种几何特性，包括界面限制区域的阶数和尾数，以及控制其最大高度波动的多对数预因子。我们的论点将来自Ising界面随机线表示的经典输入与新的局部重采样和耦合方案相结合。
{"title":"Local and global geometry of the 2D Ising interface in critical prewetting","authors":"S. Ganguly, Reza Gheissari","doi":"10.1214/21-AOP1505","DOIUrl":"https://doi.org/10.1214/21-AOP1505","url":null,"abstract":"Consider the Ising model at low temperatures and positive external field λ on an N×N box with Dobrushin boundary conditions that are plus on the north, east and west boundaries and minus on the south boundary. If λ=0, the interface separating the plus and minus phases is diffusive, having O( N) height fluctuations, and the model is fully wetted. Under an order one field, the interface fluctuations are O(1), and the interface is only partially wetted, being pinned to its southern boundary. We study the critical prewetting regime of λN↓0, where the height fluctuations are expected to scale as λ−1/3 and the rescaled interface is predicted to converge to the Ferrari–Spohn diffusion. Velenik (Probab. Theory Related Fields 129 (2004) 83–112) identified the order of the area under the interface up to logarithmic corrections. Since then, more refined features of such interfaces have only been identified in simpler models of random walks under area tilts.\u0000In this paper we resolve several conjectures of Velenik regarding the refined features of the Ising interface in the critical prewetting regime. Our main result is a sharp bound on the one-point height fluctuation, proving e−Θ(x3/2) upper tails reminiscent of the Tracy–Widom distribution, capturing a tradeoff between the locally Brownian oscillations and the global field effect. We further prove a concentration estimate for the number of points above which the interface attains a large height. These are used to deduce various geometric properties of the interface, including the order and tails of the area it confines and the polylogarithmic prefactor governing its maximum height fluctuation. Our arguments combine classical inputs from the random-line representation of the Ising interface with novel local resampling and coupling schemes.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45353176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Levy matrices) whose associated eigenvalues are sufficiently small. We show that the limiting law of any such entry is non-Gaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue. Although the latter random variable is typically nonexplicit, for the median eigenvector it is given by the inverse of a one-sided stable law. Moreover, we show that different entries of the same eigenvector are asymptotically independent, but that there are nontrivial correlations between eigenvectors with nearby eigenvalues. Our findings contrast sharply with the known eigenvector behavior for Wigner matrices and sparse random graphs.
{"title":"Eigenvector statistics of Lévy matrices","authors":"A. Aggarwal, P. Lopatto, Jake Marcinek","doi":"10.1214/20-AOP1493","DOIUrl":"https://doi.org/10.1214/20-AOP1493","url":null,"abstract":"We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Levy matrices) whose associated eigenvalues are sufficiently small. We show that the limiting law of any such entry is non-Gaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue. Although the latter random variable is typically nonexplicit, for the median eigenvector it is given by the inverse of a one-sided stable law. Moreover, we show that different entries of the same eigenvector are asymptotically independent, but that there are nontrivial correlations between eigenvectors with nearby eigenvalues. Our findings contrast sharply with the known eigenvector behavior for Wigner matrices and sparse random graphs.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41530075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We complete the analysis of the extremal eigenvalues of the adjacency matrix A of the Erdős-Rényi graph G(N, d/N) in the critical regime d log N of the transition uncovered in [2, 3], where the regimes d log N and d log N were studied. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial (excluding the trivial top eigenvalue) eigenvalues of A/ √ d outside of the asymptotic bulk [−2, 2]. This correspondence implies that the transition characterized by the appearance of the eigenvalues outside of the asymptotic bulk takes place at the critical value d = d∗ = 1 log 4−1 log N . For d < d∗ we obtain rigidity bounds on the locations of all eigenvalues outside the interval [−2, 2], and for d > d∗ we show that no such eigenvalues exist. All of our estimates are quantitative with polynomial error probabilities. Our proof is based on a tridiagonal representation of the adjacency matrix and on a detailed analysis of the geometry of the neighbourhood of the large degree vertices. An important ingredient in our estimates is a matrix inequality obtained via the associated nonbacktracking matrix and an Ihara-Bass formula [3]. Our argument also applies to sparse Wigner matrices, defined as the Hadamard product of A and a Wigner matrix, in which case the role of the degrees is replaced by the squares of the `2-norms of the rows.
{"title":"Extremal eigenvalues of critical Erdős–Rényi graphs","authors":"Johannes Alt, Raphael Ducatez, A. Knowles","doi":"10.1214/20-AOP1483","DOIUrl":"https://doi.org/10.1214/20-AOP1483","url":null,"abstract":"We complete the analysis of the extremal eigenvalues of the adjacency matrix A of the Erdős-Rényi graph G(N, d/N) in the critical regime d log N of the transition uncovered in [2, 3], where the regimes d log N and d log N were studied. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial (excluding the trivial top eigenvalue) eigenvalues of A/ √ d outside of the asymptotic bulk [−2, 2]. This correspondence implies that the transition characterized by the appearance of the eigenvalues outside of the asymptotic bulk takes place at the critical value d = d∗ = 1 log 4−1 log N . For d < d∗ we obtain rigidity bounds on the locations of all eigenvalues outside the interval [−2, 2], and for d > d∗ we show that no such eigenvalues exist. All of our estimates are quantitative with polynomial error probabilities. Our proof is based on a tridiagonal representation of the adjacency matrix and on a detailed analysis of the geometry of the neighbourhood of the large degree vertices. An important ingredient in our estimates is a matrix inequality obtained via the associated nonbacktracking matrix and an Ihara-Bass formula [3]. Our argument also applies to sparse Wigner matrices, defined as the Hadamard product of A and a Wigner matrix, in which case the role of the degrees is replaced by the squares of the `2-norms of the rows.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45011873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
n β+2 β+1 , will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the k-th smallest gap, which is proportional to xk(β+1)−1e−x β+1 . In particular, the results apply to the classical COE, CUE and CSE in random matrix theory. The essential part of the proof is to derive several identities and inequalities regarding the Selberg integral, which should have their own interest.
{"title":"Small gaps of circular β-ensemble","authors":"Renjie Feng, Dongyi Wei","doi":"10.1214/20-AOP1468","DOIUrl":"https://doi.org/10.1214/20-AOP1468","url":null,"abstract":"n β+2 β+1 , will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the k-th smallest gap, which is proportional to xk(β+1)−1e−x β+1 . In particular, the results apply to the classical COE, CUE and CSE in random matrix theory. The essential part of the proof is to derive several identities and inequalities regarding the Selberg integral, which should have their own interest.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46945302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note corrects Lemma 3.7 in our paper [1]. The main results of the paper remain correct as stated. This note corrects an error in [1, Lemma 3.7]. The lemma is not correct as stated, and the first conclusion must instead be stated as a hypothesis. This erratum corrects the statement of the lemma and then shows that the additional hypothesis is satisfied in each of the three applications of the lemma later in the paper. The main results of the paper remain unchanged. The error in [1, Lemma 3.7] is at the end of the “first step” of the proof. Specifically, the last equation before the “second step” (lines 5-6 of page 3769) is not accurate, because the preceding equation was proven only for all F t -measurable functions φ1(μ), not for all F μ T -measurable functions. We rewrite the lemma as follows, stating equivalence between its two claims as well as a third and often more convenient form: Lemma 3.7*. Let P ∈ Pp(Ω) such that (B,W ) is a Wiener process with respect to the filtration (F t )t∈[0,T ] under P , and define ρ := P ◦ (ξ,B,W, μ)−1. Suppose that (1) and (3) of Definition 3.4 are satisfied, that P (X0 = ξ) = 1, and that the state equation (3.3) holds under P . The following are equivalent: (A) For P ◦ μ−1-almost every ν ∈ Pp(X ), it holds that (Wt)t∈[0,T ] is an (F W,Λ,X t )t∈[0,T ]-Wiener process under ν. (B) Under P , F T ∨ F ξ,W,Λ t is independent of σ{Ws −Wt : s ∈ [t, T ]} for every t ∈ [0, T ). (C) P is an MFG pre-solution Proof. (A⇒ C): Let Q = P ◦ (ξ,B,W, μ,Λ)−1. Assuming (A) holds, the second and third steps of the original proof [1, Lemma 3.7] are correct and show that Q ∈ A(ρ). As all of the other defining properties of an MFG pre-solution hold by assumption, we deduce (C). (C ⇒ B): Note that (C) entails that FΛ t is conditionally independent of F ξ,B,W,μ T given F ξ,B,W,μ t under P , for every t ∈ [0, T ). Fix t ∈ [0, T ), and fix bounded functions φt, ψT , ψt, and ht+ such that φt(Λ) is FΛ t -measurable, ψT (B,μ) is F B,μ T -measurable, ψt(ξ,W ) is F ξ,W t -measurable, and ht (W ) is σ{Ws −Wt : s ∈ [t, T ]}-measurable. The conditional independence yields E [ φt(Λ)| F T ] = E [ φt(Λ)| F t ] , a.s. The independence of ξ, (B,μ), and W easily implies that F T ∨F ξ,W t is independent of σ{Ws− Wt : s ∈ [t, T ]}, and we deduce E [φt(Λ)ψT (B,μ)ψt(ξ,W )ht+(W )] = E [ E [ φt(Λ)| F t ] ψT (B,μ)ψt(ξ,W )ht+(W ) ] = E [ E [ φt(Λ)| F t ] ψT (B,μ)ψt(ξ,W ) ] E [ht+(W )] = E [φt(Λ)ψT (B,μ)ψt(ξ,W )]E [ht+(W )] .
{"title":"Errata: Mean field games with common noise","authors":"R. Carmona, F. Delarue, D. Lacker","doi":"10.1214/20-aop1432","DOIUrl":"https://doi.org/10.1214/20-aop1432","url":null,"abstract":"This note corrects Lemma 3.7 in our paper [1]. The main results of the paper remain correct as stated. This note corrects an error in [1, Lemma 3.7]. The lemma is not correct as stated, and the first conclusion must instead be stated as a hypothesis. This erratum corrects the statement of the lemma and then shows that the additional hypothesis is satisfied in each of the three applications of the lemma later in the paper. The main results of the paper remain unchanged. The error in [1, Lemma 3.7] is at the end of the “first step” of the proof. Specifically, the last equation before the “second step” (lines 5-6 of page 3769) is not accurate, because the preceding equation was proven only for all F t -measurable functions φ1(μ), not for all F μ T -measurable functions. We rewrite the lemma as follows, stating equivalence between its two claims as well as a third and often more convenient form: Lemma 3.7*. Let P ∈ Pp(Ω) such that (B,W ) is a Wiener process with respect to the filtration (F t )t∈[0,T ] under P , and define ρ := P ◦ (ξ,B,W, μ)−1. Suppose that (1) and (3) of Definition 3.4 are satisfied, that P (X0 = ξ) = 1, and that the state equation (3.3) holds under P . The following are equivalent: (A) For P ◦ μ−1-almost every ν ∈ Pp(X ), it holds that (Wt)t∈[0,T ] is an (F W,Λ,X t )t∈[0,T ]-Wiener process under ν. (B) Under P , F T ∨ F ξ,W,Λ t is independent of σ{Ws −Wt : s ∈ [t, T ]} for every t ∈ [0, T ). (C) P is an MFG pre-solution Proof. (A⇒ C): Let Q = P ◦ (ξ,B,W, μ,Λ)−1. Assuming (A) holds, the second and third steps of the original proof [1, Lemma 3.7] are correct and show that Q ∈ A(ρ). As all of the other defining properties of an MFG pre-solution hold by assumption, we deduce (C). (C ⇒ B): Note that (C) entails that FΛ t is conditionally independent of F ξ,B,W,μ T given F ξ,B,W,μ t under P , for every t ∈ [0, T ). Fix t ∈ [0, T ), and fix bounded functions φt, ψT , ψt, and ht+ such that φt(Λ) is FΛ t -measurable, ψT (B,μ) is F B,μ T -measurable, ψt(ξ,W ) is F ξ,W t -measurable, and ht (W ) is σ{Ws −Wt : s ∈ [t, T ]}-measurable. The conditional independence yields E [ φt(Λ)| F T ] = E [ φt(Λ)| F t ] , a.s. The independence of ξ, (B,μ), and W easily implies that F T ∨F ξ,W t is independent of σ{Ws− Wt : s ∈ [t, T ]}, and we deduce E [φt(Λ)ψT (B,μ)ψt(ξ,W )ht+(W )] = E [ E [ φt(Λ)| F t ] ψT (B,μ)ψt(ξ,W )ht+(W ) ] = E [ E [ φt(Λ)| F t ] ψT (B,μ)ψt(ξ,W ) ] E [ht+(W )] = E [φt(Λ)ψT (B,μ)ψt(ξ,W )]E [ht+(W )] .","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48642816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove general nonlinear large deviation estimates similar to Chatterjee-Dembo’s original bounds except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation of the traces of powers of Wigner matrices with sub-Gaussian entries, and the upper tail of cycles counts in sparse Erdős–Rényi graphs down to the sparsity threshold n−1/2.
{"title":"Nonlinear large deviation bounds with applications to Wigner matrices and sparse Erdős–Rényi graphs","authors":"F. Augeri","doi":"10.1214/20-aop1427","DOIUrl":"https://doi.org/10.1214/20-aop1427","url":null,"abstract":"We prove general nonlinear large deviation estimates similar to Chatterjee-Dembo’s original bounds except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation of the traces of powers of Wigner matrices with sub-Gaussian entries, and the upper tail of cycles counts in sparse Erdős–Rényi graphs down to the sparsity threshold n−1/2.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45062055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Connectivity properties of the adjacency graph of $mathrm{SLE}_{kappa}$ bubbles for $kappain(4,8)$","authors":"Ewain Gwynne, Joshua Pfeffer","doi":"10.1214/19-aop1402","DOIUrl":"https://doi.org/10.1214/19-aop1402","url":null,"abstract":"","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49649430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Counterexample to Remark 1.7 in [1]. Let φ be a random variable in F (i.e. φ is a random measurable mapping on a compact metric spaceM) of law Q such thatM×Ω 3 (x, ω) 7→ φ(x, ω) ∈ M is measurable. Suppose that Q is regular and let J be a regular presentation of Q. Let X be a random variable in M independent of φ. Out of φ and X, define ψ ∈ F by ψ(x) = φ(x) is x 6= X and ψ(x) = X is x = X. Then M × Ω 3 (x, ω) 7→ ψ(x, ω) ∈ M is measurable. Suppose also that the law of X has no atoms, then (reminding the definition of F) ψ and X are independent and the law of ψ is Q. Note that ψ(X) = X and (except for very special cases) we won’t have that a.s. J (ψ)(X) = ψ(X) = X.
{"title":"Flows, coalescence and noise. A correction","authors":"Y. Jan, Olivier Raimond","doi":"10.1214/19-aop1394","DOIUrl":"https://doi.org/10.1214/19-aop1394","url":null,"abstract":"Counterexample to Remark 1.7 in [1]. Let φ be a random variable in F (i.e. φ is a random measurable mapping on a compact metric spaceM) of law Q such thatM×Ω 3 (x, ω) 7→ φ(x, ω) ∈ M is measurable. Suppose that Q is regular and let J be a regular presentation of Q. Let X be a random variable in M independent of φ. Out of φ and X, define ψ ∈ F by ψ(x) = φ(x) is x 6= X and ψ(x) = X is x = X. Then M × Ω 3 (x, ω) 7→ ψ(x, ω) ∈ M is measurable. Suppose also that the law of X has no atoms, then (reminding the definition of F) ψ and X are independent and the law of ψ is Q. Note that ψ(X) = X and (except for very special cases) we won’t have that a.s. J (ψ)(X) = ψ(X) = X.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41705994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the standard first passage percolation model on Z^d with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of R^d and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut A from infinity. We prove that the rescaled maximal flow between nA and infinity φ(nA)/n^ (d−1) almost surely converges towards a deterministic constant depending on A. This constant corresponds to the capacity of the boundary ∂A of A and is the integral of a deterministic function over ∂A. This result was shown in dimension 2 and conjectured for higher dimensions by Garet in [6].
{"title":"The maximal flow from a compact convex subset to infinity in first passage percolation on $mathbb{Z}^{d}$","authors":"Barbara Dembin","doi":"10.1214/19-aop1367","DOIUrl":"https://doi.org/10.1214/19-aop1367","url":null,"abstract":"We consider the standard first passage percolation model on Z^d with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of R^d and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut A from infinity. We prove that the rescaled maximal flow between nA and infinity φ(nA)/n^ (d−1) almost surely converges towards a deterministic constant depending on A. This constant corresponds to the capacity of the boundary ∂A of A and is the integral of a deterministic function over ∂A. This result was shown in dimension 2 and conjectured for higher dimensions by Garet in [6].","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47930900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients","authors":"M. Hutzenthaler, Arnulf Jentzen","doi":"10.1214/19-aop1345","DOIUrl":"https://doi.org/10.1214/19-aop1345","url":null,"abstract":"","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66077577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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