{"title":"Scaling limits of directed polymers in spatial-correlated environment","authors":"Yingxia Chen, F. Gao","doi":"10.1214/23-ejp955","DOIUrl":"https://doi.org/10.1214/23-ejp955","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46875114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider line ensembles of non-intersecting random walks constrained by a hard wall, each tilted by the area underneath it with geometrically growing pre-factors bi where b>1. This is a model for the level lines of the (2+1)D SOS model above a hard wall, which itself mimics the low-temperature 3D Ising interface. A similar model with b=1 and a fixed number of curves was studied by Ioffe, Velenik, and Wachtel (2018), who derived a scaling limit as the time interval [−N,N] tends to infinity. Line ensembles of Brownian bridges with geometric area tilts (b>1) were studied by Caputo, Ioffe, and Wachtel (2019), and later by Dembo, Lubetzky, and Zeitouni (2022+). Their results show that as the time interval and the number of curves n tend to infinity, the top k paths converge to a limiting measure μ. In this paper we address the open problem of proving existence of a scaling limit for random walk ensembles with geometric area tilts. We prove that with mild assumptions on the jump distribution, under suitable scaling the top k paths converge to the same measure μ as N→∞ followed by n→∞. We do so both in the case of bridges fixed at ±N and of walks fixed only at −N.
我们考虑由硬墙约束的非相交随机行走的线束，每个线束都被其下方的区域倾斜，其前因子bi呈几何增长，其中b>1。这是一个(2+1)D SOS模型在硬墙上的水平线模型，它本身模仿了低温3D Ising界面。Ioffe, Velenik和Wachtel(2018)研究了b=1且曲线数量固定的类似模型，他们推导出了时间间隔[- N,N]趋于无穷时的缩放极限。Caputo, Ioffe, and Wachtel(2019)和Dembo, Lubetzky, and Zeitouni(2022+)分别研究了几何面积倾斜(b>1)的布朗桥线系。结果表明，当时间间隔和曲线数n趋近于无穷大时，k条路径收敛于一个极限测度μ。在本文中，我们讨论了具有几何面积倾斜的随机漫步集合的尺度极限证明的开放性问题。我们证明了在对跳跃分布的温和假设下，在适当的尺度下，当N→∞和N→∞时，顶部k个路径收敛到相同的测度μ。对于固定在±N的桥梁和只固定在−N的步行，我们都这样做。
{"title":"Scaling limit for line ensembles of random walks with geometric area tilts","authors":"Christian Serio","doi":"10.1214/23-ejp1026","DOIUrl":"https://doi.org/10.1214/23-ejp1026","url":null,"abstract":"We consider line ensembles of non-intersecting random walks constrained by a hard wall, each tilted by the area underneath it with geometrically growing pre-factors bi where b>1. This is a model for the level lines of the (2+1)D SOS model above a hard wall, which itself mimics the low-temperature 3D Ising interface. A similar model with b=1 and a fixed number of curves was studied by Ioffe, Velenik, and Wachtel (2018), who derived a scaling limit as the time interval [−N,N] tends to infinity. Line ensembles of Brownian bridges with geometric area tilts (b>1) were studied by Caputo, Ioffe, and Wachtel (2019), and later by Dembo, Lubetzky, and Zeitouni (2022+). Their results show that as the time interval and the number of curves n tend to infinity, the top k paths converge to a limiting measure μ. In this paper we address the open problem of proving existence of a scaling limit for random walk ensembles with geometric area tilts. We prove that with mild assumptions on the jump distribution, under suitable scaling the top k paths converge to the same measure μ as N→∞ followed by n→∞. We do so both in the case of bridges fixed at ±N and of walks fixed only at −N.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135261330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exponential asymptotics for Brownian self-intersection local times under Dalang’s condition","authors":"Xia Chen","doi":"10.1214/23-ejp985","DOIUrl":"https://doi.org/10.1214/23-ejp985","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135262285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study existence and uniqueness of solutions to the equation dXt=b(Xt)dt+dBt, where b is a distribution in some Besov space and B is a fractional Brownian motion with Hurst parameter H⩽1∕2. First, the equation is understood as a nonlinear Young equation. This involves a nonlinear Young integral constructed in the space of functions with finite p-variation, which is well suited when b is a measure. Depending on H, a condition on the Besov regularity of b is given so that solutions to the equation exist. The construction is deterministic, and B can be replaced by a deterministic path w with a sufficiently smooth local time. Using this construction we prove the existence of weak solutions (in the probabilistic sense). We also prove that solutions coincide with limits of strong solutions obtained by regularisation of b. This is used to establish pathwise uniqueness and existence of a strong solution. In particular when b is a finite measure, weak solutions exist for H< 2−1, while pathwise uniqueness and strong existence hold when H⩽1∕4. The proofs involve fine properties of the local time of the fractional Brownian motion, as well as new regularising properties of this process which are established using the stochastic sewing Lemma.
{"title":"Regularisation by fractional noise for one-dimensional differential equations with distributional drift","authors":"Lukas Anzeletti, Alexandre Richard, Etienne Tanré","doi":"10.1214/23-ejp1010","DOIUrl":"https://doi.org/10.1214/23-ejp1010","url":null,"abstract":"We study existence and uniqueness of solutions to the equation dXt=b(Xt)dt+dBt, where b is a distribution in some Besov space and B is a fractional Brownian motion with Hurst parameter H⩽1∕2. First, the equation is understood as a nonlinear Young equation. This involves a nonlinear Young integral constructed in the space of functions with finite p-variation, which is well suited when b is a measure. Depending on H, a condition on the Besov regularity of b is given so that solutions to the equation exist. The construction is deterministic, and B can be replaced by a deterministic path w with a sufficiently smooth local time. Using this construction we prove the existence of weak solutions (in the probabilistic sense). We also prove that solutions coincide with limits of strong solutions obtained by regularisation of b. This is used to establish pathwise uniqueness and existence of a strong solution. In particular when b is a finite measure, weak solutions exist for H< 2−1, while pathwise uniqueness and strong existence hold when H⩽1∕4. The proofs involve fine properties of the local time of the fractional Brownian motion, as well as new regularising properties of this process which are established using the stochastic sewing Lemma.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135448454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate percolation in the Boolean model with convex grains in high dimension. For each dimension d, one fixes a compact, convex and symmetric set K⊂Rd with non empty interior. In a first setting, the Boolean model is a reunion of translates of K. In a second setting, the Boolean model is a reunion of translates of K or ρK for a further parameter ρ∈(1,2). We give the asymptotic behavior of the percolation probability and of the percolation threshold in the two settings.
{"title":"Percolation in the Boolean model with convex grains in high dimension","authors":"Jean-Baptiste Gouéré, Florestan Labéy","doi":"10.1214/23-ejp997","DOIUrl":"https://doi.org/10.1214/23-ejp997","url":null,"abstract":"We investigate percolation in the Boolean model with convex grains in high dimension. For each dimension d, one fixes a compact, convex and symmetric set K⊂Rd with non empty interior. In a first setting, the Boolean model is a reunion of translates of K. In a second setting, the Boolean model is a reunion of translates of K or ρK for a further parameter ρ∈(1,2). We give the asymptotic behavior of the percolation probability and of the percolation threshold in the two settings.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Multi-dimensional BSDEs with mean reflection","authors":"Baoyou Qu, Falei Wang","doi":"10.1214/23-ejp991","DOIUrl":"https://doi.org/10.1214/23-ejp991","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47703775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a sprinkled decoupling inequality for increasing events of Gaussian vectors with an error that depends only on the maximum pairwise correlation. As an application we prove the non-triviality of the percolation phase transition for Gaussian fields on $mathbb{Z}^d$ or $mathbb{R}^d$ with (i) uniformly bounded local suprema, and (ii) correlations which decay at least polylogarithmically in the distance with exponent $gamma>3$; this expands the scope of existing results on non-triviality of the phase transition, covering new examples such as non-stationary fields and monochromatic random waves.
{"title":"A sprinkled decoupling inequality for Gaussian processes and applications","authors":"S. Muirhead","doi":"10.1214/23-ejp994","DOIUrl":"https://doi.org/10.1214/23-ejp994","url":null,"abstract":"We establish a sprinkled decoupling inequality for increasing events of Gaussian vectors with an error that depends only on the maximum pairwise correlation. As an application we prove the non-triviality of the percolation phase transition for Gaussian fields on $mathbb{Z}^d$ or $mathbb{R}^d$ with (i) uniformly bounded local suprema, and (ii) correlations which decay at least polylogarithmically in the distance with exponent $gamma>3$; this expands the scope of existing results on non-triviality of the phase transition, covering new examples such as non-stationary fields and monochromatic random waves.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46429317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Corrigendum to: The sum of powers of subtree sizes for conditioned Galton–Watson trees","authors":"J. A. Fill, S. Janson","doi":"10.1214/23-ejp915","DOIUrl":"https://doi.org/10.1214/23-ejp915","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41592721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the stationary distributions for the stochastic vertex models. Our main focus is the stochastic six vertex (S6V) model. We show that the extremal stationary distributions of the S6V model are given by product Bernoulli measures. Moreover, for the S6V model under a moving frame of speed 1, we show that the extremal stationary distributions are given by product Bernoulli measures and blocking measures. Finally, we generalize our results to the stochastic higher spin six vertex model. Our proof relies on the coupling of the S6V models introduced in [4], the analysis of current and the method of fusion.
{"title":"Classification of stationary distributions for the stochastic vertex models","authors":"Yier Lin","doi":"10.1214/23-ejp1022","DOIUrl":"https://doi.org/10.1214/23-ejp1022","url":null,"abstract":"In this paper, we study the stationary distributions for the stochastic vertex models. Our main focus is the stochastic six vertex (S6V) model. We show that the extremal stationary distributions of the S6V model are given by product Bernoulli measures. Moreover, for the S6V model under a moving frame of speed 1, we show that the extremal stationary distributions are given by product Bernoulli measures and blocking measures. Finally, we generalize our results to the stochastic higher spin six vertex model. Our proof relies on the coupling of the S6V models introduced in [4], the analysis of current and the method of fusion.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135261333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Belinschi et al. [Adv. Math., 226 (2011)] proved that the normal distribution is freely infinitely divisible. This paper establishes a certain monotonicity, real analyticity and asymptotic behavior of the density of the free Lévy measure. The monotonicity property strengthens the result in Hasebe et al. [Int. Math. Res. Not. (2019)] that the normal distribution is freely selfdecomposable.
{"title":"On the free Lévy measure of the normal distribution","authors":"Takahiro Hasebe, Yuki Ueda","doi":"10.1214/23-ejp1035","DOIUrl":"https://doi.org/10.1214/23-ejp1035","url":null,"abstract":"Belinschi et al. [Adv. Math., 226 (2011)] proved that the normal distribution is freely infinitely divisible. This paper establishes a certain monotonicity, real analyticity and asymptotic behavior of the density of the free Lévy measure. The monotonicity property strengthens the result in Hasebe et al. [Int. Math. Res. Not. (2019)] that the normal distribution is freely selfdecomposable.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135448460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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