We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem - without the need of renormalization - in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear Young integration.
{"title":"Regularization by random translation of potentials for the continuous PAM and related models in arbitrary dimension","authors":"F. Bechtold","doi":"10.1214/22-ecp490","DOIUrl":"https://doi.org/10.1214/22-ecp490","url":null,"abstract":"We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem - without the need of renormalization - in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear Young integration.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44368701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Stein couplings of Chen and Roellin (2010) vastly expanded the range of applications for which coupling constructions in Stein's method for normal approximation could be applied, and subsumed both Stein's classical exchangeable pair, as well as the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants.
{"title":"Zero bias enhanced Stein couplings","authors":"L. Goldstein","doi":"10.1214/22-ECP504","DOIUrl":"https://doi.org/10.1214/22-ECP504","url":null,"abstract":"The Stein couplings of Chen and Roellin (2010) vastly expanded the range of applications for which coupling constructions in Stein's method for normal approximation could be applied, and subsumed both Stein's classical exchangeable pair, as well as the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44555929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability d N . We determine the full region of delocalization by determining the critical values of d log N down to which delocalization persists: for d log N > 1 log 4−1 all eigenvectors are completely delocalized, and for d log N > 1 all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [1, 3] that localized eigenvectors exist in the corresponding spectral regions.
我们分析了Erdős-Rényi图在N个顶点上的邻接矩阵的特征向量，边缘概率为d N。我们通过确定d log N的临界值来确定脱域的整个区域:对于d log N > 1 log 4−1，所有特征向量都是完全脱域的，对于d log N > 1，所有特征值远离谱边的特征向量都是完全脱域的。在这些临界值以下，我们知道[1,3]在相应的光谱区域存在局域特征向量。
{"title":"The completely delocalized region of the Erdős-Rényi graph","authors":"Johannes Alt, Raphael Ducatez, A. Knowles","doi":"10.1214/22-ecp450","DOIUrl":"https://doi.org/10.1214/22-ecp450","url":null,"abstract":"We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability d N . We determine the full region of delocalization by determining the critical values of d log N down to which delocalization persists: for d log N > 1 log 4−1 all eigenvectors are completely delocalized, and for d log N > 1 all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [1, 3] that localized eigenvectors exist in the corresponding spectral regions.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47542430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given 1 < p < ∞ and a pair ( x, y ) of nonnegative numbers satisfying x p ≤ y , we identify the optimal upper bounds for (cid:107)| sup n f n |(cid:107) p, ∞ , for nonnegative martingales f = ( f n ) n ≥ 0 satisfying (cid:107) f (cid:107) 1 = x and (cid:107) f (cid:107) pp = y .
本文研究了一个鞅极大函数的弱型估计的尖锐扩张。给定1＜p＜∞和一对满足x p≤y的非负数（x，y），我们确定了（cid:107）|supn f n|（cid:107）p，∞的最优上界，对于非负鞅f＝（f n）n≥0满足（cid:10 7）f（cid:7）1＝x和（cid:07）f（acid:107）pp＝y。
{"title":"Weak-type estimates for martingale maximal functions","authors":"A. Osȩkowski, Mateusz Wojtas","doi":"10.1214/22-ecp494","DOIUrl":"https://doi.org/10.1214/22-ecp494","url":null,"abstract":"The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given 1 < p < ∞ and a pair ( x, y ) of nonnegative numbers satisfying x p ≤ y , we identify the optimal upper bounds for (cid:107)| sup n f n |(cid:107) p, ∞ , for nonnegative martingales f = ( f n ) n ≥ 0 satisfying (cid:107) f (cid:107) 1 = x and (cid:107) f (cid:107) pp = y .","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46138797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Consider the set of Borel probability measures on R k and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.
{"title":"The maximum domain of attraction of multivariate extreme value distributions is small","authors":"P. Leonetti, A. K. Chokami","doi":"10.1214/22-ecp501","DOIUrl":"https://doi.org/10.1214/22-ecp501","url":null,"abstract":". Consider the set of Borel probability measures on R k and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44224546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the central limit theorem for stationary random fields under L1-projective condition","authors":"H. Lin, F. Merlevède, D. Volný","doi":"10.1214/22-ecp486","DOIUrl":"https://doi.org/10.1214/22-ecp486","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41419643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this short note, we prove a central limit theorem for a type of replica overlap of the Brownian directed polymer in a Gaussian random environment, in the low temperature regime and in all dimensions. The proof relies on a superconcentration result for the KPZ equation driven by a spatially mollified noise, which is inspired by the recent work of Chatterjee [14].
{"title":"Gaussian fluctuations of replica overlap in directed polymers","authors":"Yu Gu, T. Komorowski","doi":"10.1214/22-ecp476","DOIUrl":"https://doi.org/10.1214/22-ecp476","url":null,"abstract":". In this short note, we prove a central limit theorem for a type of replica overlap of the Brownian directed polymer in a Gaussian random environment, in the low temperature regime and in all dimensions. The proof relies on a superconcentration result for the KPZ equation driven by a spatially mollified noise, which is inspired by the recent work of Chatterjee [14].","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48840622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.
{"title":"Sharp phase transition for Cox percolation","authors":"C. Hirsch, B. Jahnel, S. Muirhead","doi":"10.1214/22-ecp487","DOIUrl":"https://doi.org/10.1214/22-ecp487","url":null,"abstract":"We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47500768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We point out an error in "The remainder in the renewal theorem", and show that the result is essentially correct in two important special cases.
我们指出了“更新定理中的余数”中的一个错误，并证明了在两个重要的特殊情况下，结果是基本正确的。
{"title":"Erratum: The remainder in the renewal theorem","authors":"R. Doney","doi":"10.1214/22-ecp456","DOIUrl":"https://doi.org/10.1214/22-ecp456","url":null,"abstract":"We point out an error in \"The remainder in the renewal theorem\", and show that the result is essentially correct in two important special cases.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49507916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It has been discovered that the Kadomtsev-Petviashvili (KP) equation governs the distribution of the fluctuation of many random growth models. In particular, the Tracy-Widom distributions appear as special self-similar solutions of the KP equation. We prove that the anti-derivative of the Baik-Rains distribution, which governs the fluctuation of the models in the KPZ universality class starting with stationary initial data, satisfies the KP equation. The result is first derived formally by taking a limit of the generating function of the KPZ equation, which satisfies the KP equation. Then we prove it directly using the explicit Painlevé II formulation of the Baik-Rains distribution. the long-time fluctuations of models which belong to the KPZ universality class. A 1 ( x ) is a stationary process, whose one-point distribution is the Tracy-Widom GOE distribution. The one point marginal of A 2 ( x ) is given by the Tracy-Widom GUE distribution. The one point marginal of A stat ( x ) is given by the Baik-Rains distribution.
{"title":"The logarithmic anti-derivative of the Baik-Rains distribution satisfies the KP equation","authors":"Xincheng Zhang","doi":"10.1214/22-ecp469","DOIUrl":"https://doi.org/10.1214/22-ecp469","url":null,"abstract":"It has been discovered that the Kadomtsev-Petviashvili (KP) equation governs the distribution of the fluctuation of many random growth models. In particular, the Tracy-Widom distributions appear as special self-similar solutions of the KP equation. We prove that the anti-derivative of the Baik-Rains distribution, which governs the fluctuation of the models in the KPZ universality class starting with stationary initial data, satisfies the KP equation. The result is first derived formally by taking a limit of the generating function of the KPZ equation, which satisfies the KP equation. Then we prove it directly using the explicit Painlevé II formulation of the Baik-Rains distribution. the long-time fluctuations of models which belong to the KPZ universality class. A 1 ( x ) is a stationary process, whose one-point distribution is the Tracy-Widom GOE distribution. The one point marginal of A 2 ( x ) is given by the Tracy-Widom GUE distribution. The one point marginal of A stat ( x ) is given by the Baik-Rains distribution.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49510101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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