The Stochastic Advection by Lie Transport is a variational formulation of stochastic fluid dynamics introduced to model the effects of unresolved scales, whilst preserving the geometric structure of ideal fluid flows. In this work, we show that the SALT equations can arise from the decomposition of the fluid flow map into its mean and fluctuating components. The fluctuating component is realised as a prescribed stochastic diffeomorphism that introduces stochastic transport into the system and we construct it using homogenisation theory. The dynamics of the mean component are derived from a variational principle utilising particular forms of variations that preserve the composite structure of the flow. Using a new variational principle, we show that SALT equations can arise from random Lagrangians and are equivalent to random coefficient PDEs. We also demonstrate how to modify the composite flow and the associated variational principle to derive models inspired by the Lagrangian Averaged Euler-Poincare (LAEP) theory.
李氏输运随机吸附是随机流体动力学的一种变分公式，用于模拟未解决的尺度效应，同时保留理想流体流的几何结构。在这项工作中，我们证明了 SALT 方程可以通过将流体流图分解为平均分量和波动分量而产生。波动分量被视为一种规定的随机差分，它将随机传输引入系统，我们利用均质化理论构建了波动分量。均值分量的动力学原理来自变分原理，利用特定的变分形式保留了流动的复合结构。利用新的变分原理，我们证明了 SALT 方程可以从随机拉格朗日衍生出来，并等价于随机系数 PDE。我们还演示了如何修改复合流和相关的变分原理，以推导出受拉格朗日平均欧拉-平卡理论（LAEP）启发的模型。
{"title":"Variational closures for composite homogenised fluid flows","authors":"Theo Diamantakis, Ruiao Hu","doi":"arxiv-2409.10408","DOIUrl":"https://doi.org/arxiv-2409.10408","url":null,"abstract":"The Stochastic Advection by Lie Transport is a variational formulation of\u0000stochastic fluid dynamics introduced to model the effects of unresolved scales,\u0000whilst preserving the geometric structure of ideal fluid flows. In this work,\u0000we show that the SALT equations can arise from the decomposition of the fluid\u0000flow map into its mean and fluctuating components. The fluctuating component is\u0000realised as a prescribed stochastic diffeomorphism that introduces stochastic\u0000transport into the system and we construct it using homogenisation theory. The\u0000dynamics of the mean component are derived from a variational principle\u0000utilising particular forms of variations that preserve the composite structure\u0000of the flow. Using a new variational principle, we show that SALT equations can\u0000arise from random Lagrangians and are equivalent to random coefficient PDEs. We\u0000also demonstrate how to modify the composite flow and the associated\u0000variational principle to derive models inspired by the Lagrangian Averaged\u0000Euler-Poincare (LAEP) theory.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fueled by the influence of real-world networks both in science and society, numerous mathematical models have been developed to understand the structure and evolution of these systems, particularly in a temporal context. Recent advancements in fields like distributed cyber-security and social networks have spurred the creation of probabilistic models of evolution, where individuals make decisions based on only partial information about the network's current state. This paper seeks to explore models that incorporate emph{network delay}, where new participants receive information from a time-lagged snapshot of the system. In the context of mesoscopic network delays, we develop probabilistic tools built on stochastic approximation to understand asymptotics of both local functionals, such as local neighborhoods and degree distributions, as well as global properties, such as the evolution of the degree of the network's initial founder. A companion paper explores the regime of macroscopic delays in the evolution of the network.
{"title":"Network evolution with mesoscopic delay","authors":"Sayan Banerjee, Shankar Bhamidi, Partha Dey, Akshay Sakanaveeti","doi":"arxiv-2409.10307","DOIUrl":"https://doi.org/arxiv-2409.10307","url":null,"abstract":"Fueled by the influence of real-world networks both in science and society,\u0000numerous mathematical models have been developed to understand the structure\u0000and evolution of these systems, particularly in a temporal context. Recent\u0000advancements in fields like distributed cyber-security and social networks have\u0000spurred the creation of probabilistic models of evolution, where individuals\u0000make decisions based on only partial information about the network's current\u0000state. This paper seeks to explore models that incorporate emph{network\u0000delay}, where new participants receive information from a time-lagged snapshot\u0000of the system. In the context of mesoscopic network delays, we develop\u0000probabilistic tools built on stochastic approximation to understand asymptotics\u0000of both local functionals, such as local neighborhoods and degree\u0000distributions, as well as global properties, such as the evolution of the\u0000degree of the network's initial founder. A companion paper explores the regime\u0000of macroscopic delays in the evolution of the network.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The signature is a collection of iterated integrals describing the "shape" of a path. It appears naturally in the Taylor expansions of controlled differential equations and, as a consequence, is arguably the central object within rough path theory. In this paper, we will consider the signature of Brownian motion with time, and present both new and recently developed approximations for some of its integrals. Since these integrals (or equivalent L'{e}vy areas) are nonlinear functions of the Brownian path, they are not Gaussian and known to be challenging to simulate. To conclude the paper, we will present some applications of these approximations to the high order numerical simulation of stochastic differential equations (SDEs).
{"title":"Approximating the signature of Brownian motion for high order SDE simulation","authors":"James Foster","doi":"arxiv-2409.10118","DOIUrl":"https://doi.org/arxiv-2409.10118","url":null,"abstract":"The signature is a collection of iterated integrals describing the \"shape\" of\u0000a path. It appears naturally in the Taylor expansions of controlled\u0000differential equations and, as a consequence, is arguably the central object\u0000within rough path theory. In this paper, we will consider the signature of\u0000Brownian motion with time, and present both new and recently developed\u0000approximations for some of its integrals. Since these integrals (or equivalent\u0000L'{e}vy areas) are nonlinear functions of the Brownian path, they are not\u0000Gaussian and known to be challenging to simulate. To conclude the paper, we\u0000will present some applications of these approximations to the high order\u0000numerical simulation of stochastic differential equations (SDEs).","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a comprehensive review of stochastic processes, with a particular focus on Markov chains and jump processes. The main results related to queuing systems are analyzed. Additionally, conditions that ensure the stability, or ergodicity, of such systems are presented. The paper also discusses stability results for queuing networks and their extension to visiting systems. Finally, key contributions concerning the Probability Generating Function, an essential tool in the analysis of the aforementioned processes, are introduced. The review is conducted from the perspective of queuing theory, grounded in the Kendall-Lee notation, emphasizing stability results and the computation of performance measures based on the specific characteristics of each process.
{"title":"Notas sobre Teoría de colas y algunas aplicaciones","authors":"Carlos E. Martínez-Rodríguez","doi":"arxiv-2409.10735","DOIUrl":"https://doi.org/arxiv-2409.10735","url":null,"abstract":"This paper presents a comprehensive review of stochastic processes, with a\u0000particular focus on Markov chains and jump processes. The main results related\u0000to queuing systems are analyzed. Additionally, conditions that ensure the\u0000stability, or ergodicity, of such systems are presented. The paper also\u0000discusses stability results for queuing networks and their extension to\u0000visiting systems. Finally, key contributions concerning the Probability\u0000Generating Function, an essential tool in the analysis of the aforementioned\u0000processes, are introduced. The review is conducted from the perspective of\u0000queuing theory, grounded in the Kendall-Lee notation, emphasizing stability\u0000results and the computation of performance measures based on the specific\u0000characteristics of each process.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ${mathfrak{G}}subsetmathbb{R}^{3}$ with $vol(mathfrak{G})sim L^{3}$. Let ${mathscr{T}}(x)$ be a Gaussian random field $forall~xinmathfrak{G}$ with expectation $mathbf{E}[{mathscr{T}}(x)]=0$ and correlation $mathbf{E}[{mathscr{T}}(x)otimes{mathscr{T}}(y)]=K(x,y;lambda)$, an isotropic and regulated kernel with correlation length $lambda$. The field has a Karhunen-Loeve spectral representation ${mathscr{T}}(x)=sum_{I=1}^{infty}mathrm{Z}^{1/2}_{I}f_{I}(x)otimesmathscr{Z}_{I}$, with eigenvalues $lbracemathrm{Z}_{I}rbrace$, eigenfunctions $lbrace f_{I}(x)rbrace $ and Gaussian random variables $mathscr{Z}_{I}$ with $mathbf{E}[mathscr{Z}_{I}]=0$ and $mathbf{E}[mathscr{Z}_{I}otimesmathscr{Z}_{J}]=delta_{IJ}$. If $mathfrak{G}$ contains incompressible fluid of viscosity $nu$ with velocity $u_{a}(x,t)$ that evolves via the Navier-Stokes equations with a high 'Reynolds function' $mathsf{RE}(x,t)=tfrac{|u_{a}(x,t)|L}{nu} $ then aspects of a turbulent flow with $mathsf{RE}(x,t)gg mathsf{RE}_{*}$, a critical Reynolds number, might be represented by the 'weighted' random field $mathscr{U}_{a}(x,t)= u_{a}(x,t)+mathrm{A}u_{a}(x,t)big(mathsf{RE}(x,t)-mathsf{RE}_{*}big)^{beta}sum_{I=1}^{infty} mathrm{Z}^{1/2}_{I}f_{I}(x)otimesmathscr{Z}_{I}$ where random fluctuations and amplitude scale nonlinearly with $mathsf{RE}(x,t)$, with mean $mathbf{E}[{mathscr{U}}_{a}(x,t)] =u_{a}(x,t)$. In the inviscid limit one can prove an anomalous dissipation-type law begin{align} lim_{nurightarrow 0}bigg(lim_{u_{a}(x,t)rightarrow {u}_{a}}sup~nu int_{mathfrak{G}}int_{0}^{T}{mathbf{E}}bigg[bigg|{nabla}_{a}{mathscr{U}}_{a}(x,s)bigg|^{2}bigg]dmathcal{V}(x) dsbigg)>0 end{align} iff $beta=tfrac{1}{2}$ and $sum_{I=1}^{infty}mathrm{Z}_{I}int_{{mathfrak{G}}}{nabla}_{a}f_{I}(x){nabla}^{a}f_{I}(x)dmathcal{V}(x)>0$.
{"title":"A Spectral Representation of a Weighted Random Vectorial Field: Potential Applications to Turbulence and the Problem of Anomalous Dissipation in the Inviscid Limit","authors":"Steven D Miller","doi":"arxiv-2409.10636","DOIUrl":"https://doi.org/arxiv-2409.10636","url":null,"abstract":"Let ${mathfrak{G}}subsetmathbb{R}^{3}$ with $vol(mathfrak{G})sim L^{3}$.\u0000Let ${mathscr{T}}(x)$ be a Gaussian random field $forall~xinmathfrak{G}$\u0000with expectation $mathbf{E}[{mathscr{T}}(x)]=0$ and correlation\u0000$mathbf{E}[{mathscr{T}}(x)otimes{mathscr{T}}(y)]=K(x,y;lambda)$, an\u0000isotropic and regulated kernel with correlation length $lambda$. The field has\u0000a Karhunen-Loeve spectral representation\u0000${mathscr{T}}(x)=sum_{I=1}^{infty}mathrm{Z}^{1/2}_{I}f_{I}(x)otimesmathscr{Z}_{I}$,\u0000with eigenvalues $lbracemathrm{Z}_{I}rbrace$, eigenfunctions $lbrace\u0000f_{I}(x)rbrace $ and Gaussian random variables $mathscr{Z}_{I}$ with\u0000$mathbf{E}[mathscr{Z}_{I}]=0$ and\u0000$mathbf{E}[mathscr{Z}_{I}otimesmathscr{Z}_{J}]=delta_{IJ}$. If\u0000$mathfrak{G}$ contains incompressible fluid of viscosity $nu$ with velocity\u0000$u_{a}(x,t)$ that evolves via the Navier-Stokes equations with a high 'Reynolds\u0000function' $mathsf{RE}(x,t)=tfrac{|u_{a}(x,t)|L}{nu} $ then aspects of a\u0000turbulent flow with $mathsf{RE}(x,t)gg mathsf{RE}_{*}$, a critical Reynolds\u0000number, might be represented by the 'weighted' random field\u0000$mathscr{U}_{a}(x,t)=\u0000u_{a}(x,t)+mathrm{A}u_{a}(x,t)big(mathsf{RE}(x,t)-mathsf{RE}_{*}big)^{beta}sum_{I=1}^{infty}\u0000mathrm{Z}^{1/2}_{I}f_{I}(x)otimesmathscr{Z}_{I}$ where random fluctuations\u0000and amplitude scale nonlinearly with $mathsf{RE}(x,t)$, with mean\u0000$mathbf{E}[{mathscr{U}}_{a}(x,t)] =u_{a}(x,t)$. In the inviscid limit one can\u0000prove an anomalous dissipation-type law begin{align} lim_{nurightarrow\u00000}bigg(lim_{u_{a}(x,t)rightarrow {u}_{a}}sup~nu\u0000int_{mathfrak{G}}int_{0}^{T}{mathbf{E}}bigg[bigg|{nabla}_{a}{mathscr{U}}_{a}(x,s)bigg|^{2}bigg]dmathcal{V}(x)\u0000dsbigg)>0 end{align} iff $beta=tfrac{1}{2}$ and\u0000$sum_{I=1}^{infty}mathrm{Z}_{I}int_{{mathfrak{G}}}{nabla}_{a}f_{I}(x){nabla}^{a}f_{I}(x)dmathcal{V}(x)>0$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive the KPZ equation as a continuum limit of height functions in asymmetric simple exclusion processes with a hyperbolic-scale drift that depends on the local particle configuration. To our knowledge, it is a first such result for a general class of particle systems with neither duality nor explicit invariant measures. The new tools to handle the lack of an invariant measure are estimates for Kolmogorov equations that produce a more robust proof of the Kipnis-Varadhan inequality. These tools are not exclusive to KPZ.
{"title":"KPZ equation from ASEP plus general speed-change drift","authors":"Kevin Yang","doi":"arxiv-2409.10513","DOIUrl":"https://doi.org/arxiv-2409.10513","url":null,"abstract":"We derive the KPZ equation as a continuum limit of height functions in\u0000asymmetric simple exclusion processes with a hyperbolic-scale drift that\u0000depends on the local particle configuration. To our knowledge, it is a first\u0000such result for a general class of particle systems with neither duality nor\u0000explicit invariant measures. The new tools to handle the lack of an invariant\u0000measure are estimates for Kolmogorov equations that produce a more robust proof\u0000of the Kipnis-Varadhan inequality. These tools are not exclusive to KPZ.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers the problem of decentralized submodular maximization subject to partition matroid constraint using a sequential greedy algorithm with probabilistic inter-agent message-passing. We propose a communication-aware framework where the probability of successful communication between connected devices is considered. Our analysis introduces the notion of the probabilistic optimality gap, highlighting its potential influence on determining the message-passing sequence based on the agent's broadcast reliability and strategic decisions regarding agents that can broadcast their messages multiple times in a resource-limited environment. This work not only contributes theoretical insights but also has practical implications for designing and analyzing decentralized systems in uncertain communication environments. A numerical example demonstrates the impact of our results.
{"title":"Optimality Gap of Decentralized Submodular Maximization under Probabilistic Communication","authors":"Joan Vendrell, Solmaz Kia","doi":"arxiv-2409.09979","DOIUrl":"https://doi.org/arxiv-2409.09979","url":null,"abstract":"This paper considers the problem of decentralized submodular maximization\u0000subject to partition matroid constraint using a sequential greedy algorithm\u0000with probabilistic inter-agent message-passing. We propose a\u0000communication-aware framework where the probability of successful communication\u0000between connected devices is considered. Our analysis introduces the notion of\u0000the probabilistic optimality gap, highlighting its potential influence on\u0000determining the message-passing sequence based on the agent's broadcast\u0000reliability and strategic decisions regarding agents that can broadcast their\u0000messages multiple times in a resource-limited environment. This work not only\u0000contributes theoretical insights but also has practical implications for\u0000designing and analyzing decentralized systems in uncertain communication\u0000environments. A numerical example demonstrates the impact of our results.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"122 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Martin Hairer, Seiichiro Kusuoka, Hirotatsu Nagoji
Building on the notes [Hai17], we give a sufficient condition for the marginal distribution of the solution of singular SPDEs on the $d$-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearised equation. As applications we obtain the singularity of the $Phi^4_3$-measure with respect to the Gaussian free field measure and the border of parameters for the fractional $Phi^4$-measure to be singular with respect to the Gaussian free field measure. Our approach is applicable to quite a large class of singular SPDEs.
{"title":"Singularity of solutions to singular SPDEs","authors":"Martin Hairer, Seiichiro Kusuoka, Hirotatsu Nagoji","doi":"arxiv-2409.10037","DOIUrl":"https://doi.org/arxiv-2409.10037","url":null,"abstract":"Building on the notes [Hai17], we give a sufficient condition for the\u0000marginal distribution of the solution of singular SPDEs on the $d$-dimensional\u0000torus to be singular with respect to the law of the Gaussian measure induced by\u0000the linearised equation. As applications we obtain the singularity of the\u0000$Phi^4_3$-measure with respect to the Gaussian free field measure and the\u0000border of parameters for the fractional $Phi^4$-measure to be singular with\u0000respect to the Gaussian free field measure. Our approach is applicable to quite\u0000a large class of singular SPDEs.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We first prove a mimicking theorem (also known as a Markovian projection theorem) for the marginal distributions of an Ito process conditioned to not have exited a given domain. We then apply this new result to the proof of a conjecture of P.L. Lions for the optimal control of conditioned processes.
{"title":"Mimicking and Conditional Control with Hard Killing","authors":"Rene Carmona, Daniel Lacker","doi":"arxiv-2409.10650","DOIUrl":"https://doi.org/arxiv-2409.10650","url":null,"abstract":"We first prove a mimicking theorem (also known as a Markovian projection\u0000theorem) for the marginal distributions of an Ito process conditioned to not\u0000have exited a given domain. We then apply this new result to the proof of a\u0000conjecture of P.L. Lions for the optimal control of conditioned processes.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study mesoscopic fluctuations of orthogonal polynomial ensembles on the unit circle. We show that asymptotics of such fluctuations are stable under decaying perturbations of the recurrence coefficients, where the appropriate decay rate depends on the scale considered. By directly proving Gaussian limits for certain constant coefficient ensembles, we obtain mesoscopic scale Gaussian limits for a large class of orthogonal polynomial ensembles on the unit circle. As a corollary we prove mesocopic central limit theorems (for all mesoscopic scales) for the $beta=2$ circular Jacobi ensembles with real parameter $delta>-1/2$.
{"title":"Mesoscopic Universality for Circular Orthogonal Polynomial Ensembles","authors":"Jonathan Breuer, Daniel Ofner","doi":"arxiv-2409.09803","DOIUrl":"https://doi.org/arxiv-2409.09803","url":null,"abstract":"We study mesoscopic fluctuations of orthogonal polynomial ensembles on the\u0000unit circle. We show that asymptotics of such fluctuations are stable under\u0000decaying perturbations of the recurrence coefficients, where the appropriate\u0000decay rate depends on the scale considered. By directly proving Gaussian limits\u0000for certain constant coefficient ensembles, we obtain mesoscopic scale Gaussian\u0000limits for a large class of orthogonal polynomial ensembles on the unit circle. As a corollary we prove mesocopic central limit theorems (for all mesoscopic\u0000scales) for the $beta=2$ circular Jacobi ensembles with real parameter\u0000$delta>-1/2$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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