FRG: Collaborative Research: Integrable Probability

FRG:协作研究:可积概率

基本信息

  • 批准号:
    1664617
  • 负责人:
  • 金额:
    $ 19.35万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2017
  • 资助国家:
    美国
  • 起止时间:
    2017-07-01 至 2022-06-30
  • 项目状态:
    已结题

项目摘要

Much of modern probability research seeks to understand the behavior of large and complex random systems (for instance, growth in disordered media, cracking, turbulent fluids, or traffic flow) with an aim towards developing theories with predictive and statistical value. While one can try to directly model such systems on computers, their size and complexity often render such attempts fruitless. Instead, one can look for models of such systems that are complex enough to display all of the phenomena under study, yet simple enough to admit exact mathematical computation to probe that behavior. Integrable probability is the theory behind discovering and subsequently analyzing such models. This project seeks to unify the area and various recent breakthroughs and in so doing discover a host of new types of integrable probability systems, new tools for their analysis, and new large-scale universal phenomena.Integrable probability is an area of research at the interface between probability, mathematical physics, and statistical physics on the one hand, and representation theory and integrable systems on the other. Integrable probabilistic systems are characterized by two properties: It is possible to write down concise and exact formulas for expectations of a variety of interesting observables of the systems; and asymptotics of the systems, observables, and formulas provide access to exact descriptions of new phenomena and universality classes (containing more than just integrable examples). The discovery and analysis of integrable probabilistic systems hinges upon underlying algebraic structure. These integrable probabilistic systems can be viewed as projections of powerful objects whose origins lie in representation theory and integrable systems. There is a rich history of major breakthroughs in the study of integrable probabilistic systems, including the six-vertex model, Ising model, and more recently certain models in the KPZ universality class. The basic mechanisms at the heart of many of these existing results are Schur / Macdonald processes (built off the structure of symmetric polynomials) and quantum integrable systems (built off solutions to the Yang-Baxter equation and the Bethe ansatz). Each mechanism has produced breakthrough results, such as the recent resolution of the 25-year-old physics conjecture that the KPZ stochastic partial differential equation is in the KPZ universality class. Until recently, these two routes to integrable probability have existed relatively separately. The goal of the proposed project is to create a unified theory of integrable probability, combining and generalizing the methods of Schur / Macdonald processes and quantum integrable systems and, complementarily, extracting new analyzable models and uncovering new probabilistic or physical phenomena.
许多现代概率研究试图了解大型和复杂随机系统的行为(例如,无序介质中的增长、裂解、湍流或交通流),目的是开发具有预测和统计价值的理论。虽然人们可以尝试在计算机上直接对这类系统进行建模,但它们的规模和复杂性往往使这种尝试徒劳无功。取而代之的是,人们可以寻找这样的系统的模型,这些模型足够复杂,可以显示研究中的所有现象,但又足够简单,可以进行精确的数学计算来探测这种行为。可积概率是发现和随后分析这些模型背后的理论。这个项目试图将这个领域和最近的各种突破统一起来,并在这样做的过程中发现一系列新型的可积概率系统、分析它们的新工具和新的大规模普遍现象。可积概率是一个一方面在概率、数学物理和统计物理之间,另一方面在表象理论和可积系统之间的研究领域。可积概率系统具有两个性质:可以写下系统的各种有趣的可观测数据的期望的简明而精确的公式;系统、可观测数据和公式的渐近性提供了对新现象和普适性类的精确描述(不仅包含可积实例)。可积概率系统的发现和分析取决于基本的代数结构。这些可积概率系统可以看作是强大对象的投影,其起源在于表示理论和可积系统。可积概率系统的研究有着丰富的突破历史,包括六点模型、Ising模型,以及最近在KPZ普适性类中的某些模型。许多现有结果的核心机制是Schur/Macdonald过程(建立在对称多项式的结构上)和量子可积系统(建立在杨-巴克斯特方程和Bethe ansatz解的基础上)。每一种机制都产生了突破性的结果,例如最近解决了25年来的物理猜想,即KPZ随机偏微分方程属于KPZ普适性类。直到最近,这两条通向可积概率的途径一直相对独立地存在。该项目的目标是创建一个统一的可积概率理论,将Schur/Macdonald过程和量子可积系统的方法结合起来并加以推广,作为补充,提取新的可分析模型,并揭示新的概率或物理现象。

项目成果

期刊论文数量(14)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Mapping TASEP back in time
PushTASEP in inhomogeneous space
  • DOI:
    10.1214/20-ejp517
  • 发表时间:
    2019-10
  • 期刊:
  • 影响因子:
    0
  • 作者:
    L. Petrov
  • 通讯作者:
    L. Petrov
Parameter Symmetry in Perturbed GUE Corners Process and Reflected Drifted Brownian Motions
  • DOI:
    10.1007/s10955-020-02652-7
  • 发表时间:
    2019-12
  • 期刊:
  • 影响因子:
    1.6
  • 作者:
    L. Petrov;M. Tikhonov
  • 通讯作者:
    L. Petrov;M. Tikhonov
YANG–BAXTER FIELD FOR SPIN HALL–LITTLEWOOD SYMMETRIC FUNCTIONS
YANG—BAXTER 旋转霍尔场—LITTLEWOOD 对称函数
  • DOI:
    10.1017/fms.2019.36
  • 发表时间:
    2019
  • 期刊:
  • 影响因子:
    0
  • 作者:
    BUFETOV, ALEXEY;PETROV, LEONID
  • 通讯作者:
    PETROV, LEONID
Parameter Permutation Symmetry in Particle Systems and Random Polymers
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Leonid Petrov其他文献

Rewriting History in Integrable Stochastic Particle Systems
$\mathfrak{sl}(2)$ operators and Markov processes on branching graphs
Coarsening Model on $${\mathbb{Z}^{d}}$$ with Biased Zero-Energy Flips and an Exponential Large Deviation Bound for ASEP
  • DOI:
    10.1007/s00220-018-3180-2
  • 发表时间:
    2018-06-18
  • 期刊:
  • 影响因子:
    2.600
  • 作者:
    Michael Damron;Leonid Petrov;David Sivakoff
  • 通讯作者:
    David Sivakoff

Leonid Petrov的其他文献

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{{ truncateString('Leonid Petrov', 18)}}的其他基金

Random Systems from Symmetric Functions and Vertex Models
对称函数和顶点模型的随机系统
  • 批准号:
    2153869
  • 财政年份:
    2022
  • 资助金额:
    $ 19.35万
  • 项目类别:
    Continuing Grant
Workshop on Representation Theory, Combinatorics, and Geometry
表示论、组合学和几何研讨会
  • 批准号:
    1839534
  • 财政年份:
    2018
  • 资助金额:
    $ 19.35万
  • 项目类别:
    Standard Grant

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