Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
arXiv:1403.6877
Abstract
Integrable probability has emerged as an active area of research at the interface of probability/mathematical physics/statistical mechanics on the one hand, and representation theory/integrable systems on the other. Informally, integrable probabilistic systems have two properties: 1) It is possible to write down concise and exact formulas for expectations of a variety of interesting observables (or functions) of the system. 2) Asymptotics of the system and associated exact formulas provide access to exact descriptions of the properties and statistics of large universality classes and universal scaling limits for disordered systems. We focus here on examples of integrable probabilistic systems related to the Kardar-Parisi-Zhang (KPZ) universality class and explain how their integrability stems from connections with symmetric function theory and quantum integrable systems.
arXiv commentProceedings of the ICM, 31 pages, 10 figures
References in corpus (30)
- Level-Spacing Distributions and the Airy Kernel
- Quantum Inverse Scattering Method and Correlation Functions
- Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
- How Algebraic Bethe Ansatz works for integrable model
- Free-energy distribution of the directed polymer at high temperature
- Asymptotics in ASEP with Step Initial Condition
- Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
- Exact Solution of the Master Equation for the Asymmetric Exclusion Process
- Scaling for a one-dimensional directed polymer with boundary conditions
- Effects of Particle Shape on Growth Dynamics at Edges of Evaporating Colloidal Drops
- The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1
- A Fredholm Determinant Representation in ASEP
- The Continuum Directed Random Polymer
- Correlation functions for a strongly correlated boson system
- Renormalization fixed point of the KPZ universality class
- Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
- Bounds for scaling exponents for a 1+1 dimensional directed polymer in a Brownian environment
- A multi-layer extension of the stochastic heat equation
- A q-weighted version of the Robinson-Schensted algorithm
- KPZ Scaling Theory and the Semi-discrete Directed Polymer Model
- Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers
- Lectures on the integrability of the 6-vertex model
- General beta Jacobi corners process and the Gaussian Free Field
- Limits of Multilevel TASEP and similar processes
- Probabilistic measures and algorithms arising from the Macdonald symmetric functions
- The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension
- Diffusivity of lattice gases
- Law of Large Numbers for Infinite Random Matrices over a Finite Field
- Multilevel Dyson Brownian motions via Jack polynomials
- Correlation Functions of the Schur Process Through Macdonald Difference Operators
Cited by in corpus (15)
- The one-dimensional KPZ equation and its universality class
- Stochastic six-vertex model
- Current Fluctuations of the Stationary ASEP and Six-Vertex Model
- Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
- Half-space Macdonald processes
- Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments
- Large Deviations For Sticky Brownian Motions
- Height fluctuations for the stationary KPZ equation
- The lower tail of the half-space KPZ equation
- Correlation Functions of the Pfaffian Schur Process Using Macdonald Difference Operators
- Some recent progress in singular stochastic PDEs
- Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
- Free field theory and observables of periodic Macdonald processes
- Magnus expansions and pseudospectra of Master Equations
- Disordered statistical physics in low dimensions: extremes, glass transition, and localization