Lectures on Integrable Probability: stochastic vertex models and symmetric functions
Name
1605.01349.pdf
Description
Submitted version
Size
1.58 MB
Format
Adobe PDF
Checksum (MD5)
009dbcceffd4a9b6c3c73f88f4edcc04
Author(s)
Borodin, Alexei
Petrov, Leonid
Alternative Title
Integrable probability: stochastic vertex models and symmetric functions
Date Issued
January 2018
Journal
Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School: Volume 104, July 2015
Publisher
Oxford University Press
Citation
Borodin, Alexei and Leonid Petrov. "Integrable probability: stochastic vertex models and symmetric functions." Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School: Volume 104, July 2015. Edited by Gregory Schehr, Alexander Altland, Yan V. Fyodorov, Neil O'Connell, and Leticia F. Cugliandolo. Oxford University Press, 2017
Version
Original manuscript
Abstract
We consider a homogeneous stochastic higher spin six-vertex model in a quadrant. For this model concise integral representations for multipoint q-moments of the height function and for the q-correlation functions are derived. At least in the case of the step initial condition, these formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six-vertex model, ASEP, various q-TASEPs, and associated zero-range processes. The arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the higher spin six-vertex model for suitable domains; they generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six-vertex model. Keywords: six-vertex model; ASEP; TASEP; Kardar–Parisi–Zhang equation; Yang–Baxter equation; integrable probability
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1093/oso/9780198797319.003.0002
