Riemann surfaces for KPZ with periodic boundaries

2019/08/31 by Sylvain Prolhac
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Combinatorics #Equilateral triangle #Geometry #Hypercube #Korteweg–de Vries equation #Mathematical analysis #Mathematical physics #Mathematics #Meromorphic function #Nonlinear system #Physics #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Riemann hypothesis #Riemann surface #Stochastic processes and statistical mechanics #Universality (dynamical systems) #cond-mat.stat-mech #math-ph #math.MP

paper · pdf · doi:10.21468/scipostphys.8.1.008

published as SciPost Phys. 8, 008 (2020) · 81 pages, 23 figures

arxiv created 2019/11/14 · openalex publication_date 2020/01/22 · arxiv updated 2020/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The Riemann surface for polylogarithms of half-integer index, which has the topology of an infinite dimensional hypercube, is studied in relation to one-dimensional KPZ universality in finite volume. Known exact results for fluctuations of the KPZ height with periodic boundaries are expressed in terms of meromorphic functions on this Riemann surface, summed over all the sheets of a covering map to an infinite cylinder. Connections to stationary large deviations, particle-hole excitations and KdV solitons are discussed.

Citations

• Probability distribution of the free energy of a directed polymer in a random medium
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice-first excited states
• Critical resonance in the non-intersecting lattice path model
• Tree structures for the current fluctuations in the exclusion process
• The completeness of the Bethe ansatz for the periodic ASEP
• Finite-Time Fluctuations for the Totally Asymmetric Exclusion Process
• The KPZ fixed point
• Multi-point distribution of periodic TASEP
• Lectures notes on compact Riemann surfaces
• Lectures on nonlinear integrable equations and their solutions