RESUMEN

The KPZ fixed point
Daniel Remenik, Universidad de Chile

The Kardar-Parisi-Zhang (KPZ) universality class is a broad class of models coming from mathematical physics which includes random interface growth, directed random polymers, interacting particle systems, and random stirred fluids. These models share a very special and rich asymptotic fluctuation behavior, which is loosely characterized by fluctuations which grow like t^{1/3} as time t evolves, decorrelate at a spatial scale of t^{2/3}, and have certain very special limiting distibutions; this fluctuation behavior is model independent but depends on the initial data, and in some prominent cases it is connected with distributions coming from random matrix theory.

A somewhat vague conjecture in the field was that there should be a universal, scaling invariant, limit for all models in the KPZ class, containing all the fluctuation behavior seen in the class. In these lectures I will describe joint work with K. Matetski and J. Quastel [5] where we were able to construct and give a complete description of this limiting process, known as the KPZ fixed point. This limiting universal process is a Markov process, taking values in real valued functions which look locally like Brownian motion.

The construction follows from an novel exact solution for one of the most basic models in the KPZ class, the totally asymmetric exclusion process (TASEP), for arbitrary initial condition. This formula is given as the Fredholm determinant of a kernel involving the transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.

Two sets of lecture notes [6,7] serve as a good complement to the mini-course.

Tentative schedule:

  • Lecture 1. This lecture will be devoted mostly to an introduction to the KPZ universality class and a description of the basic asymptotic fluctuation behavior for TASEP. I will also introduce the KPZ fixed point and discuss its main properties.
  • Lecture 2. I will provide some background on Fredholm determinants and determinantal processes, and discuss some earlier work [2,3,4] on continuum statistics for Airy processes which played an important role in solving TASEP.
  • Lecture 3. In this lecture I will explain how the explicit TASEP solution was obtained, starting from the biorthogonal ensemble representation found by Sasamoto in 2005 [1,8].
  • Lecture 4. Finally I will show how the scaling limit of the TASEP formulas leads to the KPZ fixed point and how its main properties follow from this construction.

References:

  • A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129.5-6 (2007), pp. 10551080.
  • A. Borodin, I. Corwin, and D. Remenik. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincaré Probab. Statist. 51.1 (2015), pp. 28–58.
  • I. Corwin, J. Quastel, and D. Remenik. Continuum statistics of the Airy 2 process. Comm. Math. Phys. 317.2 (2013), pp. 347–362.
  • J. Quastel, D. Remenik. How flat is flat in random interface growth? To appear in Trans. AMS. arXiv:1606.09228. 
  • K. Matetski, J. Quastel and D. Remenik. The KPZ fixed point. arXiv:1701.00018.
  • K. Matetski, J. Quastel. From TASEP to the KPZ fixed point. arXiv:1710.02635.
  • D. Remenik. Course notes on the KPZ fixed point. http://www.dim.uchile.cl/ ~dremenik/KPZFixedPointNotes.pdf.
  • T. Sasamoto. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38.33 (2005), p. L549.