Experimental Realization Of Tracy–Widom Distributions And Beyond: Kpz Interfaces In Turbulent Liquid Crystal

Summary

Analytical studies have shown the Tracy–Widom distributions and the Airy processes in the asymptotics of a few growth models in the Kardar–Parisi– Zhang (KPZ) universality class. Here the author shows evidence that these mathematical objects arise even in a real experiment: more specifically, in growing interfaces of turbulent liquid crystal. The present article is devoted to overviewing the current status of this experimental approach to the KPZ class, which directly concerns random matrix theory and related fields of mathematical physics. In particular, the author summarizes those statistical properties which were derived rigorously for simple solvable models and realized here experimentally, and those which were evidenced in the experiment and remain to be explained by further mathematical or theoretical studies.

1.Introduction

The first decade of the 21st century and a couple of preceding years have been marked by a series of remarkable analytical developments, which have revealed profound and rigorous connections among random matrix theory, combinatorial problems, and the physical problem of the fluctuating interface growth ([Baik and Rains 2001; Kriecherbauer and Krug 2010; Sasamoto and Spohn 2010a; Corwin 2012] and references therein). Their primary conclusions in terms of the interface growth problem, first pointed out by Johansson [2000] for TASEP and by Prähofer and Spohn [2000] for the PNG model, are the following [Kriecherbauer and Krug 2010; Sasamoto and Spohn 2010a; Corwin 2012]: (i) The distribution function and the spatial correlation function were obtained rigorously for the asymptotic interface fluctuations. (ii) The results depend on the global shape of the interfaces, or on the initial condition. For the two prototypical cases of the curved and flat growing interfaces, the distribution function is given by the Tracy–Widom distribution [Tracy and Widom 1994; 1996] for GUE and GOE, respectively, and the spatial two-point correlation function by the covariance of the Airy2 and Airy1 process [Prähofer and Spohn 2002; Sasamoto 2005],

respectively. (iii) All or some of these conclusions were reached for a number of models, namely the TASEP [Johansson 2000; Borodin et al. 2007; 2008; Sasamoto 2005] and the PASEP [Tracy and Widom 2009], the PNG model [Prähofer and Spohn 2000; 2002; Borodin et al. 2008], and the KPZ equation [Sasamoto and Spohn 2010c; 2010b; Amir et al. 2011; Calabrese et al. 2010; Dotsenko 2010; Calabrese and Le Doussal 2011; Prolhac and Spohn 2011].