Alexei Borodin (Boston), Viktor Dotsenko (Paris), Vadim Gorin (Moscow), Alan Hammond (Oxford), Fabio Toninelli (Lyon), Yvan Velenik (Genève).

I will review some of the instances when the system of size N undergoes fluctuations of size N^{1/3}. Included will be 2D Ising model, random walking around the walls, deviations away from the convex envelope, etc.

Alexei Borodin (Boston): KPZ 1/3 and beyond via Macdonald processes. pdf

Vadim Gorin (Moscow): Integrable probability on interlacing particles. pdf

Families of particles subject to interlacing conditions often arise in statistical mechanics and random matrix theory. Examples include eigenvalues of Hermitian matrix and its submatrices, lozenge tilings, six-vertex model with domain wall boundary conditions. We will discuss recent results on the asymptotic behavior of random interlacing particle configurations. One distinction of the models we study is the existence of explicit formulas for various distributions and averages. This is a manifestation of the connection of our models to the representation theory: There families of interlacing particles are known under the name Gelfand-Tsetlin patterns.

Alan Hammond (Oxford): No! Three different one-thirds. (Or, three instances of the competition between curvature and fluctuation)

Many planar models of random interfaces exhibit the scaling characteristic of the KPZ class, namely that, for an interface of diameter t, typical facets in the convex boundary of the interface have length t^{2/3}, and typical excursions of the interface between consecutive convex boundary extreme points have an inward deviation of t^{1/3}. In many such models, this characteristic scaling may be viewed as arising due to a competition between a Gaussian fluctuation dominant on smaller scales and a constraint of curvature which is significant on large scales.

We will discuss three examples:

1. Consider one-dimensional Brownian bridge B on [-T,T] conditioned to remain above the semicircle of radius T centred at the origin. The upper convex envelope of the conditioned motion has facets of length T^{2/3}.

2. Consider the subcritical random cluster model conditioned on the presence of an interface surrounding the origin and trapping area n^2. In a series of papers on "phase separation" two years ago, I identified the longitudinal n^{2/3} and latitudinal n^{1/3} scalings along with logarithmic corrections.

3. The polynuclear growth model is an interface model describing the maximal number of planar Poisson points that may be visited by an oriented path.

In each case, one may scale space parabolically to obtain a new coordinate system suited to studying the limiting fluctuations. In case 1, Ferrari and Spohn have studied an SDE with a drift term expressed in terms of the Airy function which arises in such a limit; in case 2, a limiting description would be Markovian in the variables of profile height and area captured; in case 3, Prahofer and Spohn identified the Airy process as the correct limiting object.

In all three models, Gaussian fluctuation and curvature determinine the 2/3-1/3 scaling, but it seems likely that the limiting scaled stochastic processes are all different, with KPZ universality being present only in the third case.

Victor Dotsenko (Paris): Bethe ansatz derivations of the Tracy-Widom distributions in one-dimensional directed polymers.

The derivation of the free energy distribution functions in one-dimensional directed polymers (which belong to the KPZ universality class) is discussed. By mapping original problem to the N-particle quantum boson system with attractive interactions the resulting Tracy-Widom distributions are derived in terms of the Bethe ansatz replica technique. The cases of the GUE, GOE and GSE Tracy-Widom distributions are discussed.

Fabio Toninelli (Lyon): L^{1/3} fluctuations for the contours of the 2D SOS model. pdf

We consider the two-dimensional discrete SOS model at low temperature T in a L x L box. In presence of a wall at height zero, the surface is typically at height (T/4)log(L) (due to entropic repulsion). We show that the level lines of the interface, once rescaled by 1/L, have a deterministic limit described by a suitable Wulff shape, and that the fluctuations around the limit shape are of order L^{1/3} (before rescaling). This seems to be connected with the "1/3-type fluctuations" one finds in growth models and directed polymers in (1+1) dimensions. (joint work with P. Caputo, E. Lubetzky, F. Martinelli and A. Sly)

Yvan Velenik (Genève): Equilibrium properties of a layer of unstable phase

Consider a 1+1-dimensional effective model describing an interface above a hard wall and penalized according to the area between it and the wall. I'll recall some results about this model, obtained nearly 10 years ago with Ostap Hryniv. I'll then describe some work in progress, with Dima Ioffe and Seyna Shlosman, concerning the scaling limit of this model, as the penalization is sent to zero. I'll then describe a variety of situations in which this problem has appeared, sometimes somewhat unexpectedly.