# Exponential Convergence for 3D Stochastic Primitive Equations of the Large Scale Ocean

*By Zhao DONG, Institute of Applied Mathematics, AMSS, CAS*

In this paper, we consider the ergodicity for the three-dimensional stochastic primitive equations of the large scale oceanic motion. We proved that if the noise is at the same time sufficiently smooth and non-degenerate, then the weak solutions converge exponentially fast to equilibrium. Moreover, the uniqueness of invariant measure is stated.

# Long-range Order in Random 3-colorings of Z^{d}

*By Yinon Spinka, Tel-Aviv University*

Consider a random coloring of a bounded domain in Z^{d} with the probability of each coloring F proportional to exp(-β*N(F)), where β>0 is a parameter (representing the inverse temperature) and N(F) is the number of nearest neighboring pairs colored by the same color. This is the anti-ferromagnetic 3-state Potts model of statistical physics, used to describe magnetic interactions in a spin system. The Kotecký conjecture is that in such a model, for d≥3 and high enough β, a sampled coloring will typically exhibit long-range order, placing the same color at most of either the even or odd vertices of the domain. We give the first rigorous proof of this fact for large d. This extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the case β=infinity.

No background in statistical physics will be assumed and all terms will be explained thoroughly.

*Joint work with Ohad Feldheim.*

# Mean-field Bound on the 1-arm Exponent for Ising Ferromagnets in High Dimensions

*By Akira Sakai, Hokkaido University*

The Ising model is a statistical-mechanical model for magnets. It is now known that, if the spin-spin coupling is non-negative and reflection-positive, then it exhibits a continuous phase transition. In particular, the critical 1-spin expectation at the center of a ball of

radius $r$ vanishes as $r$ goes to infinity. It is believed to decay in powers of $r$, with an exponent $\rho$ called the 1-arm exponent. Presumably this exponent takes on the mean-field value 1 in high dimensions, but the best possible bound so far is $(d-2)/2$, due to a

hyperscaling inequality.

I will show how we achieve the mean-field bound on the Ising 1-arm exponent, i.e., $\rho\le1$.

*This talk is based on my ongoing project with Satoshi Handa and Markus Heydenreich.*

# Hydrodynamics and Quenched Strong Equilibrium for Asymmetric Zero Range Process with Sitewise Disorder

*By Krishnamurthi Ravishankar, State University of New York at New Paltz*

# #Cancelled# - Funky CLTs for Critical Interacting Particle Systems in High Dimensions

*By Mark Holmes, The University of Auckland*

We will discuss recent and ongoing work involving the proof of "functional central limit theorems for measure-valued processes" relevant to some well-known interacting particle systems (such as the voter model) in high dimensions.

# An Elementary Model of Percolation on Trees

*By Zhan SHI, Université Paris VI*

I am going to discuss a few questions, with or without answers, about the free energy in a simplified model of depinning transition in the limit of strong disorder. The study, initiated by Derrida and Retaux in 2014, can also be formulated for an elementary percolation model on trees.

*Joint work with Bernard Derrida, Nina Gantert and Yueyun Hu.*

# Absorbing-State Phase Transitions

*By Leonardo Rolla, Universidad de Buenos Aires*

Modern statistical mechanics offers a large class of driven-dissipative stochastic systems that naturally evolve to a critical state, of which activated random walks is perhaps the best example. The main pursuit in this field is to describe the critical behavior, the scaling relations and critical exponents of such systems, and whether the critical density in the infinite system is the same as the equilibrium density in their driven-dissipative finite-volume version. These questions are however far beyond the reach of existing techniques. In this talk we will report on the progress obtained in recent years, and discuss some of the open problems.