Dynamical Evolution of Interacting Particle Systems: Mixing Times, Interface Fluctuations and Universality

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This research project aims to study a range of problems on dynamical aspects of systems of interacting particles, and en route to develop new tools in Probability Theory to advance our understanding of their features in and out of equilibrium. The motivating problems - several of which have been intensively studied in statistical and mathematical physics as well as in computer science - touch fundamental questions such as the convergence time to equilibrium at the critical point of phase transition for classical models such as 2D and 3D Ising and Potts models under different boundary conditions, cluster interfaces at equilibrium (both interesting on its own accord, and crucial for the analysis of the dynamics), and the role of the underlying geometry on the dynamics, or the lack thereof - whereby dynamical features are universal on a broad class of graphs. The first research direction focuses on the classical Ising, Potts and FK model on d-dimensional tori, and examines two of the most common Markov chains used to sample them/model their evolution - Glauber dynamics and Swendsen-Wang dynamics. The phase transition that both the dynamical and static models undergo has received much attention, yet various basic problems have so far been out of reach of rigorous analysis in all three temperature regimes (high/low/critical). The PI proposes to study several such problems, including a power-law for mixing at criticality for the extremal 4-color Potts model in 2D vs. slow mixing in 3D, and establishing the cutoff phenomenon for the FK model off-criticality. A second research direction aims to understand properties of cluster interfaces in these models under various boundary conditions (b.c.): The PI plans to study rigidity and fluctuations of 3D Ising and 2D Potts interfaces under prescribed b.c., towards showing fast mixing for the 3D Ising model with plus b.c. and critical 2D q-state Potts model with free b.c. for all q. The final proposed topic compares the behavior of systems of interacting particles on lattices to other environments. For the SK spin glass model, various features of the static measure are known to be universal in the law of the interactions, and the PI proposes to establish an analogous statement for Langevin dynamics; for branching Brownian motion, the PI will investigate the law of the maximum in the presence of a periodic environment affecting the particles; and for the stochastic Ising model, the PI aims to show an aspect of sensitivity to initial conditions that is special to lattices, contrary to the behavior on typical random regular graphs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.