Probability and Mathematical Physics Seminar

Probability and the City seminar

Speaker: Michael Aizenman (Princeton University), Daniel Stein (NYU) and Ivan Corwin (Columbia University)

Location: Warren Weaver Hall 1302

Date: Friday, March 8, 2024, 10 a.m.


Michael Aizenman (10am)
Title: Stochastic geometry of correlations in statistical and quantum mechanics

Abstract: The talk will start by recalling some of the pivotal contributions made by Charles M. Newman and collaborators in their rigorous analyses of critical phenomena in statistical mechanic, in which  non-perturbative results of fundamental interest were derived through a broad range of analytic and probabilistic methods. Among those are techniques enabled  by percolation-type stochastic geometric representations of the spread of correlations in systems of local interactions.  The discussion will expand into such methods which, when available, are enabled by   local conditional symmetries.  Their more recent applications include proofs of discontinuity, or its absence, in symmetry breaking phase transitions  in some paradigmatic classical and quantum spin models, and bounds on the quantum entanglement in the ground states of related quantum spin arrays.   
(Results mentioned will include also speaker’s joint works with S. Warzel and H. Duminil-Copin).

Daniel Stein (11am)
Title: The Thermodynamic Structure of Short-Range Spin Glasses

Abstract: The thermodynamics of classical spin glasses, and more generally the statistical mechanics of quenched disorder, is a problem of general interest to physicists and mathematicians and one to which Chuck has made numerous contributions. Its importance was recognized in 2021 with the awarding of the Nobel Prize in Physics to Giorgio Parisi for his replica symmetry breaking solution of the Sherrington-Kirkpatrick model (a mean-field model of spin glasses) and its application to other problems in complex systems.

I will begin with a brief review of the main features of replica symmetry breaking, including an application to physical systems. I will then turn to the problem of understanding the nature of the spin glass phase in nearest-neighbor classical spin glass models in finite dimensions. The central question to be addressed is the nature of broken symmetry in these systems. This is still a subject of controversy, and although the issues surrounding it have become more sharply defined in recent years, it remains an open question.  I will explore this problem, introducing a mathematical construct called the metastate, to arrive at a unified picture of our current understanding of the structure of the spin glass phase in finite dimensions.
Ivan Corwin (12pm)
Title: Scaling limit of colored ASEP.
Abstract: Each site x in Z is initially occupied by a particle of color -x. Across each bond (x,x+1) particles swap places at rate 1 or q<1 depending on whether they are in reverse order (e.g. color 2 then 1) or order (color 1 then 2). This process describes a bijection of Z-->Z which starts maximally in reverse order and randomly drifts towards being ordered. Another name for this model is the "colored asymmetric simple exclusion process". I will explain how to use the Yang-Baxter equation along with techniques involving Gibbs line ensemble to extract the space-time scaling limit of this process, as well as a discrete time analog, the "colored stochastic six vertex model". The limit is described by objects in the Kardar-Parisi-Zhang universality class, namely the Airy sheet, directed landscape and KPZ fixed point. This is joint work with Amol Aggarwal and Milind Hegde.