Marek Biskup (UCLA)
Title: A scaling limit of random walks among de-randomized conductances
Abstract: The theory of random walks in disordered reversible environments, a.k.a. conductance models, has advanced remarkably under the assumption that the environment is drawn at random from a law that is stationary and ergodic under translates. Indeed, the concept of the ``point of view of the particle'’ then enables ergodic theorems that conveniently produce limits that would be difficult to establish otherwise and the corrector method from stochastic homogenization yields a proof of scaling to Brownian motion. Unfortunately, the fact that both the input and the statement are stochastic make it all but impossible to decide whether the stated conclusion applies to a given (non-periodic) conductance configuration. I will show how to overcome this by de-randomizing both the mixing theory for the ``point of view of the particle'' and the corrector method. An invariance principle will then hold for any conductance configuration (modulo certain growth/decay restrictions) whose block averages converge and define an ergodic law on the space of environments.
Johannes Baeulmer (UCLA)
Title: The truncation problem for long-range percolation
Abstract: Consider long-range percolation on the integer lattice, where each pair of points x and y is connected with probability p(x,y). When are the long edges necessary for the existence of an infinite cluster? We discuss this question in the strong-decay regime and in the non-summable regime in dimension d >= 3.
Marco Carfagnini (UCSD)
Title: Onsager-Machlup functional for SLE loop measure
Abstract: Onsager-Machlup functional measures how likely a stochastic process stays close to a given path. This functional can be viewed as an infinite dimensional analogue of a probability density function. SLE is a family of measures on simple paths in the plane introduced by O. Schramm obtained from the Loewner transform of a multiple of Brownian motion. It is well-known that the Onsager-Machlup functional for Brownian motion is the Dirichlet energy. We show that the Onsager-Machlup of the SLE_k loop measure, for any 0 < k <4, is expressed using the Loewner energy and the central charge c(k) of SLE_k. Loewner energy is defined as the Dirichlet energy of the Loewner driving function of the loop, but it also has tight links to many other fields of mathematics. Our proof relies on the conformal restriction covariance of SLE, and on a relation between the renormalized Brownian loop measure and Werner measure. This is based on the joint work (arXiv: 2311.00209) with Yilin Wang (IHES).
Evgeni Dimitrov (USC)
Title: Global asymptotics for β-Krawtchouk corners processes
Abstract: I will discuss a two-parameter family of probability measures, that are called β-Krawtchouk corners processes. These measures are related to Jack symmetric functions, and can be thought of as integrable discretizations of β- corners processes from random matrix theory, or alternatively as positive temperature measures on lozenge tilings of infinite trapezoidal domains. For such models we show that the height function asymptotically concentrates around an explicit limit shape and prove that its limiting fluctuations are described by a pull-back of the Gaussian free field, which agrees with the one for Wigner matrices. If time permits, I will discuss the main tools we use to establish our results, which are certain multi-level loop equations. The talk is based on joint work with Alisa Knizel.
Puja Pandey (UCSB)
Title: On the Equivalence of Statistical Distances for Isotropic Convex Measures in Discrete and Continuous setting
Abstract: In convex geometry and its probabilistic aspects, many fundamental inequalities can be reversed up to universal constants in the presence of geometric properties. In this talk we will see that distances between probability measures are equivalent for convex measures. Distances include total variation distance, Wasserstein distance, Kullback-Leibler distance and Levy-Prokhorov distance. This extends a result of Meckes and Meckes (2014).
Jorge Garza Vargas (Caltech)
Title: From qualitative to quantitative results in Random Matrix Theory
Abstract: Since the 90's free probability has proven to be a helpful tool for predicting the asymptotic (i.e. as the dimension goes to infinity) spectral behavior of random matrices. More recent results show that, in fact, free probability (strongly) predicts the behavior of random matrices even for dimensions that are not too large.
I will present a new elementary proof technique that takes as input qualitative asymptotic results for random matrices (such as the aforementioned ones from the 90's) and turns them into quantitative rates of convergence (such as the aforementioned more recent results). This is joint work with Chi-Fang Chen and Joel Tropp.
Hao Wu (Tsinghua University)
Title: Multiple SLEs and Dyson Brownian motion
Abstract:
Multiple SLEs come naturally as the scaling limit of multiple interfaces in 2-dimensional statistical physics models. Dyson Brownian motion usually describes the movement of trajectory of independent Brownian motions under mutual repulsion. In this talk, we will describe the connection between multiple SLEs and Dyson Brownian motion. The talk has two parts.
In the first part, we take critical Ising model as an example and explain the emergence of multiple SLEs. We give the connection probabilities of multiple SLEs. Such probabilities are related to solutions to BPZ equations in conformal field theory.
In the second part, we explain the connection between multiple SLEs and Dyson Brownian motion. It turns out that, under proper time-parameterization, and conditioning on a rare event, the driving function of multiple SLEs becomes Dyson Brownian motion. Using such a connection, we may translate estimates on Dyson Brownian motion to estimates on multiple SLEs.