Seminar Schedule:
The seminar meets on Tuesdays from 4:30 to 5:30 pm EST.
When a new point Xn arrives, it connects to the nearest point among X1,…,Xn-1.
This forms a sequence of trees (Tn). If you are given the sequence of unlabeled trees (Tn), can you recover information about the underlying space, in particular the dimension d?
If initially two points colored in red and blue are placed on the space, using the same procedure as above we color the new arriving points according to the color of its parent. We shall look at the interfaces between the blue points and the red points answering a few conjectures of Aldous.
Based on ongoing joint works with Jerome Casse and Alice Contat and with Anne-Laure Badesvant, Guillaume Blanc and Arvind Singh.
Abstract: While the tail behavior of linear functions of i.i.d. random variables is well understood by now, in many natural settings one encounters non-linear functions of the same. Two important examples are triangle or other subgraph counts in Erdos-Renyi random graphs, where each edge in a graph of size n occurs independently with probability p, and eigenvalues of random matrices whose entries are i.i.d random variables up to symmetry constraints. A particularly central subclass of examples include sparse instances, i.e., when the edge density in a random graph goes to zero with the graph size, or analogously when only a vanishing fraction of the random matrix entries are non-zero. However, while the study of such non-linear random variables have led to spectacular progress in the theory of concentration of measure and large deviations, both for dense and sparse models, arguably the most interesting, owing to their connections to various models of statistical mechanics, sparse cases of constant average degree (e.g., Erdos-Renyi random graph of size n with p=O(1/n )), have generally remained out of reach.
Focusing on this particular sparsity regime, we will present several recent results pinning down the tail behavior of various non-linear observables, thereby settling long standing questions.
Abstract: We consider the Erdős–Rényi graph on N vertices with edge probability d/N. It is well known that the structure of this graph changes drastically when d is of order log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of the eigenvectors remains delocalized. In this talk, I will present the phase diagram depicting these localized and delocalized phases and our recent progress in establishing it rigorously.This is based on joint works with Raphael Ducatez and Antti Knowles.
potential densities u(x,y) with respect to some ψ-finite measure on T. Let g and f be finite
excessive functions for Y . Set ug,f(x,y) = u(x,y) + g(x)f(y), x,y ∈ T. Let η = {η(x),x ∈ T} be a mean zero Gaussian process with covariance u(x,y) and
set ψ2(x) = E(η(x) – η(0))2.In this talk many explicit examples are given. In most of them Y is a symmetric Lévy
process or a diffusion, that is killed at the end of an independent exponential time or the first
time it hits 0.
Under general smoothness conditions on the corresponding functions g, f and ψ2, laws of
the iterated logarithm are found for Xk∕2 = {Xk∕2(t),t ∈ T}, a k∕2–permanental process with kernel {ug,f(x,y),x,y ∈ T}, of the following form:
For any d ∈ T and all integers k ≥ 1,
lim sup x↓0(|Xk∕2(d + x) – Xk∕2(d)|)(2ψ2(x) log log 1∕x)–1∕2 = (2Xk∕2(d))1∕2, a.s.
Using these limit theorems and the Eisenbaum Kaspi Isomorphism Theorem, laws of the
iterated logarithm are found for the local times of Markov processes with potential densities
the same as or closely related to {ug,f(x,y),x,y ∈ T}.
Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, closely connected to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.
Joint work with Shirshendu Ganguly.
Title: Algorithms for the Potts model on expander graphs
Abstract: We give algorithms to sample and approximate the ferromagnetic Potts model on d-regular expander graphs. We require much weaker expansion than in previous works. The main improvements come from a significantly sharper analysis of abstract polymer models, using extremal graph theory and applications of Karger’s algorithm to counting cuts. It is #BIS-hard to approximate the partition function at low temperatures on d-regular graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. This is joint work with Charlie Carlson, Ewan Davies, Alexandra Kolla, Aditya Potukuchi and Corrine Yap.
Speaker: Duncan Dauvergne
*Online seminar*
Title: Infection spread in a sea of random walks
Abstract: We consider a class of interacting particle systems where particles perform independent random walks on and spread an infection according to a susceptible-infected-recovered model. I will discuss a new method for understanding this model and some variants. A highlight of this method is that if recovery rate is low, then the infection survives forever with positive probability, and spreads outwards linearly leaving a herd immunity region in its wake. Based on joint work with Allan Sly.
Abstract: Abelian networks a class of models from statistical physics introduced as toy models to help understand the complex behavior exhibited by forest fires and avalanches. In this talk we will introduce a few different Abelian networks, give examples of the techniques used to study them and discuss some recent results.
Abstract: We discuss a more recent approach to proving Selberg’s central limit theorem for the Riemann zeta function on the critical line. We will also see that it is possible to extend this approach to a multivariate context. The rate of convergence involved will come up, but there is still more work to be done here.
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