Southeastern Probability Conference

Curie-Weiss Model under $l^p$ constraint

We consider the Curie-Weiss model on a complete graph with $n$ vertices, but spin configurations $(\sigma_i)_{i=1}^n$ constrained to satisfy $\sum_{i=1}^n|\sigma_i|^p = n$ for some $p>0$. For $p=\infty$, this reduces to the classical Ising Curie-Weiss model. The model behaves differently depending on the value of $p$ -

 i) There exists an increasing sequence of real numbers $0=p_1<p_2<p_3<\cdots <1$ converging to $1$, such that for $p \in (p_{k-1},p_k)$, a typical configuration has exactly $k$ 'big' values.

ii) For $p\ge 1$, there exists a critical inverse temperature $\beta_c(p)$, such that the magnetization is concentrated around $0$ for $\beta<\beta_c(p)$ and satisfies a Gaussian CLT after scaling. However, $\beta_c(p)\to 3$ as $p\to\infty$ and $\beta_c(\infty)=1$.

iii) For $p\ge 2$ and $\beta > beta_c$, the magnetization is not concentrated at zero.

We also generalize the model using a self-scaled Hamiltonian for a large class of symmetric spin distributions. One main ingredient is a generalized Hubbard-Stratonovich transform, developed using a coupling of log-gamma distributions. Based on joint work with Daesung Kim.

Fractal Geometry of the Parabolic Anderson Model in higher dimension

Parabolic Anderson model (PAM) is one of the prototypical frameworks for modelling conduction of electrons in crystals filled with defects. Intermittency of the peaks of the PAM is one of the widely studied topics in the last few decades and it holds close ties with the phenomenon of Anderson localization.  We show that the peaks of the PAM in dimension 2 and 3 are macroscopically multifractal. More precisely, we prove that the spatial peaks of the PAM have infinitely many distinct values and we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor) of those peaks. As a byproduct, we obtain the exact spatial asymptotics of the solution of the PAM. We also study the spatio-temporal peaks of the PAM and show their macroscopic multifractality. Some of the major tools used in our proof techniques include paracontrolled calculus and tail probabilities of the largest point in the spectrum of the Anderson Hamiltonian. This talk is based on a joint work with Jaeyun Yi.

Polymer phase transitions at fixed temperature

We consider the standard directed polymer model in space dimensions strictly greater than 2. It is a classical fact that the polymer is in the weak-disorder phase at high temperature and in strong disorder at low temperature. We introduce a natural notion of directionality into the model and show that at some fixed temperatures, the polymer is simultaneously in the weak disorder phase in some directions, while being in strong disorder in others. 

Generalized Front Propagation for Stochastic Spatial Models

I will begin by presenting a framework introduced by Barles and Souganidis which provides a global-in-time interpretation of front propagation, past the time of singularities. Using moment duality, this framework can be used to prove that a variety of stochastic spatial population models asymptotically exhibit phase separation governed by generalized mean curvature flow. This talk is based on joint work in progress with Thomas Hughes. 

Hole radii for the Kac polynomials

The Kac polynomial is one of the most studied models of random polynomials. It has the form $$f_n(x) = \sum_{i=0}^{n} xi_i x^i$$. It is known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. In a beautiful work, Michelen showed that the holes at $\pm 1$ are of order $1/n$. We show that in fact, all the hole radii are of the same order.  

Joint work with Hoi Nguyen. 

Generalized Ray-Knight theorems, and convergence and non-convergence of self-interacting random walks

A number random walk models with spatially local self-interactions have been successfully studied by analyzing the embedded Markovian structure of the directed edge local times. Often this takes the form of proving generalized Ray-Knight theorems for the random walk. Over 20 years ago, Tóth proved such generalized Ray-Knight theorems for a large class of self-interacting random walks and noted that for two models which he called asymptotically free walks and polynomially self-repelling walks the generalized Ray-Knight theorems were consistent with convergence of the walks to a process called Brownian motion perturbed at extrema (BMPE). I will discuss a recent work with Elena Kosygina and Tom Mountford where we prove that (1) the asymptotically free walks do converge to a BMPE, and (2) the polynomially self-repelling random walks cannot converge to a BMPE. The negative result for the polynomially self-repelling walk is somewhat unexpected and comes as a result of improved joint Ray-Knight like theorems for the random walk. 

Zeros of lacunary random polynomials

Zeros of Kac polynomials are asymptotically uniformly distributed near the unit circumference with probability one, under very mild assumptions on random coefficients. We consider random polynomials with gaps, i.e., some missing powers of the variable. Polynomials with gaps are often named as lacunary or sparse, and sometimes as Müntz polynomials or fewnomials. We show how the gap structure enhances the phenomenon of clustering to the unit circle for the zeros of lacunary random polynomials, with larger gaps resulting into stronger clustering. We also discuss how the gaps influence the number of real zeros for random polynomials with i.i.d. Gaussian coefficients.

Stochastic synchronization of solutions to the KPZ equation

I will present recent results on the uniqueness, ergodicity, and attractiveness of stationary solutions to the Kardar-Parisi-Zhang (KPZ) equation on the real line. It is known that this equation admits Brownian motion with a linear drift as a stationary solution. We show that these solutions are attractive, a principle known as one force--one solution (1F1S): the solution to the KPZ equation started in the distant past from an initial condition with a given velocity will converge almost surely to a Brownian motion with that same drift.  As a result, we deduce that these stationary measures are in fact totally ergodic. Furthermore, we can couple all these stationary solutions so that the above attractiveness holds simultaneously (i.e. on a single full measure event) for all but a countably infinite (random) set of asymptotic velocities.   This is joint work with Chris Janjigian and Timo Seppalainen. Part of the work is also joint with Tom Alberts.