# CUNY Probability Seminar, Fall 2016

The CUNY Probability Seminar is typically held on Tuesdays from 4:15 to 5:15 pm at the CUNY Graduate Center Math Department. The exact dates, times and locations are mentioned below.

Sept 13, 2016 at 4:15pm, Room 5417
Speaker: Eliran Subag, Weizmann Institute of Science
Title: Critical points and the Gibbs measure of pure spherical spin glasses

Abstract: For each $N$ let $H_N(x)$ be an isotropic Gaussian field on the $N$-dimensional unit sphere, meaning that $Cov(H_N(x),H_N(y))$ is a function, say $f_N$, of the inner product of $x$ and $y$. The spherical spin glass models of statistical mechanics are exactly the class of all such sequences of random fields where one assumes in addition that $f_N = Nf$ for some $f$ independent of the dimension $N$. To investigate the intricate landscape $H_N(x)$ one may study its critical points and values. Focusing on the pure p-spin models, I will review recent developments concerning the distribution of the number of critical values at a given height and the associated extremal point process. Combining these results with a local investigation of the behaviour of $H_N(x)$ in neighborhoods around the critical points, we obtain a detailed geometric picture for the Gibbs measure at low enough temperature: the measure concentrates on spherical “bands” around the deepest critical points. The main focus of the talk will be describing the latter and consequences of it. The talk is based on a joint work with Ofer Zeitouni.

Sept 20m 2016, at 4:15pm, Room 5417 Speaker: Haya Kaspi, Technion
Title: TBD

Sept 27, 2016, at 4:15pm, Room 5417
Speaker: Andrey Sarantsev, University of California – Santa Barbara
Title: Competing Brownian Particles

Abstract: Consider a finite or infinite system of Brownian particles on the real line. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank relative to other particles. These systems were introduced in Banner, Fernholz, Karatzas (2005). Since then, extensive theory was developed for finite systems. However, infinite systems proved to be much more difficult. We survey the latest results.

October 7, 2016, at 3:30pm, Math Thesis Room
NOTE special day/time/room !
Speaker: Marius Schmidt, Frankfurt
Title: From Derrida’s random energy model to branching random walks: from 1 to 3

Abstract: We study the extremes of a class of Gaussian fields with in-built hierarchical structure. The number of scales in the underlying trees depends on a parameter alpha in [0,1]: choosing alpha=0 yields the random energy model by Derrida (REM), whereas alpha=1 corresponds to the branching random walk (BRW). When the parameter alpha increases, the level of the maximum of the field decreases smoothly from the REM- to the BRW-value. However, as long as alpha<1 strictly, the limiting extremal process is always Poissonian.

Oct 18, 2016, at 4:15pm, Room 5417
Speaker: Wei Wu, NYU
Title: Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension.

Abstract: Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension.  This has been (partially) established for Ising and Phi^4 models for d \geq 4. We describe a simple spin model from uniform spanning forests in $\Z^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in d=4, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some “continuum escaping probability”. Based on joint works with Greg Lawler and Xin Sun.

Oct 25, 2016, at 4:15 pm, Room 5417                                                                                              Speaker:  Philip Matchett Wood, University of Wisconsin, Madison                                                       Title: Low-degree factors of random polynomials

Abstract: We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers. It is known that certain models are very likely to produce random polynomials that are irreducible, and our project can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random polynomial models. Interestingly, though the question comes from algebra and number theory, we primarily use tools from combinatorics, including additive combinatorics, and probability theory. We prove for a variety of models that it is very unlikely for a random polynomial with integer coefficients to have a low-degree factor—suggesting that this is, in fact, a universal behavior.  For example, we show that the characteristic polynomial of random matrix with independent +1 or −1 entries is very unlikely to have a factor of degree up to $n^{1/2-\epsilon}$.  The talk will discuss joint work with Sean O’Rourke and also joint work with Christian Borst, Evan Boyd, Claire Brekken, Samantha Solberg, and Melanie Matchett Wood.

Nov 1, 2016, at 4:15pm, Room 5417
Speaker: Jonathon Peterson, Purdue University                                                                                         Title: Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment

Abstract: For one-dimensional random walks in a random environment with positive limiting speed $v_0>0$ and with environments having both local drifts to the right and to the left, it is known that the large deviation probabilities of moving at a speed $v$ in $(0,v_0)$ which is slower than the typical speed decays slower than exponentially fast. In this talk I will consider precise asymptotics of these slowdown probabilities under the quenched measure. We will show that these quenched probabilities decay like $e^{-C_n(\omega) n^{-\gamma}}$ for some fixed $\gamma \in (0,1)$ and for some environment-dependent sequence $C_n(omega)$ which oscillates between $0$ and $\infty$. This confirms a conjecture of Gantert and Zeitouni. This talk is based on joint work with Sung Won Ahn.

Nov 8, 2016, at 4:15pm, Room 5417
Speaker: Chiranjib Mukherjee, WIAS                                                                                                            Title: Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in $d\geq 3$.

Abstract: We consider the smoothed multiplicative-noise stochastic heat equation                      $$d u_{\eps,t}= \frac 12 \Delta u_{\eps,t} d t + \beta \eps^{\frac{d-2}2} u_{\eps,t} d B_{\eps,t}$$
in dimension $d\geq 3$, where $B_{\eps,t}$ is a spatially smoothed space-time white noise, and $\beta>0$ is a parameter. We show the existence of a $\beta_c \in (0,\infty)$ so that the solution exhibits weak disorder when $\beta<\beta_c$ and strong disorder when $\beta>\beta_c$. The novelty of our proof is the introduction of the theory of the Gaussian multiplicative chaos (GMC) in the context of the continuous directed polymer.  (Joint work with A. Shamov (Weizmann Institute) and O. Zeitouni (Weizmann Institute/ NYU).

Nov 22, 2016, at 4:15 pm, Room  5417                                                                                        Speaker: Vladislav Vysotskiy, Imperial College, London, and St. Petersburg Division of Steklov Mathematical Institute)                                                                                                                                   Title: Large deviations of convex hulls of planar random walks

Abstract: We give logarithmic asymptotic bounds for large deviations probabilities for the perimeter of the convex hull of a planar random walk. These bounds are sharp for increments of many types, including Gaussian distributions and shifted or linearly transformed rotationally invariant distributions. For such random walks, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. These results on the perimeter are easily extended to the mean width of convex hulls of random walks in higher dimensions. Our method also allows to find the logarithmic asymptotics of large deviations probabilities for the area of the convex hull of planar random walks with rotationally invariant distributions of increments. This is a joint work with Arseniy Akopyan (IST Austria).

Nov 29, 2016, at 4:15 pm, Room 5417                                                                                             Speaker: Ilya Vinogradov, Princeton University                                                                                       Title: Point processes of number-theoretic origin

Abstract: The study of randomness of fixed objects is an area of active research with many exciting developments in the last few years. We will discuss recent results about affine and hyperbolic lattices, as well as lattices with points removed. Theorems about these processes address convergence of moments as well as rates of convergence, and their proofs
showcase a beautiful interplay between dynamical systems, mathematics physics, and number theory.

Dec 6, 2016, at 4:15 pm, Room 5417                                                                                            Speaker: Lisa Hartung, Courant Institute, NYU                                                                                          Title: The phase diagram of the branching Brownian motion energy model at complex temperature

Abstract: Branching Brownian motion (BBM) is a classical process in probability, describing a population of particles performing independent Brownian motion and branching according to a Galton Watson process. In this talk I will describe the phase diagram of the branching Brownian motion energy model at complex temperature. It turns out that there are three distinct phases in which the limiting behaviour of the partition function can be described (joint work with A. Klimovsky).