# CUNY Probability Seminar, Fall 2017

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. If you want to talk at the seminar, or want to be added to/removed from the seminar mailing list, then please contact the Seminar Coordinator

Ivan Matic (http://mfe.baruch.cuny.edu/maticivan)

Tuesday, September 12, 2017, 4:15 PM, Rm. 5417

Speaker: Jack Hanson, City College, CUNY

Title: The chemical distance in 2d critical percolation

Abstract: In 2d critical percolation, a fraction 1/2 of edges are removed from Z^2, and one considers the resulting subgraph. The intrinsic or “chemical” distance in this graph is known to grow superlinearly in the Euclidean distance. For instance, the length S_n of the shortest crossing of a sidelength n square is typically at least n^{1+\delta}. A trivial upper bound is provided by L_n, the length of the lowest crossing of the square; on the triangular lattice, L_n scales as n^{4/3}. We will discuss work providing the first nontrivial power law improvement to this upper bound, showing in particular that E[S_n] < n^{-\delta} E[L_n].

Tuesday, September 26, 2017, 4:15 PM, Rm. 5417

Speaker: Atilla Yilmaz, NYU and Koc University

Title: Nonconvex homogenization of a class of one-dimensional stochastic viscous Hamilton-Jacobi equations

Abstract: I will present a homogenization result for a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs have representations involving exponential expectations of controlled Brownian motion in random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and it turns out to be explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in random potential. The proof of homogenization relies on identifying asymptotically optimal control policies and constructing correctors from the PDE perspective. Based on recent joint work with Elena Kosygina and Ofer Zeitouni.

Tuesday, October 3, 2017, 4:15 PM, Rm. 5417

Speaker: Toby Johnson, College of Staten Island

Title: Concentration via size biased couplings

Abstract: The spectral gap of a random d-regular graph on n vertices has been a major topic of research over the last thirty years. We show that the second-highest eigenvalue is O(sqrt(d)) with high probability so long as d = O(n^(2/3)), an improvement on previous results which held only up to d = o(n^(1/2)). The focus of this talk isn’t this result, which has since been improved on by Tikhomirov and Youssef, but rather the framework we developed along the way to prove concentration inequalities using the method of size biased couplings. This method is general, easy to apply, and should be broadly useful. Joint work with Nicholas Cook and Larry Goldstein.

Tuesday, October 10, 2017, 4:15 PM, Rm. 5417

Speaker: Joe Po-Chou Chen, Colgate University

Title: Local ergodicity in the exclusion process on an infinite weighted graph

Abstract: Hydrodynamic limit of the weakly asymmetric exclusion process has been well studied in the Euclidean setting. In this talk I will describe recent progress on extending this program to non-translationally-invariant settings. My focus will be on the local ergodic (or coarse graining) argument which justifies the replacement of the microscopic observables by their macroscopic averages, with superexponentially small cost.

The two main parts of the argument are:
1) A Sobolev-type inequality called the “moving particle lemma,” based on the octopus inequality of Caputo, Liggett, and Richthammer, for the exclusion process on any finite weighted graph. We believe that the lemma is optimal in the absence of spatial symmetries.
2) Using the moving particle lemma, we can implement the so-called “two-blocks estimate” to carry out the coarse graining procedure, on any strongly recurrent weighted graph. This includes fractals, many trees, and random graphs arising from critical percolation.

These form the technical backbone of the proof of the hydrodynamic limit of the boundary-driven exclusion process on the Sierpinski gasket, which is being written in collaboration with Michael Hinz and Alexander Teplyaev.

Tuesday, October 24, 2017, 4:15 PM, Rm. 5417

Speaker: Michael Carlisle, Baruch College, CUNY

Title: Conditioning while Forgetting or Misremembering: “Faulty Memory” Filtrations

Abstract: We generalize the definition of a filtration of a stochastic process. This explores and continues work on filtration expansions and shrinkages, and “loss of information” of a filtration, in the vein of Stricker, Yor, Föllmer, Protter, and others. We refer to these objects as “faulty memory” structures, as opposed to the “perfect memory” of a filtration, regardless of “how much” information is present in such a filtration relative to the “universal” sigma-algebra. We examine the martingale and Markov properties, and stationarity and independence of increments, when evaluating common processes under such “faulty” filtrations. We close with some examples.

Tuesday, November 7, 2017, 4:15 PM, Rm. 5417

Speaker: Thomas Leble, NYU

Title: Statistical physics approach for one and two-dimensional log-gases

Abstract: Log-gases are systems of particules with logarithmic interaction. They can be shown to coincide with the law of eigenvalues for certain random matrices, but are also interesting statistical physics models.
I will present results, obtained with Sylvia Serfaty, concerning their *microscopic* behavior as well as the fluctuations around the *macroscopic* equilibrium. These rely on a “energy” approach, some of which can be extended to more general interactions and dimensions. In the one-dimensional (Hermitian random matrix) case, the limit microscopic behavior is known as the Sine-beta point process, whose definition involves stochastic analysis. I will mention a possible description of Sine-beta in more physical terms.

Tuesday, November 21, 2017, 4:15 PM, Rm. 8203 (Please note the non-standard location!)

Speaker: Michael Damron, Georgia Tech

Title: TBA

Abstract: TBA

Tuesday, November 28, 2017, 4:15 PM, Rm. 5417

Speaker: Reza Gheissari, NYU

Title: Mixing Times of Critical Potts Models

Abstract: The Potts model is a generalization of the Ising model to $q\geq 3$ states; on $\mathbb Z^d$ it is an extensively studied model of statistical mechanics, known to exhibit a rich phase transition for $d=2$ at some $\beta_c(q)$. Specifically, the Gibbs measure on $\mathbb Z^2$ exhibits a sharp transition between a disordered regime at high temperature ($\beta>\beta_c$ and an ordered regime at low temperature ($\beta>\beta_c(q)$). At $\beta=\beta_c(q)$, when $q\leq 4$, the Potts model has a continuous phase transition and its scaling limit is believed to be conformally invariant; when $q>4$, the phase transition is discontinuous and the ordered and disordered phases coexist.

I will discuss recent progress, joint with E. Lubetzky, in analyzing the time to equilibrium (mixing time) of natural Markov chains (e.g., heat-bath/Metropolis) for the 2D Potts model, where the mixing time on an $n\times n$ torus should transition from $O(\log n)$ at high temperatures to exponential in $n$ at low temperatures, via a critical slowdown at $\beta=\beta_c$ of $n^z$ when $q\leq 4$ and exponential in $n$ when $q>4$.

Tuesday, December 5, 2017, 4:15 PM, Rm. 5417

Speaker: Dan Pirjol, JPM

Title: Large deviations for time-averaged diffusions and Asian options

Abstract: Time integrals of one dimensional diffusions appear in the statistical mechanics of disordered systems, actuarial science and mathematical finance. The talk presents large deviations asymptotics for the time-average of a diffusion in the small time limit. This is derived using the classical pathwise large deviations result for diffusions obtained by Varadhan in 1967, and the contraction principle. The rate function is expressed as a variational problem, which is solved explicitly. The result is applied to the short maturity asymptotics of Asian options in mathematical finance. Based on work with Lingjiong Zhu from Florida State University.