# Seminars & Events for Probability Seminar

##### Increasing subsequences on the plane and the Slow Bond Conjecture

For a Poisson process in R^2 with intensity 1, the distribution of the maximum number of points on an oriented path from (0,0) to (N,N) has been studied in detail, culminating in Baik-Deift-Johansson's celebrated Tracy-Widom fluctuation result. We consider a variant of the model where one adds, on the diagonal, additional points according to an independent one dimensional Poisson process with rate \lambda. The question of interest here is whether for all positive values of \lambda, this results in a change in the law of large numbers for the the number of points in the maximal path. A closely related question comes from a variant of Totally Asymmetric Exclusion Process, introduced by Janowsky and Lebowitz. Consider a TASEP in 1-dimension, where the bond at the origin rings at a slower rate r<1.

##### Extrema of the planar Gaussian Free Field: convergence of the maximum using hidden tree structures

In a recent work, Bramson, Ding and the speaker proved that the maximum of the Gaussian free field in a discrete box of side $N$, centered around its mean, converges in distribution to a shifted Gumbel. The proof uses branching random walks, modified branching random walks, and a modification of the classical second moment method. Underlying the proof is a hidden rough tree structure. I will explain the terms in the abstract, the structure of the proof, and will sketch applications to other related problems.

##### Fluctuations of the stationary Kardar-Parisi-Zhang equations

Up to a random height shift, two-sided Brownian motion is invariant for the Kardar-Parisi-Zhang equation. In this talk we describe recent results with Borodin, Ferrari and Veto through which we use Macdonald processes and the geometric Robinson-Schensted-Knuth correspondence to compute the distribution of this height shift and demonstrate cube-root fluctuations in large time, with a universal limit law. This also relates to the two-point correlation function and super-diffusivity of the stochastic Burgers equation.-

##### Infinite Dimensional Stochastic Differential Equations for Dyson's Brownian Motion

Dyson's Brownian Motion (DBM) describes the evolution of the spectra of certain random matrices, and is governed by a system of stochastic differential equations (SDEs) with a singular, long-range interaction. In this talk I will outline a construction of the strong solution of the infinite dimensional SDE that corresponds to the bulk limit of DBM. This is a pathwise construction that allows an explicit space with generic configurations. The ideas used further lead to a proof of the pathwise uniqueness of the solution and of the convergence of the finite to infinite dimensional SDE.

##### Equivalence of decay of correlations, the log-Sobolev inequality, and of the spectral gap

In this talk we consider a lattice system of unbounded real-valued spins, which is described by its Gibbs measure mu. We discuss how a known result for finite-range interaction is generalized to infinite-range. The result states that it is equivalent: The correlations of the Gibbs measure mu decay, the Gibbs measure mu satisfies a log-Sobolev inequality uniformly in the systems size and the boundary values, and mu satisfies a uniform spectral gap. Such a statement is interesting, because it connects a static property of the equilibrium state mu of the system to a dynamic property of the system i.e. how fast the Glauber dynamics converges to equilibrium.

##### Central Limit Theorems and Lee-Yang Zeros

I will describe some old and some new results (joint work with Pittel, Ruelle, Speer) on how to derive Central Limit Theorems and even Local Central Limit Theorems from information about the location of zeros of the generating function. Applications to the distribution of eigenvalues of random matrices, graph counting polynomials, and statistical mechanical systems will be given.

##### Extremal individuals in branching systems

Branching processes have been subject of intense and fascinating studies for a long time. In my talk I will present two problems in order to highlight their rich structure and various technical approaches in the intersection of probability and analysis. Firstly, I will present results concerning a branching random walk with the time-inhomogeneous branching law. We consider a system of particles, which at the end of each time unit produce offspring randomly and independently. The branching law, determining the number and locations of the offspring is the same for all particles in a given generation. Until recently, a common assumption was that the branching law does not change over time. In a pioneering work, Fang and Zeitouni (2010) considered a process with two macroscopic time intervals with different branching laws.

##### Diffusions with Rough Drifts and Stochastic Symplectic Maps

**This is a joint seminar with Analysis of Fluids and Related Topics. **According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a $d$-dimensional diffusion with a drift in $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $d/r+2/q<1$. As an application one show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function $H$ satisfies $\nabla H\in L^{r,q}$ with $d/r+2/q<1$.

##### Effect of initial conditions on mixing for the Ising Model

Recently, the ``information percolation'' framework was introduced as a way to obtain sharp estimates on mixing for the high temperature Ising model, and in particular, to establish cutoff in three dimensions up to criticality from a worst starting state. I will describe how this method can be used to understand the effect of different initial states on the mixing time, both random (''warm start'') and deterministic. Joint work with Allan Sly.

##### Roots of random polynomials: Universality and Number of Real roots

Estimating the number of real roots of a polynomial is among the oldest and most basic questions in mathematics.The answer to this question depends very strongly on the structure of the coefficients, of course. What happens if we choose the coefficients randomly? In this case, the number of real roots become a random variable, whose value is between 0 and the degree of the polynomial. Can one understand this random variable? What is its mean, variance, and limiting distribution? Random polynomials were first studied by Waring in the 18th century. In the 1930s, Littlewood and Offord started their famous studied which led to the surprising fact that in general a random polynomials with iid coefficients have (with high probability), order log n real roots.

##### Loop-Erased Random Walk

I will discuss recent work with Christian Benes and Fredrik Viklund on the convergence of the Green's function of planar loop-erased random walk and the relationship to the convergence to the Schramm-Loewner evolution in the natural parametrization (Minkowski content). I will not assume that people know what the words in the last sentence mean --- I will describe the result and its importance.

##### Tails of Random Projections and the Atypicality of Cramer’s Theorem

In recent years, there has been much interest in the interplay between geometry and probability in high-dimensional spaces. One striking result that has been established is a central limit theorem for random projections of random variables that are uniformly distributed in high-dimensional convex sets. It is therefore natural to ask if such random projections also exhibit other properties satisfied by sums of iid random variables, such as large deviation principles. As a step in this direction, we establish (quenched and annealed) large deviation principles for random projections of sequences of random vectors that are uniformly distributed on $l^p$ balls in $n$-dimensional Euclidean space. We also prove several interesting consequences of these principles, including the perhaps surprising fact that the well known Cramer’s theo