# Limit shapes of restricted integer partitions under non—multiplicative conditions

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Abstract: Limit shapes are an increasingly popular way to understand

the large—scale characteristics of a random ensemble. The limit shape

of unrestricted integer partitions has been studied by many authors

primarily under the uniform measure and Plancherel measure. In

addition, asymptotic properties of integer partitions subject to

restrictions has also been studied, but mostly with respect to

\emph{independent} conditions of the form ``parts of size $i$ can

occur at most $a_i$ times.” While there has been some progress on

asymptotic properties of integer partitions under other types of

restrictions, the techniques are mostly ad hoc. In this talk, we will

present an approach to finding limit shapes of restricted integer

partitions which extends the types of restrictions currently

available, using a class of asymptotically stable bijections. This is

joint work with Igor Pak.

# Rare events for point process limits of random matrices.

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The Gaussian Unitary and Orthogonal Ensembles (GUE, GOE) are some of the most studied Hermitian random matrix models. When appropriately rescaled the eigenvalues in the interior of the spectrum converge to a translation invariant limiting point process called the Sine process. On large intervals one expects the Sine process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We solve the large deviation problem which asks about the asymptotic probability of seeing a different density in a large interval as the size of the interval tends to infinity. Our proof works for a one-parameter family of models called beta-ensembles which contain the Gaussian orthogonal, unitary and symplectic ensembles as special cases.

# Brownian Motion in Three Dimensions Conditioned to have the Origin as a Recurrent Point

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I will discuss aspects of a polymer model based on three-dimensional Brownian motion conditioned to hit (and keep returning to) the origin

introduced by Mike Cranson and co-authors. The construction and certain properties of this conditioned Brownian motion will be approached from two points of view (i) Dirichlet forms, and (ii) excursion theory. The latter gives a nice interpretation of the Johnson-Helms example from martingale theory. It turns out that this diffusion process is not a semimartingale, even though its radial part is just a one-dimensional Brownian motion reflected at the origin.

(This talk is based on joint work with Liping Li of Fudan University.)

# Dissipation and high disorder

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We show that the total mass, i.e. the sum over all points in the d-dimensional integer lattice of the solution to the parabolic Anderson model with initial function the point mass at the origin goes to zero in the high disorder regime. This talk is basedon joint work with L. Chen, D. Khoshnevisan, and K. Kim.

# Heavy tails and one-dimensional localization.

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In this talk we address a question posed several years ago by G. Zaslovski: what is the effect of heavy tails of one-dimensional random potentials on the standard objects of localization theory: Lyapunov exponents, density of states, statistics of eigenvalues, etc. Professor G. Zaslovski always expressed a special interest in the models of chaos containing strong fluctuations, e.g. L ́evy flights? We’ll consider several models of potentials constructed by the use of iid random variables which belong to the domain of attraction of the stable distribution with parameter α < 1. This is a report on joint work with S. Molchanov.

# Random zero sets under repeated differentiation of an analytic function

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We study the result of repeatedly differentiating a random entire function whose zeros are the points of a Poisson process of intensity 1 on $\mathbb{R}$. Based on joint work with Robin Pemantle.

# Conditional Speed of Branching Brownian Motion, Skeleton Decomposition and Application to Random Obstacles

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We study a d-dimensional branching Brownian motion, among obstacles scattered

according to a Poisson random measure with a radially decaying intensity. Obstacles

are balls with constant radius and each one works as a trap for the whole motion when

hit by a particle. Considering a general offspring distribution, we derive the decay

rate of the annealed probability that none of the particles hits a trap,

asymptotically, in time.

This proves to be a rich problem, motivating the proof of a general result about the speed

of branching Brownian motion conditioned on

non-extinction. We provide an appropriate `skeleton-decomposition' for the underlying

Galton-Watson process when supercritical, and show that the `doomed particles' do

not contribute to the asymptotic decay rate.

This is joint work with Mine Caglar and Mehmet Oz.

# Asymmetric Cauchy distribution and the destruction of large random recursive trees

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The probability mass function $1/j(j+1)$ for $j\geq 1$ belongs to the domain of attraction of a completely asymmetric Cauchy distribution.

The purpose of the talk is to review some of applications of this simple observation to limit theorems related to the destruction of random recursive trees.

Specifically, a random recursive tree of size $n+1$ is a tree chosen uniformly at random amongst the $n!$ trees on the set of vertices $\{0,1, 2, ..., n\}$ such that the sequence of vertices along any segment starting from the root $0$ increases. One destroys this tree by removing its edges one after the other in a uniform random order. It was first observed by Iksanov and M\"ohle that the central limit theorems for the random walk with step distribution given above explains the fluctuations of the number of cuts needed to isolate the root. We shall discuss further results in the same vein.

# Viscosity Solutions for Forward SPDEs and PPDEs

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In this talk we introduce a notion of stochastic viscosity solution

for a class of fully nonlinear SPDEs and the corresponding Path-dependent

PDEs (PPDEs). The definition is based on our new accompanying work

on the pathwise stochastic Taylor expansion, using a variation of the path-

derivatives initiated by Dupire. As a consequence this new definition of the

viscosity solution is directly in the pathwise sense, without having to invoke

the stochastic characteristics for the localization. The issues of consistency,

stability, comparison principles, and ultimately the well-posedness of the

stochastic viscosity solutions will be discussed under this new framework.

This is a joint work with Rainer Buckdahn and Jianfeng Zhang.