UVA Probability Seminar
Wednesdays, 4:30 - 5:30pm
Kerchof 317
Organizers: Christian Gromoll, Tai Melcher, & Leonid Petrov
Mathematics Recent Seminars Other Seminars Maps & Directions

Fall 2015

16 Sep Anirban Basak (Duke)
Limiting spectral distribution of non-Hermitian matrices
23 Sep Russell Lyons* (Indiana Bloomington) special colloquium, meets at 4pm
Random walks on groups and the Kaimanovich-Vershik conjecture
30 Sep Lionel Levine (Cornell)
Threshold state of the abelian sandpile
7 Oct Alexander Drewitz (Columbia)
The maximal particle of branching random walk in random environment
28 Oct Brian Rider (Temple)
A universality result for the random matrix hard edge
4 Nov Alexey Bufetov (MIT)
Asymptotics of probabilistic particle systems via Schur-generating functions
11 Nov Jonathon Peterson (Purdue)
Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment
* please note time, date, and/or location change

Abstracts

Limiting spectral distribution of non-Hermitian matrices

Anirban Basak (Duke)

Lately, there has been a huge interest in identifying the limiting distribution of the empirical measure of eigenvalues of non-Hermitian matrices. In this talk, we will consider sparse matrices. One example of sparse matrices, is the adjacency matrix of random oriented $d$-regular graphs. Motivated by this example, we (with Amir Dembo) consider sum of $d$ i.i.d. Unitary or Orthogonal matrices, and show that the limit is the circular version of Kesten-Mckay measure.

During the second part of the talk, I will discuss about an ongoing work with Mark Rudelson. Here we consider another class of sparse matrices, where entries are i.i.d., multiplied by Bernoulli($p_n$), where $p_n \rightarrow 0$. Under some moment assumptions on the entries, we show that the limit is the circular law.
top of page

Random walks on groups and the Kaimanovich-Vershik conjecture

Russell Lyons* (Indiana Bloomington) special colloquium, meets at 4pm

In the 1980s, much progress was made in understanding random walks on groups. In particular, characterizations of when there are non-constant bounded harmonic functions were given using asymptotic entropy. Later, Kaimanovich gave criteria for identifying all bounded harmonic functions. However, a conjecture of Kaimanovich and Vershik from 1979 remained open, with the first breakthrough by Erschler in 2011. We present a simple proof in joint work with Yuval Peres.
top of page

Threshold state of the abelian sandpile

Lionel Levine (Cornell)

A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile s_0 if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a "threshold state'' $s_T$ that topples forever. Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in $s_T$ in the limit as $s_0$ tends to negative infinity. I will outline how this conjecture was proved in http://arxiv.org/abs/1402.3283 by means of a Markov renewal theorem. This talk will be elementary and all sandpile terms will be defined.
top of page

The maximal particle of branching random walk in random environment

Alexander Drewitz (Columbia)

One-dimensional branching Brownian motion has been the subject of intensive research, in particular during the last decade. We consider the discrete space version of branching random walk and investigate the setting of a spatially random branching environment; in particular we are interested in the position of the maximal particle. Via the Feynman-Kac formula this is connected to fluctuations of the solutions to the parabolic Anderson model (i.e., the heat equation with a random potential) as well as to a randomized version of the Fisher-KPP equation. The Fisher-KPP equation is a fundamental reaction-diffusion partial differential equation which had originally been introduced in order to model the spread of an advantageous alelle in a population of a one-dimensional habitat.
top of page

A universality result for the random matrix hard edge

Brian Rider (Temple)

The hard edge refers to the distribution of the smallest singular value for certain ensembles of random matrices, or, and what is the same, that of the minimal point of a logarithmic gas constrained to the positive half line. For any "inverse temperature" and "quadratic" potential the possible limit laws (as the dimension, or number of particles, tends to infinity) was characterized by J. Ramirez and myself in terms of the spectrum of a (random) diffusion generator. Here we show this picture persists for more general convex polynomial potentials. Joint work with P. Waters.
top of page

Asymptotics of probabilistic particle systems via Schur-generating functions

Alexey Bufetov (MIT)

We study the asymptotic behavior of probabilistic particle systems coming from representation theory and statistical mechanics. In particular, we study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers and the Central Limit Theorem for the random counting measures describing the decomposition. It turns out that this problem is intrinsically connected with random lozenge and domino tilings, and also with free probability.
top of page

Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment

Jonathon Peterson (Purdue)

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.
top of page

Mathematics Recent Seminars Other Seminars Maps & Directions