Probability seminar

• Title: FINITELY DEPENDENT GRAPH HOMOMORPHISMS
• Speaker: Avi Levy (University of Washington)
• Time: 2:30 p.m., Monday, December 7, 2015
• Room: THO 325
• Abstract: When a child randomly paints a coloring book, adjacent regions receive distinct colors whereas distant regions remain independent. It took mathematicians until 2014 to replicate this effect, when Holroyd and Liggett discovered the first stationary $k$-dependent $q$-colorings. In this talk, I will discuss an extension of Holroyd and Liggett's construction which associates a canonical insertion procedure to every finite graph. The known colorings turn out to be diamonds in the rough: apart from multipartite analogues, they are the only $k$-dependent processes which arise from finite graphs in this manner. Time permitting, I will present extensions of these results to weighted graphs and shifts of finite type. Joint work with Alexander Holroyd.

• Title: COALESCENCE IN BRANCHING TREES AND BRANCHING RANDOM WALKS
• Speaker: Krishna B. Athreya (Iowa State University)
• Time: 2:30 p.m., Monday, November 30, 2015
• Room: THO 325
• Abstract: Consider a branching tree with one root at the origin. Assuming that at the $n$-th level there are at least two vertices. Pick two of them by simple random sampling without replacement. Now trace their lines back till they meet. Call that level $X_n$. In this talk we discuss the behavior of the distribution of $X_n$ as $n$ goes to infinity for the supercritical, critical, and subcritical Galton Watson branching trees. We also discuss the explosive case when the offspring mean is infinite and the offspring distribution is heavy tailed. We apply these results to study branching random walks. Some open problems will be described.

• Title: SOME PROPERTIES OF SUPER-BROWNIAN MOTION IN RANDOM ENVIRONMENTS
• Speaker: Guohuan Zhao (Peking University)
• Time: 2:30 p.m., Monday, November 23, 2015
• Room: THO 325
• Abstract: We consider a superprocess $X=\{X_t, t\ge 0\}$ in a random environment described by a Gaussian field $W(t,x)$ whose covariance function is given by $g(x,y)(t\wedge s)$. Suppose there exists a positive function $\bar{g}$ such that $g(x,y)\leq \bar{g}(x-y)$ and the process $X$ starts from $m$, the Lebesgue measure on $\mathbb{R}^d$. We first prove that for dimension $d\geq 3$ there exists $\delta>0$ such that if $\sup_x\int_{\mathbb{R}^d} G(x,y)\bar{g}(y)dy\le\delta$ then the distribution of $X_t$ converges weakly to a non-trivial distribution $\pi^m$ as $t\to\infty$ and $\int \mu\pi^m(d\mu)=m.$ Moreover $\pi^m$ is an invariant probability distribution of $X_t$. This result implies the Conjecture 1.4 in Mytnik and Jie Xiong's paper Local extinction for superprocesses in random environments'' is true. We also show if $g(x,y)=g(x-y)$ with $g\in C^2(\mathbb{R}^d)$ and $g(0)$ being large enough, then $X$ suffers local extinction. The third result is: if $d=1$, then $X_t$ has compact support for all $t$ if the initial measure does.

• Title: RANDOM GAMES
• Speaker: Alexander E. Holroyd (Microsoft Research)
• Time: 2:30 p.m., Monday, November 16, 2015
• Room: THO 325
• Abstract: Alice and Bob compete in a game of skill, making moves alternately until one or other reaches a winning position, at which the game ends. Or, perhaps neither player can force a win, in which case optimal play continues forever, and we say that the game is drawn.

What is the outcome of a typical game? That is, what happens if the game itself is chosen randomly, but is known to both players, who play optimally?

I will provide some answers (any many questions) in several settings, including trees, directed and undirected lattices, and point processes. The competitive nature of game play frequently brings out some of the subtlest and most fundamental properties of probabilistic models. We will encounter continuous and discontinuous phase transitions, hard-core models, probabilistic cellular automata, bootstrap percolation, maximum matching, and stable marriage.

Based on joint works with Riddhipratim Basu, Maria Deijfen, Irene Marcovici, James Martin and Johan Wastlund.

• Title: BOUNDARY HARNACK PRINCIPLE AND MARTIN BOUNDARY AT INFINITY FOR FELLER PROCESSES
• Speaker: Zoran Vondracek (University of Illinois, Urbana-Champaign, and University of Zagreb)
• Time: 2:30 p.m., Monday, November 9, 2015
• Room: THO 325
• Abstract: A boundary Harnack principle (BHP) was recently proved for Feller processes in metric measure spaces by Bogdan, Kumagai and Kwasnicki. In this talk I will first show how their method can be modified to obtain a BHP at infinity - a result which roughly says that two non-negative function which are harmonic in an unbounded set decay at the same rate at infinity. With BHP at hand, one can identify the Martin boundary of an unbounded set at infinity with a single Martin boundary point and show that, in case infinity is accessible, this point is minimal. I will also present analogous result for a finite Martin boundary point. The local character of these results implies that minimal thinness of a set at a minimal Martin boundary point is also a local property of that set near the boundary point. Joint work with Panki Kim and Renming Song.

• Title: THRESHOLD STATE OF THE ABELIAN SANDPILE
• Speaker: Lionel Levine (Cornell University)
• Time: 2:30 p.m., Monday, November 2, 2015
• Room: THO 325
• Abstract: 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.

• Title: CUTOFF FOR NON-BACKTRACKING RANDOM WALKS ON SPARSE RANDOM GRAPHS
• Speaker: Anna Ben-Hamou (University of Paris-Diderot and Microsoft)
• Time: 2:30 p.m., Monday, October 19, 2015
• Room: THO 325
• Abstract: A finite ergodic Markov chain exhibits cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. In this talk, we will consider the case of non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we will establish the cutoff phenomenon, determine its precise window, and prove that the cutoff profile approaches a remarkably simple, universal shape. This is a joint work with Justin Salez (Paris-Diderot).

• Title: BROWNIAN MOTION AND AN INERT PARTICLE
• Speaker: Clayton Barnes (University of Washington)
• Time: 2:30 p.m., Monday, October 12, 2015
• Room: THO 325
• Abstract: In his 2001 paper, Frank Knight constructed a process that moves with inertia and is impinged upon by a Brownian motion. Later, Bass, Burdzy, Chen and Hairer found the stationary distribution for the process extended to multi-dimensions. In this talk we give a discrete version of the inert particle, showing it converges in distribution after interpolation. In addition, we characterize the distribution of the maximum, and show that the sequence of local maximums forms a submartingale when a reflected boundary is introduced.

• Title: THE SMALLEST SINGULAR VALUE OF RANDOM MATRICES WITH INDEPENDENT ROWS
• Speaker: Konstantin Tikhomirov (University of Alberta)
• Time: 2:30 p.m., Monday, October 5, 2015
• Room: THO 325
• Abstract: We consider a classical problem of estimating the least singular value of random rectangular and square matrices with independent identically distributed entries as well as a more general model of random matrices with i.i.d. rows. The novelty of our results consists primarily in imposing very weak, or nonexisting, moment assumptions on the distribution of the entries. In particular, we show that the smallest singular value of a sufficiently tall $N\times n$ rectangular matrix with i.i.d. entries with certain condition on the Levy concentration function is of order $\sqrt{N}$ with a large probability. Further, we extend a fundamental result of Bai and Yin from early 1990-es on the limiting behaviour of the smallest singular value of rectangular matrices, by dropping the assumption of bounded 4th moment (a problem whether the assumption was necessary was discussed, in particular, in a book of Bai and Silverstein). Finally, we will discuss very recent joint results with Elizaveta Rebrova (University of Michigan) and Djalil Chafai (Paris Dauphine) regarding invertibility of random square matrices with i.i.d. heavy-tailed entries and Bai--Yin type convergence for matrices with i.i.d. log-concave rows, respectively. To obtain the results, we have used various techniques including special versions of the $\varepsilon$-net argument and rank one update approach of Batson--Spielman--Srivastava.

• Title: INVARIANT TRANSPORTS AND MATCHINGS OF RANDOM MEASURES
• Speaker: Guenter Last (Karlsruhe Institute of Technology)
• Time: 2:30 p.m., Monday, September 21, 2015
• Room: THO 134
• Abstract: The extra head problem formulated and solved by Liggett (2001) requires finding a head in an infinite sequence of independent fair coin tosses, without changing the (joint) distribution of the rest of the sequence. Holroyd and Peres (2005) constructed a a stable and shift invariant transport (marriage) of Lebesgue measure and an ergodic point process with unit intensity. Invariant matchings of stationary point processes were studied by Holroyd, Pemantle, Peres and Schramm (2008). In this talk we shall discuss matching and embedding problems for a two-sided standard Brownian motion and explain the close relationship of all these matching problems with shift invariant transports and Palm measures of stationary random measures. This talk is based on joint work with Peter Morters (Bath) and Hermann Thorisson (Reykjavik).