# Spring 2015

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.

## Thursday, January 15, Miklos Racz, UC-Berkeley Stats

Title: Testing for high-dimensional geometry in random graphs

Abstract: I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels; we are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from an Erdos-Renyi random graph. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we use a bound on the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, and Ronen Eldan.

## Thursday, January 29, Arnab Sen, University of Minnesota

Title: Double Roots of Random Littlewood Polynomials

Abstract: We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions.

This is joint work with Ron Peled and Ofer Zeitouni.

## Thursday, February 19, Xiaoqin Guo, Purdue

Title: Quenched invariance principle for random walks in time-dependent random environment

Abstract: In this talk we discuss random walks in a time-dependent zero-drift random environment in $Z^d$. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators. This is a joint work with Jean-Dominique Deuschel and Alejandro Ramirez.

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## Wednesday, March 11, Sam Stechmann, UW-Madison, 2:25

Please note the unusual time and room.

Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena

Abstract: In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.

## Thursday, March 19, Mark Huber, Claremont McKenna Math

Title: Understanding relative error in Monte Carlo simulations

Abstract: The problem of estimating the probability $p$ of heads on an unfair coin has been around for centuries, and has inspired numerous advances in probability such as the Strong Law of Large Numbers and the Central Limit Theorem. In this talk, I'll consider a new twist: given an estimate $\hat p$, suppose we want to understand the behavior of the relative error $(\hat p - p)/p$. In classic estimators, the values that the relative error can take on depends on the value of $p$. I will present a new estimate with the remarkable property that the distribution of the relative error does not depend in any way on the value of $p$. Moreover, this new estimate is very fast: it takes a number of coin flips that is very close to the theoretical minimum. Time permitting, I will also discuss new ways to use concentration results for estimating the mean of random variables where normal approximations do not apply.

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