After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy-Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Péché.

A Graph Construction Game is a Maker-Breaker game. Maker and Breaker take turns in choosing previously unoccupied edges of the complete graph K N . Maker's aim is to claim a copy of a given target graph G while Breaker's aim is to prevent Maker from doing so. In this paper we show that if G is a d-degenerate graph on n vertices and N > d 11 2 2d+9 n, then Maker can claim a copy of G in at most d 11 2 2d+7 n rounds. We also discuss a lower bound on the number of rounds Maker needs to win, and the gap between these bounds.

Abstract. We study the probability of a real-valued stationary process to be positive on a large interval [0, N ]. We show that if in some neighborhood of the origin the spectral measure of the process has density which is bounded away from zero and infinity, then the decay of this probability is bounded between two exponential functions in N . This generalizes similar bounds obtained for particular cases, such as a recent result by Artezana, Buckley, Marzo, Olsen.

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing?We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.

Let f be a zero mean continuous stationary Gaussian process on R whose spectral measure vanishes in a δ-neighborhood of the origin. Then the probability that f stays non-negative on an interval of length L is at most e −cδ 2 L 2 with some absolute c > 0 and the result is sharp without additional assumptions.1991 Mathematics Subject Classification. 60G10, 60G15.