Event Detail

Event Type: 
Department Colloquium
Monday, April 22, 2013 - 09:00
Kidder 364

Speaker Info

Courant Institute of Mathematical Sciences, New York University

Stein's method is a semi-classical tool for establishing distributional convergence with explicit rates, particularly effective in problems involving complex dependencies. I will briefly describe the main idea behind the method and explain a new approach for applying the method for Gaussian approximation. Our main example will be the number of local maxima in the energy landscape for `large average' sub-matrices of an $n\times n$ Gaussian random matrix. Two sub-matrices, with the same size, are neighbors if they share either the same set of rows or the same set of columns. We define the `height' of a sub-matrix to be its average entry and given an integer $k$, we look at the random landscape on the graph structure with all possible $k\times k$ submatrices. For fixed $k$, behavior of the number of local maxima has very atypical behavior with mean  $\approx n^k$ and fluctuation $\approx n^{k - k/(2k+2)}$, with logarithmic corrections. We will use the new variant of Stein's method to prove a Central Limit Theorem in this super-diffusive case. Based partly on joint work with Shankar Bhamidi and Andrew Nobel.