Georgia Institute of Technology College of Sciences

We study the spectral properties of matrices of long-range percolation model. These are N*N random real symmetric matrices H whose elements are independent random variables taking zero value with probability 1-\psi((i-j)/b), b\in \R^{+}, where \psi is an even positive function with \psi(t)<1 and vanishing at infinity. We show that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the ensemble we study in the limit N,b\to\infty. In the second part, we study the leading term of the correlation function of the resolvent G(z)=(H-z)^{-1} with large enough |Imz| in the limit N,b\to\infty, b=O(N^{\alpha}), 1/3<\alpha<1. We show that this leading term, when considered in the local spectral scale leads to an expression found earlier by other authors for band random matrix ensembles. This shows that the ensemble we study and that of band random matrices belong to the same class of spectral universality.

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n^h) copies of H can be made H-free by removing o(n^2) edges. We give a new proof which avoids Szemeredi’s regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.

This will be an expository talk on the study of orthogonal polynomials on the real line and on the unit circle. Topics include recurrence relations, recurrence coefficients and simple examples. The talk will conclude with applications of orthogonal polynomials to other areas of research.