Large deviations for random matrices

Large deviations and stochastic calculus for large random matrices

Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their s...

Large Deviations for Functions of Two Random Projection Matrices

In this paper two independent and unitarily invariant projection matrices P (N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size N converges to in nity. The result is formulated on the tracial state space TS(A) of the universal C -algebra A generated by two selfadjoint projections. The random pair (P (N); Q(N)) de...

Fluctuations and large deviations of some perturbed random matrices

General statement of the problem. The following algebraic problem is very classical: Let A and B be two Hermitian matrices of the same size. Assume we know the spectrum of each of them. What can be said about the spectrum of their sum A + B? The problem was posed by Weyl [1912]. He gave a series of necessary conditions, known as Weyl’s interlacing inequalities: if λ1(A)≥ · · · ≥ λn(A), λ1(B)≥ ·...

Large Deviations for Random Trees.

We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.

Large Deviations for Random Walk in Random