Unitary matrix integrals, symmetric polynomials, and long-range random walks

Non Unitary Random Walks

Motivated by the recent refutation of information loss paradox in black hole by Hawking, we investigate the new concept of non unitary random walks. In a non unitary random walk, we consider that the state 0, called the black hole, has a probability weight that decays exponentially in e−λt for some λ > 0. This decaying probabilities affect the probability weight of the other states, so that the...

Repeated Quantum Interactions and Unitary Random Walks

Among the discrete evolution equations describing a quantum system HS undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed by classical random variables in RN . The characterization we obtain is entirely algebraical in terms of the unitary operator driving the elementary interaction. We show that the solutions of these equ...

RANDOM WALKS ON SYMMETRIC SPACESAND INEQUALITIES FOR MATRIX SPECTRAAlexander

Unitary Integrals and Related Matrix Models

Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t’Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. O...

The Replica Limit of Unitary Matrix Integrals

We investigate the replica trick for the microscopic spectral density, ρs(x), of the Euclidean QCD Dirac operator. Our starting point is the low-energy limit of the QCD partition function for n fermionic flavors (or replicas) in the sector of topological charge ν. In the domain of the smallest eigenvalues, this partition function is simply given by a U(n) unitary matrix integral. We show that t...