Orthogonal and Symplectic Random Matrix Ensembles: Universality and Asymptotics of the Partition Function

University Of Rochester, Rochester NY

Investigators

Abstract

Abstract Gioev The PI will work on four problems, all of which involve asymptotic techniques developed in Random Matrix Theory (RMT). Problems I and II are concerned with the proof of universality for orthogonal and symplectic ensembles of random matrices with varying polynomial weights on the line, and with polynomial weights on the half-line, respectively. Problem III deals with the rigorous proof of the asymptotic expansion for the partition function (free energy) for orthogonal and symplectic ensembles with varying polynomial weights on the line which has applications in graph enumeration problems arising in 2D quantum gravity and string theory. Finally, Problem IV concerns proving various asymptotic formulae for the entanglement entropy of fermion systems on the lattice and in the continuum (this problem has connections with RMT). Over the past fifty years it has become clear that Random Matrix Theory (RMT), initially introduced into the theoretical physics community by Wigner in 1950's as a model for the scattering of neutrons off large nuclei, has a long and extraordinarily varied list of applications not only in physics, but also in pure and applied mathematics, and in other sciences, such as cardiology. More precisely, it turns out that (the statistical properties of) various seemingly unrelated objects, such as vibration frequencies of elastic plates, bus arrival times when the drivers try to optimize the traffic flow, zeros of the Riemann zeta function, the distribution of phone numbers in a large phone directory, and heartbeat peaks, for example, are all remarkably well-described by eigenvalues of a large random matrix (RM). The fundamental question is: Why does RMT model such a broad variety of phenomena? The answer is that there should be certain limiting distribution laws for the eigenvalues of large RM's that are independent of the precise distribution of the matrix ensembles. This loose statement is known as the Universality Conjecture, and it can be viewed as an analog of the Central Limit Theorem in probability theory, but now for certain classes of correlated random variables (which are the eigenvalues of a RM). Universality questions arose very early on in RMT, and universality is widely believed to be true. However, a mathematical proof of universality for the so-called unitary ensembles was found only in the late 1990's. The PI and Deift have very recently proved universality in the remaining two cases, viz., the orthogonal and symplectic ensembles, in great generality. Problems I and II, which deal with continuation and extension of these results, will significantly extend our understanding of universality to the important case where the underlying equilibrium measure consists of more than one interval. The asymptotic expansion of the partition function for the three types of RM ensembles has important applications in mathematics and also in 2D quantum gravity and string theory. Very recently, in the unitary case, the expansion was rigorously established by Ercolani and McLaughlin. In Problem III, the PI plans to prove rigorously the expansion for general orthogonal and symplectic ensembles. This will have implications in 2D quantum gravity models which are not covered by the unitary case. In the emerging field of quantum computation, a subject currently of great interest, one of the basic quantities of study is the so-called entanglement entropy which measures the number of maximally entangled pairs that can be extracted from a given quantum state. In Problem IV, the PI plans to prove various results concerning asymptotic expansions of the entanglement entropy for fermions as the size of a subsystem becomes large. These results will considerably extend the understanding of the entanglement entropy for the lattice and also continuous fermion systems in higher dimensions.