Abstract
We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called L^2-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher–Hartwig singularities. Using Riemann–Hilbert methods, we prove a rather general Fisher–Hartwig formula for one-cut regular unitary invariant ensembles.
Highlights
1.1 Main resultLog-correlated Gaussian fields, namely Gaussian random generalized functions whose covariance kernels have a logarithmic singularity on the diagonal, are known to show up in various models of modern probability and mathematical physics—e.g. in combinatorial models describing random partitions of integers [35], random matrix theory [31,34,60], lattice models of statistical mechanics [41], the construction of conformally invariant random planar curves such as stochastic Loewner evolution [4,63], and growth models [9] just to name a few examples
While our results do not extend to the full range of values of β where one expects the result to be valid, we believe that an appropriate modification of the methods of this paper eventually will yield the result in its full generality
One could consider the behavior of the characteristic polynomial in a microscopic neighborhood of a fixed point, where one might expect it to be asymptotically a random analytic function as it is for the CUE—see [13], or one could consider the logarithm of the absolute value of the characteristic polynomial on a macroscopic scale inside or outside the support of the equilibrium measure
Summary
Log-correlated Gaussian fields, namely Gaussian random generalized functions whose covariance kernels have a logarithmic singularity on the diagonal, are known to show up in various models of modern probability and mathematical physics—e.g. in combinatorial models describing random partitions of integers [35], random matrix theory [31,34,60], lattice models of statistical mechanics [41], the construction of conformally invariant random planar curves such as stochastic Loewner evolution [4,63], and growth models [9] just to name a few examples. A fundamental tool in describing these geometric properties of the fields is a class of random measures, which can be formally written as an exponential of the field As these fields are distributions instead of functions, exponentiation is not an operation one can naively perform, but through a suitable limiting and normalization procedure, these random measures can be rigorously constructed and they are known as Gaussian multiplicative chaos measures. For a large class of models of random matrix theory, the following is true: when the size of the matrix tends to infinity, the logarithm of the characteristic polynomial behaves like a log-correlated field This is essentially equivalent to a suitable central limit theorem for the global linear statistics of the random matrix—see [31,34,60] for results concerning the GUE, Haar distributed random unitary matrices, and the complex Ginibre ensemble. This settles some conjectures due to Forrester and Frankel—see Remark 2.11 and [28, Conjecture 5 and Conjecture 8] for further information about their conjectures
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