Liouville quantum gravity from random matrix dynamics

Abstract

We establish the first connection between 2d Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if (Ut) is a Brownian motion on the unitary group at equilibrium, then the measuresconverge in the limit of large dimension to the 2d LQG measure, a properly normalized exponential of the 2d Gaussian free field. Gaussian free field type fluctuations associated with these dynamics were first established by Spohn (1998) and convergence to the LQG measure in 2d settings was conjectured since the work of Webb ( 2014), who proved the convergence of related one dimensional measures by using inputs from Riemann-Hilbert theory.The convergence follows from the first multi-time extension of the result by Widom (1973) on Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols. To prove these, we develop a general surgery argument and combine determinantal point processes estimates with stochastic analysis on Lie group, providing in passing a probabilistic proof of Webb's 1d result. We believe the techniques will be more broadly applicable to matrix dynamics out of equilibrium, joint moments of determinants for classes of correlated random matrices, and the characteristic polynomial of non-Hermitian random matrices. PreliminariesBasic notations. In this paper, dλ denotes the Lebesgue measure on the unit circle U, and dm the Lebesgue measure on C. We remind that the Fourier coefficients of f are defined as fk = 1

Convergence of the logarithm of the characteristic polynomial of unitary Brownian motion in Sobolev space

“…. ., l, converge in 3 Several Sobolev spaces are involved because we can improve the regularity with respect to one of the parameters at the cost of sacrificing some regularity with respect to the other parameter.…”

“…The relation between the two normalisations is Ũn (2t) = U n (t). We chose our normalisation to be consistent with the work of Spohn [24], and Bourgade and Falconet [3].…”

“…Characteristic polynomials of random matrices are fundamental objects in random matrix theory. They are closely related to the theory of log-correlated fields and to Gaussian multiplicative chaos, see [2][3][4][5][6][7][8][9][10] for the circular ensembles, and [11][12][13] for Hermitian ensembles and the Ginibre ensemble, among others. In the case of Haar-distributed matrices from the classical compact groups, there are also remarkable similarities between the statistics of the characteristic polynomial and those of the Riemann zeta function and other number-theoretic L-functions, see for example [14][15][16][17][18][19][20][21] for a review.…”

“…The goal of this paper is to specify Sobolev spaces of optimal regularity in which the convergence of the real and imaginary part of the logarithm of the characteristic polynomial to Gaussian free fields holds 3 . This is the natural dynamical version of the corresponding stationary result for Haar-distributed unitary matrices by Hughes et al [2], who proved that for any fixed time the logarithm of the characteristic polynomial converges to a generalized Gaussian field on the unit circle.…”

Benford's law and the C$β$E

“…For β = 2 this is still open and only proved for a gentle regularization of |D N,β (θ)| in [20]. For further results on the relation between CβE and Gaussian Multiplicative Chaos we refer to [3,6].…”