The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(\beta = 1)$, complex ($\beta = 2)$ and real quaternion $(\beta = 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $\beta > 0$. In the Gaussian case a further construction valid for $\beta > 0$ is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. The resulting functional forms are shown to be a special cases of duality formulas due to Desrosiers. New derivations of the general case of Desrosiers' dualities are given. A soft edge scaling limit of the averaged characteristic polynomial is identified, and an explicit evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page

Random matrices, log-gases and Holder regularity

The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner's original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices.Comment: Proceedings of ICM 201

Fully Adaptive Gaussian Mixture Metropolis-Hastings Algorithm

Markov Chain Monte Carlo methods are widely used in signal processing and communications for statistical inference and stochastic optimization. In this work, we introduce an efficient adaptive Metropolis-Hastings algorithm to draw samples from generic multi-modal and multi-dimensional target distributions. The proposal density is a mixture of Gaussian densities with all parameters (weights, mean vectors and covariance matrices) updated using all the previously generated samples applying simple recursive rules. Numerical results for the one and two-dimensional cases are provided

Directed polymer in a random medium of dimension 1+3 : multifractal properties at the localization/delocalization transition

We consider the model of the directed polymer in a random medium of dimension 1+3, and investigate its multifractal properties at the localization/delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyse the statistics of the position weights $w_L(\vec r)$ of the end-point of the polymer of length $L$ via the moments $Y_q(L) = \sum_{\vec r} [w_L(\vec r)]^q$. We measure the generalized exponents $\tau(q)$ and $\tilde \tau(q)$ governing the decay of the typical values $Y^{typ}_q(L) = e^{\bar{\ln Y_q(L)}} \sim L^{- \tau(q)}$ and disorder-averaged values $\bar{Y_q(L)} \sim L^{- \tilde \tau(q)}$ respectively. To understand the difference between these exponents, $\tau(q) \neq \tilde \tau(q)$ above some threshold $q>q_c \sim 2$, we compute the probability distributions of $y=Y_q(L)/Y^{typ}_q(L)$ over the samples : we find that these distributions becomes scale invariant with a power-law tail $1/y^{1+x_q}$. These results thus correspond to the Ever-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of Inverse Participation Ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent $\nu \sim 2$.Comment: 10 pages, 15 figure

Phase-averaged transport for quasi-periodic Hamiltonians

For a class of discrete quasi-periodic Schroedinger operators defined by covariant re- presentations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is proven. This result is established under a Diophantine condition on the incommensuration parameter. The relevant class of operators is distinguished by invariance with respect to symmetry automorphisms of the rotation algebra. It includes the critical Harper (almost-Mathieu) operator. As a by-product, a new solution of the frame problem associated with Weyl-Heisenberg-Gabor lattices of coherent states is given