The Localization Transition in the Ultrametric Ensemble

We study the hierarchical analogue of power-law random band matrices, a symmetric ensemble of random matrices with independent entries whose variances decay exponentially in the metric induced by the tree topology on $mathbb{N}$. We map out the entirety of the localization regime by proving the localization of eigenfunctions and Poisson statistics of the suitably scaled eigenvalues. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model.

Edge statistics of Dyson Brownian motion

We consider the edge statistics of Dyson Brownian motion with deterministic initial data. Our main result states that if the initial data has a spectral edge with rough square root behavior down to a scale $eta_* geq N^{-2/3}$ and no outliers, then after times $t gg sqrt{ eta_*}$, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge $t = N^{varepsilon} / N^{1/3}$ for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bourgade-ErdH{o}s-Yau-Yin] and the energy estimate of [Bourgade-ErdH{o}s-Yau].

Phase transitions for products of characteristic polynomials under Dyson Brownian motion

We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $beta$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.

The Brown measure of the free multiplicative Brownian motion

The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $mathsf{GL}(N;mathbb{C}),$ in the sense of $ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $Sigma_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $bar{Sigma}_{t},$ which is strictly positive and real analytic on $Sigma_{t}$. This density has a simple form in polar coordinates: [ W_{t}(r,theta)=frac{1}{r^{2}}w_{t}(theta), ] where $w_{t}$ is an analytic function determined by the geometry of the region $Sigma_{t}$. We show also that the spectral measure of free unitary Brownian motion $u_{t}$ is a shadow of the Brown measure of $b_{t}$, precisely mirroring the relationship between Wigners semicircle law and Ginibres circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

Local constants of motion imply information propagation

Interacting quantum many-body systems are usually expected to thermalise, in the sense that the evolution of local expectation values approach a stationary value resembling a thermal ensemble. This intuition is notably contradicted in systems exhibiting many-body localisation, a phenomenon receiving significant recent attention. One of its most intriguing features is that, in stark contrast to the non-interacting case, entanglement of states grows without limit over time, albeit slowly. In this work, we establish a novel link between quantum information theory and notions of condensed matter, capturing the phenomenon in the Heisenberg picture. We show that the existence of local constants of motion, often taken as the defining property of many-body localisation, together with a generic spectrum, is sufficient to rigorously prove information propagation: These systems can be used to send a signal over arbitrary distances, in that the impact of a local perturbation can be detected arbitrarily far away. We perform a detailed perturbation analysis of quasi-local constants of motion and also show that they indeed can be used to construct efficient spectral tensor networks, as recently suggested. Our results provide a detailed and model-independent picture of information propagation in many-body localised systems.