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We consider the local statistics of $$H = V^* X V + U^* Y U$$ where V and U are independent Haar-distributed unitary matrices, and X and Y are deterministic real diagonal matrices. In the bulk, we prove that the gap statistics and correlation functions coincide with the GUE in the limit when the matrix size $$N \rightarrow \infty $$ under mild assumptions on X and certain rigidity assumptions on Y (the latter being an assumption on the convergence of the eigenvalues of Y to the quantiles of its limiting spectral measure which we assume to have a density). Our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to Dyson Brownian motion. Our method also applies to the case when V and U are drawn from the orthogonal group. Our proof relies on the local law for H proved in Bao et al. (Commun Math Phys 349(3):947–990, 2017; J Funct Anal 271(3):672–719, 2016; Adv Math 319:251–291, 2017) as well as the DBM convergence results of Landon and Yau (Commun Math Phys 355(3):949–1000, 2017) and Landon et al. (Adv Math 346:1137–1332, 2019).

We consider Dyson Brownian motion for classical values of β with deterministic initial data V. We prove that the local eigenvalue statistics coincide with the GOE/GUE in the fixed energy sense after time t ≳ 1 / N if the density of states of V is bounded above and below down to scales η ≪ t in a window of size L ≫ t . Our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known. Our methodology builds on the homogenization theory developed in [16] which reduces the microscopic problem to a mesoscopic problem. As an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical Dyson Brownian motion.

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erdős–Rényi graph $G(N,p)$. Tracy–Widom fluctuations of the extreme eigenvalues for $p\gg N^{-2/3}$ was proved in (Probab. Theory Related Fields 171 (2018) 543–616; Comm. Math. Phys. 314 (2012) 587–640). We prove that there is a crossover in the behavior of the extreme eigenvalues at $p\sim N^{-2/3}$. In the case that $N^{-7/9}\ll p\ll N^{-2/3}$, we prove that the extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean zero condition and when $p=CN^{-2/3}$, we find that the fluctuations of the extreme eigenvalues are given by a combination of the Gaussian and the Tracy–Widom distribution. These results show that the eigenvalues at the edge of the spectrum of sparse Erdős–Rényi graphs are less rigid than those of random $d$-regular graphs (Bauerschmidt et al. (2019)) of the same average degree.

We study Dyson Brownian motion with general potential V and for general $$\beta \ge 1$$ . For short times $$t = o (1)$$ and under suitable conditions on V we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale $$\eta _* \ll t \ll 1$$ , we prove a mesoscopic central limit theorem for linear statistics at all scales $$\eta $$ with $$N^{-1}\ll \eta \ll t$$ .

We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times $t=o(1)$ with deterministic initial data V . Our main result states that if the density of states of $V$ is bounded both above and away from $0$ down to scales $\ell \ll t$ in a small interval of size $G \gg t$ around an energy $E_0$, then the local statistics coincide with the GOE/GUE near the energy $E_0$ after time $t$. Our methods are partly based on the idea of coupling two Dyson Brownian motions from [6], the parabolic regularity result of [15], and the eigenvalue rigidity results of [21]., Comment: 43 pages. Second draft contains improvements of the hypotheses of main result

The dynamic reflection probability of Davies and Simon (Commun Math Phys 63(3):277–301, 1978) and the spectral reflection probability of Gesztesy et al. (Diff Integral Eqs 10(3):521–546, 1997) and Gesztesy and Simon (Helv Phys Acta 70:66–71, 1997) for a one-dimensional Schrodinger operator \(H = - \Delta + V\) are characterized in terms of the scattering theory of the pair \((H, H_\infty )\) where \(H_\infty \) is the operator obtained by decoupling the left and right half-lines \(\mathbb {R}_{\le 0}\) and \(\mathbb {R}_{\ge 0}\). An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of Breuer et al. (Commun Math Phys 295(2):531–550, 2010). This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black-box model and follows a strategy parallel to Jaksic (Commun Math Phys 332:827–838, 2014).

We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of "homogenization" for Dyson Brownian Motion, this yields the universality of quantities which depend on the behavior of single eigenvalues of Wigner matrices and $\beta$-ensembles. Among the results we obtain are the Gaussian fluctuations of single eigenvalues for Wigner matrices (without an assumption of 4 matching moments) and classical $\beta$-ensembles ($\beta=1, 2, 4$), Gaussian fluctuations of the eigenvalue counting function, and an asymptotic expansion up to order $o(N^{-1})$ for the expected value of eigenvalues in the bulk of the spectrum. The latter result solves a conjecture of Tao and Vu., Comment: Typos corrected, revisions made in accordance with referee suggestions

A positive temperature analogue of the scattering length of a potential $V$ can be defined via integrating the difference of the heat kernels of $-\Delta$ and $-\Delta + \frac 12 V$, with $\Delta$ the Laplacian. An upper bound on this quantity is a crucial input in the derivation of a bound on the critical temperature of a dilute Bose gas \cite{SU}. In \cite{SU} a bound was given in the case of finite range potentials and sufficiently low temperature. In this paper, we improve the bound and extend it to potentials of infinite range., Comment: LaTeX, 6 pages

Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associated to the coupled and decoupled operators are derived. In particular, it is shown that a CMV matrix is reflectionless iff the scattering matrix is off-diagonal which in turn provides a short proof of an important result of [Breuer-Ryckman-Simon]. These developments parallel those recently obtained for Jacobi matrices., Comment: 14 pages

The property that a Jacobi matrix is reflectionless is usually characterized either in terms of Weyl m-functions or the vanishing of the real part of the boundary values of the diagonal matrix elements of the resolvent. We introduce a characterization in terms of stationary scattering theory (the vanishing of the reflection coefficients) and prove that this characterization is equivalent to the usual ones. We also show that the new characterization is equivalent to the notion of being dynamically reflectionless, thus providing a short proof of an important result of [Breuer-Ryckman-Simon]. The motivation for the new characterization comes from recent studies of the non-equilibrium statistical mechanics of the electronic black box model and we elaborate on this connection. To appear in Commun. Math. Phys., Comment: 10 pages

We study entropic functionals/fluctuations of the XY chain with Hamiltonian $$\begin{array}{ll} \frac{1}{2} \sum\limits_{x \in \mathbb{Z}}J_x( \sigma_x^{(1)} \sigma_{x+1}^{(1)} +\sigma_x^{(2)} \sigma_{x+1}^{(2)}) + \lambda_x \sigma_x^{(3)}\end{array}$$ where initially the left (x ≤ 0)/right (x > 0) part of the chain is in thermal equilibrium at inverse temperature β l /β r . The temperature differential results in a non-trivial energy/entropy flux across the chain. The Evans–Searles (ES) entropic functional describes fluctuations of the flux observable with respect to the initial state while the Gallavotti–Cohen (GC) functional describes these fluctuations with respect to the steady state (NESS) the chain reaches in the large time limit. We also consider the full counting statistics (FCS) of the energy/entropy flux associated with a repeated measurement protocol, the variational entropic functional (VAR) that arises as the quantization of the variational characterization of the classical Evans–Searles functional and a natural class of entropic functionals that interpolate between FCS and VAR. We compute these functionals in closed form in terms of the scattering data of the Jacobi matrix hu x = J x u x+1 + λ x u x + J x−1 u x−1 canonically associated with the XY chain. We show that all these functionals are identical if and only if h is reflectionless (we call this phenomenon entropic identity). If h is not reflectionless, then the ES and GC functionals remain equal but differ from the FCS, VAR and interpolating functionals. Furthermore, in the non-reflectionless case, the ES/GC functional does not vanish at α = 1 (i.e., the Kawasaki identity fails) and does not have the celebrated α ↔ 1 − α symmetry. The FCS, VAR and interpolating functionals always have this symmetry. In the Schrodinger case, where J x = J for all x, the entropic identity leads to some unexpected open problems in the spectral theory of one-dimensional discrete Schrodinger operators.

We consider the Fr\"ohlich $N$-polaron Hamiltonian in the strong coupling limit and bound the ground state energy from below. In particular, our lower bound confirms that the ground state energy of the Fr\"ohlich polaron and the ground state energy of the associated Pekar-Tomasevich variational problem are asymptotically equal in the strong coupling limit. We generalize the operator approach that was used to prove a similar result in the N=1 case in Lieb and Thomas (1997) and apply a Feynman-Kac formula to obtain the same result for an arbitrary particle number $N \geq 1$., Comment: 15 pages