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1. Dynamical Localization for Random Band Matrices Up to $$W\ll N^{1/4}$$

   https://doi.org/10.1007/s00220-024-04948-1  

   Cipolloni, Giorgio; Peled, Ron; Schenker, Jeffrey; Shapiro, Jacob (March 2024, Communications in Mathematical Physics)

   We prove that a large class of N × N Gaussian random band matrices with band width W exhibits dynamical Anderson localization at all energies when $$W < 

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2. Universality for 1d Random Band Matrices: Sigma-Model Approximation

   https://doi.org/10.1007/s10955-018-1969-1  

   Shcherbina, Mariya; Shcherbina, Tatyana (January 2018, Journal of Statistical Physics)

   The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:1233–1260, 2016; Commun Math Phys 351:1009–1044, 2017). We consider random Hermitian block band matrices consisting of $$W\times W$$ random Gaussian blocks (parametrized by $$j,k \in\Lambda=[1,n]^d\cap \mathbb{Z}^d$$) with a fixed entry's variance $$J_{jk}=\delta_{j,k}W^{-1}+\beta\Delta_{j,k}W^{-2}$$, $$\beta>0$$ in each block. Taking the limit $$W\to\infty$$ with fixed $$n$$ and $$\beta$$, we derive the sigma-model approximation of the second correlation function similar to Efetov's one. Then, considering the limit $$\beta, n\to\infty$$, we prove that in the dimension $d=1$ the behaviour of the sigma-model approximation in the bulk of the spectrum, as $$\beta\gg n$$, is determined by the classical Wigner -- Dyson statistics. 

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3. A “lifting” method for exponential large deviation estimates and an application to certain non-stationary 1D lattice Anderson models

   https://doi.org/10.1063/5.0150430  

   Hurtado, Omar (June 2023, Journal of Mathematical Physics)

   Proofs of localization for random Schrödinger operators with sufficiently regular distribution of the potential can take advantage of the fractional moment method introduced by Aizenman–Molchanov [Commun. Math. Phys. 157(2), 245–278 (1993)] or use the classical Wegner estimate as part of another method, e.g., the multi-scale analysis introduced by Fröhlich–Spencer [Commun. Math. Phys. 88, 151–184 (1983)] and significantly developed by Klein and his collaborators. When the potential distribution is singular, most proofs rely crucially on exponential estimates of events corresponding to finite truncations of the operator in question; these estimates in some sense substitute for the classical Wegner estimate. We introduce a method to “lift” such estimates, which have been obtained for many stationary models, to certain closely related non-stationary models. As an application, we use this method to derive Anderson localization on the 1D lattice for certain non-stationary potentials along the lines of the non-perturbative approach developed by Jitomirskaya–Zhu [Commun. Math. Physics 370, 311–324 (2019)] in 2019. 

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4. Multi-level loop equations for $$\beta $$-corners processes

   https://doi.org/10.1007/s00029-024-01006-5  

   Dimitrov, Evgeni; Knizel, Alisa (February 2025, Selecta Mathematica)

   Abstract The goal of the paper is to introduce a new set of tools for the study of discrete and continuous$$\beta $$ $\beta$-corners processes. In the continuous setting, our work provides a multi-level extension of the loop equations (also called Schwinger–Dyson equations) for$$\beta $$ $\beta$-log gases obtained by Borot and Guionnet in (Commun. Math. Phys. 317, 447–483, 2013). In the discrete setting, our work provides a multi-level extension of the loop equations (also called Nekrasov equations) for discrete$$\beta $$ $\beta$-ensembles obtained by Borodin, Gorin and Guionnet in (Publications mathématiques de l’IHÉS 125, 1–78, 2017). 

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5. On the stability of homogeneous equilibria in the Vlasov–Poisson system on R3

   https://doi.org/10.1088/1361-6382/acebb0  

   Ionescu, A D; Pausader, B; Wang, X; Widmayer, K (August 2023, Classical and Quantum Gravity)

   The goal of this article is twofold. First, we investigate the linearized Vlasov–Poisson system around a family of spatially homogeneous equilibria in the unconfined setting. Our analysis follows classical strategies from physics (Binney and Tremaine 2008, Galactic Dynamics,(Princeton University Press); Landau 1946, Acad. Sci. USSR. J. Phys.10,25–34; Penrose 1960,Phys. Fluids,3,258–65) and their subsequent mathematical extensions (Bedrossian et al 2022, SIAM J. Math. Anal.,54,4379–406; Degond 1986,Trans. Am. Math. Soc., 294,435–53; Glassey and Schaeffer 1994,Transp. Theory Stat. Phys.,23, 411–53; Grenier et al 2021, Math. Res. Lett., 28,1679–702; Han-Kwan et al, 2021, Commun. Math. Phys. 387, 1405–40; Mouhot and Villani 2011, Acta Math., 207, 29–201). The main novelties are a unified treatment of a broad class of analytic equilibria and the study of a class of generalized Poisson equilibria. For the former, this provides a detailed description of the associated Green’s functions, including in particular precise dissipation rates (which appear to be new), whereas for the latter we exhibit explicit formulas. Second, we review the main result and ideas in our recent work (Ionescu et al, 2022 on the full global nonlinear asymptotic stability of the Poisson equilibrium in R3 

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