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DTSTAMP:20260424T143454Z
UID:1761141600@ist.ac.at
DTSTART:20251022T160000
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DESCRIPTION:Speaker: Volodymyr Riabov\nhosted by Florian Schur\nAbstract: T
 his thesis deals with eigenvalue and eigenvector universality results for 
 random matrix ensembles equipped with non-trivial spatial structure. We co
 nsider both mean-field models with a general variance profile (Wigner-type
  matrices) and correlation structure (correlated matrices) among the entri
 es\, as well as non-mean-field random band matrices with bandwidth W >> N^
 (1/2).        To extract the universal properties of random matrix sp
 ectra and eigenvectors\, we obtain concentration estimates for their resol
 vent\, the local laws\, which generalize the celebrated Wigner semicircle 
 law for a broad class of random matrices to much finer spectral scales. Th
 e local laws hold for both a single resolvent as well as for products of m
 ultiple resolvents\, known as resolvent chains\, and express the remarkabl
 e approximately-deterministic behavior of these objects down to the micros
 copic scale.         Our primary tool for establishing the local law
 s is the dynamical Zigzag strategy\, which we develop in the setting of sp
 atially-inhomogeneous random matrices. Our proof method systematically add
 resses the challenges arising from non-trivial spatial structures and is r
 obust to all types of singularities in the spectrum\, as we demonstrate in
  the correlated setting. Furthermore\, we incorporate the analysis of the 
 deterministic resolvent chain approximations into the dynamical framework 
 of the Zigzag strategy\, synthesizing a unified toolkit for establishing m
 ulti-resolvent local laws.        Using these methods\, we prove comp
 lete eigenvector delocalization\, the Eigenstate Thermalization Hypothesis
 \, and Wigner-Dyson universality in the bulk for random band matrices down
  to the optimal bandwidth W >> N^(1/2). For mean-field ensembles\, we esta
 blish universality of local eigenvalue statistics at the cups for random m
 atrices with correlated entries\, and the Eigenstate Thermalization Hypoth
 esis for Wigner-type matrices in the bulk of the spectrum.          
 Finally\, this thesis also contains other applications of the multi-resolv
 ent local laws to spatially-inhomogeneous random matrices\, obtained prior
  to the development of the Zigzag strategy.  In particular\, we provide a
  complete analysis of mesoscopic linear-eigenvalue statistics of Wigner-ty
 pe matrices in all spectral regimes\, including the novel cusps\, and rigo
 rously establish the prethermalization phenomenon for deformed Wigner matr
 ices.
LOCATION:Central Bldg / O1 / Mondi 3 (I01.O1.010) and Zoom\, ISTA
ORGANIZER:
SUMMARY:Volodymyr Riabov: Thesis Defense: Universality in Random Matrices w
 ith Spatial Structure
URL:https://talks-calendar.ista.ac.at/events/6017
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