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SUMMARY:Nodal statistics from quantum graphs to random matrices: universal
 ity and phase transitions - Lior Alon (Massachusetts Institute of Technolo
 gy)
DTSTART:20260115T153000Z
DTEND:20260115T163000Z
UID:TALK238192@talks.cam.ac.uk
DESCRIPTION:Motivated by Berry&rsquo\;s random wave model for chaotic doma
 ins\, high-energy eigenfunctions are expected\, at the scale of the wavele
 ngth\, to behave like random waves. Accordingly\, the size of nodal sets i
 s determined to first order by the Weyl law\, while its second-order fluct
 uations are expected to be universal. In analogy with spectral statistics\
 , we refer to these fluctuations of the nodal set size as nodal statistics
 .\n&nbsp\;\nBeginning with work of Berry\, and later of Blum&ndash\;Gnutzm
 ann&ndash\;Smilansky and Bogomolny&ndash\;Schmit\, it was conjectured that
  in chaotic systems nodal statistics are asymptotically Gaussian. While st
 riking mathematical results support this picture on the sphere (Nazarov&nd
 ash\;Sodin)\, arithmetic symmetries on the torus prevent chaoticity and le
 ad to a breakdown of universality\, as shown by Wigman and collaborators a
 nd by Marinucci\, Rossi\, and Peccati.\n&nbsp\;\nIn this talk I will follo
 w Smilansky&rsquo\;s insight and focus on nodal statistics in graph-based 
 models. For quantum graphs and discrete operators on graphs\, Berkolaiko&r
 squo\;s nodal magnetic theorem relates nodal statistics to the stability o
 f eigenvalues under magnetic perturbations. This connection yields Gaussia
 n limiting statistics for quantum graphs with disjoint cycles (Alon&ndash\
 ;Band&ndash\;Berkolaiko)\, and\, combined with Morse inequalities\, extend
 s to random discrete operators on finite graphs with disjoint cycles and t
 o complete graphs with strong on-site disorder (Alon&ndash\;Goresky).\n&nb
 sp\;\nI will conclude with recent joint work on nodal statistics for rando
 m matrices (Alon&ndash\;Mikulincer&ndash\;Urschel)\, showing that nodal st
 atistics for GOE matrices obey a semicircle law rather than the conjecture
 d Gaussian behaviour. If time permits\, I will briefly discuss ongoing wor
 k on the Rosenzweig&ndash\;Porter model\, where adding a random on-site po
 tential to a GOE matrix leads to nodal statistics interpolating from semic
 ircle behaviour at low disorder to Gaussian behaviour at strong disorder\,
  suggesting a phase transition between universality classes and a connecti
 on to localization&ndash\;delocalization phenomena.
LOCATION:Seminar Room 1\, Newton Institute
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