Top eigenvalue of a random matrix: large deviations and third order phase transition
By Satya N. Majumdar, G. Schehr. 2013
182 citations recorded by the source corpus.
We study the fluctuations of the largest eigenvalue λmax of N × N random matrices in the limit of large N. The main focus is on Gaussian β ensembles, including in particular the Gaussian orthogonal (β = 1), unitary (β = 2) and symplectic (β = 4) ensembles. The probability density function (PDF) of λmax consists, for large N, of a central part described by Tracy–Widom distributions flanked, on both sides, by two large deviation tails. While the central part characterizes the typical fluctuations of λmax—of order O(N−2/3)?> —the large deviation tails are instead associated with extremely rare fluctuations—of order O(1)?>. Here we review some recent developments in the theory of these extremely rare events using a Coulomb gas approach. We discuss in particular the third order phase transition which separates the left tail from the right tail, a transition akin to the so-called Gross–Witten–Wadia phase transition found in 2-d lattice quantum chromodynamics. We also discuss the occurrence of similar third order transitions in various physical problems, including non-intersecting Brownian motions, conductance fluctuations in mesoscopic physics and entanglement in a bipartite system.
Subfields: Applied Mathematics, Condensed Matter Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/327bf9cf2becbf6d054e3347e9b0f16e17f20495
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In Applied Mathematics
In Condensed Matter Physics
- Jean-Philippe Bouchaud
- Marc Potters
- Michael W. Mahoney
- Charles H. Martin
- Gautier Marti
- Philippe Donnat
- Mikolaj Binkowski
- Cars H. Hommes
In Statistics and Probability
Citations in the corpus
- An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, unknown): “With the same g(x), and f (x) such that f ′(x) = −u(x) and f (x) → 0 as x → ∞, the GOE Tracy-Widom distribution F1(x) ≡ Prob[χTW, β=1 ≤ x] is given by F1(x) = e−[ f (x)+g(x)]/2 [50, 60].”
- The Effect of Gradient Noise on the Energy Landscape of Deep Networks (2015, unknown): “The proof of Theorem S-1 uses asymptotic expressions of the density of λmax for regions beyond √ 2±O(n−1/3) (Majumdar and Schehr, 2014).”
- Cleaning large correlation matrices: tools from random matrix theory (2016, background): “The Tracy-Widom result is actually a very nice manifestation of the universality phenomenon that describes the fluctuations of macroscopic observables in many large dimensional systems (see the recent paper [114] on this topic).”