The Noise in the Circular Law and the Gaussian Free Field

By Brian Rider, B'alint Vir'ag. 2006

189 citations recorded by the source corpus.

Fill an n × n matrix with independent complex Gaussians of variance 1/n. As n → ∞, the eigenvalues {zk} converge to a sum of an H1-noise on the unit disk and an independent H1/2-noise on the unit circle. More precisely, for C1 functions of suitable growth, the distribution of ∑n k=1(f(zk) − Ef(zk)) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the disk H1 and the circle H1/2 norms of f . As a consequence, with pn the characteristic polynomial, it is found that log |pn| −E log |pn| tends to the planar Gaussian free field conditioned to be harmonic outside the unit disk. Further, for polynomial test functions f , we prove that the limiting covariance structure is universal for a class of models including Haar distributed unitary matrices.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/21be4899936a22c23d9e97c4aeaaf7e3987c7be9

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Citations in the corpus

Fluctuations of eigenvalues of random normal matrices (2008, methodology): “Following Rider–Virág [30], we can write the cumulants as integrals involving the cyclic product (1.14) Rm,n,k(λ1, . . . , λk) = Km,n(λ1, λ2) Km,n(λ2, λ3) · · · Km,n(λk, λ1) e−m(Q(λ1)+...+Q(λk)).”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, background): “Rider and Virág [34] for a connection with the circular law for Gaussian non-Hermitian random matrices.”
Monte Carlo with determinantal point processes (2016, unknown): “See for instance [Johansson, 1997, 1998, Diaconis and Evans, 2001, Soshnikov, 2002, Pastur, 2006, Rider and Virág, 2007, Popescu, 2009, Kriecherbauer and Shcherbina, 2010, Ameur et al., 2011, 2015, Berman, 2012, Shcherbina, 2013, Breuer and Duits, 2013, 2014, Johansson and Lambert, 2015, Lambert,…”
Fluctuations of eigenvalues and second order Poincaré inequalities (2007, methodology): “Again, the basic approach relies on the method of moments, but the computations require new ideas because of the lack of independence between the matrix entries.”
Around the circular law (2011, background): “The square of the smallest singular value sn(n −1/2G)2 of the Complex Ginibre Ensemble follows an exponential law [40] and this result is asymptotically universal [149]; • Gap probabilities and Voronöı cells (see [3] and [62] for the Ginibre Ensemble); • Linear statistics of μX (some results are ava”
The circular law for random matrices (2007, background): “Rider proved in [21] and [20] results about the spectral radius and about linear statistics of eigenvalues of non-Hermitian matrices with Gaussian entries.”
Circular law theorem for random Markov matrices (2008, unknown): “Another very interesting open problem is the fluctuation of the global and extremal spectrum for which one can find some partial answers in [43], [36], [42], [38].”
Imaginary geometry I: interacting SLEs (2012, background): “Just as many random walk models have Brownian motion as a scaling limit, many random (real or integer valued) functions on two dimensional lattices have the GFF as a scaling limit [1,11,18,23,25].”