39 research outputs found
On the Spectrum of Random Anti-symmetric and Tournament Matrices
We consider a discrete, non-Hermitian random matrix model, which can be
expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We
show that, asymptotically almost surely, the real parts of the eigenvalues of
the non-Hermitian matrix around any fixed index remain interlaced with those of
the anti-symmetric matrix. Along the way, we show that some tools recently
developed to study the eigenvalue distributions of Hermitian matrices extend to
the anti-symmetric setting.Comment: Presentation revised, 35 page
Applications of mesoscopic CLTs in random matrix theory
We present some applications of central limit theorems on mesoscopic scales
for random matrices. When combined with the recent theory of "homogenization"
for Dyson Brownian Motion, this yields the universality of quantities which
depend on the behavior of single eigenvalues of Wigner matrices and
-ensembles. Among the results we obtain are the Gaussian fluctuations of
single eigenvalues for Wigner matrices (without an assumption of 4 matching
moments) and classical -ensembles (), Gaussian
fluctuations of the eigenvalue counting function, and an asymptotic expansion
up to order for the expected value of eigenvalues in the bulk of
the spectrum. The latter result solves a conjecture of Tao and Vu.Comment: Typos corrected, revisions made in accordance with referee
suggestion
Convergence of the eigenvalue density for beta-Laguerre ensembles on short scales
In this note, we prove that the normalized trace of the resolvent of the
beta-Laguerre ensemble eigenvalues is close to the Stieltjes transform of the
Marchenko-Pastur (MP) distribution with very high probability, for values of
the imaginary part greater than m^{-1+\epsilon}. As an immediate corollary, we
obtain convergence of the one-point density to the MP law on short scales. The
proof serves to illustrate some simplifications of the method introduced in our
previous work to prove a local semi-circle law for Gaussian beta-ensembles.Comment: Various corrections based on referee comments. To appear in Electron.
J. Proba
Concentration for integrable directed polymer models
In this paper, we consider four integrable models of directed polymers for
which the free energy is known to exhibit KPZ fluctuations. A common framework
for the analysis of these models was introduced in our recent work on the
O'Connell-Yor polymer.
We derive estimates for the central moments of the partition function, of any
order, on the near-optimal scale , using an iterative method.
Among the innovations exploiting the invariant structure, we develop formulas
for correlations between functions of the free energy and the boundary weights
that replace the Gaussian integration by parts appearing in the analysis of the
O'Connell-Yor case.Comment: 34 page
Fluctuations of the overlap at low temperature in the 2-spin spherical SK model
We describe the fluctuations of the overlap between two replicas in the
2-spin spherical SK model about its limiting value in the low temperature
phase. We show that the fluctuations are of order and are given by a
simple, explicit function of the eigenvalues of a matrix from the Gaussian
Orthogonal Ensemble. We show that this quantity converges and describe its
limiting distribution in terms of the Airy1random point field (i.e., the joint
limit of the extremal eigenvalues of the GOE) from random matrix theory.Comment: 32 page
Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
We show that the variance of centred linear statistics of eigenvalues of GUE
matrices remains bounded for large for some classes of test functions less
regular than Lipschitz functions. This observation is suggested by the limiting
form of the variance (which has previously been computed explicitly), but it
does not seem to appear in the literature. We combine this fact with comparison
techniques following Tao-Vu and Erd\"os, Yau, et al. and a Littlewood-Paley
type decomposition to extend the central limit theorem for linear eigenvalue
statistics to functions in the H\"older class in the case of
matrices of Gaussian convolution type. We also give a variance bound which
implies the CLT for test functions in the Sobolev space and
for general Wigner matrices satisfying moment conditions.
Previous results on the CLT impose the existence and continuity of at least one
classical derivative.Comment: 51 pages, 3 figures. Minor correction
Central limit theorem near the critical temperature for the overlap in the 2-spin spherical SK model
We prove a central limit theorem for the normalized overlap between two
replicas in the spherical SK model in the high temperature phase. The
convergence holds almost surely with respect to the disorder variables, and the
inverse temperature can approach the criticial value at a polynomial rate with
any exponent strictly greater than .Comment: 16 page
On the chemical distance in critical percolation
We consider two-dimensional critical bond percolation. Conditioned on the
existence of an open circuit in an annulus, we show that the ratio of the
expected size of the shortest open circuit to the expected size of the
innermost circuit tends to zero as the side length of the annulus tends to
infinity, the aspect ratio remaining fixed. The same proof yields a similar
result for the lowest open crossing of a rectangle. In this last case, we
answer a question of Kesten and Zhang by showing in addition that the ratio of
the length of the shortest crossing to the length of the lowest tends to zero
in probability. This suggests that the chemical distance in critical
percolation is given by an exponent strictly smaller than that of the lowest
path.Comment: 60 pages, 12 figure
Sublinear variance in first-passage percolation for general distributions
We prove that the variance of the passage time from the origin to a point x
in first-passage percolation on Z^d is sublinear in the distance to x when d
\geq 2, obeying the bound Cx/(log x), under minimal assumptions on the
edge-weight distribution. The proof applies equally to absolutely continuous,
discrete and singular continuous distributions and mixtures thereof, and
requires only 2+log moments. The main result extends work of
Benjamini-Kalai-Schramm and Benaim-Rossignol.Comment: 32 pages. We added a proof sketch and fixed the proof of Theorem 2.3
and the bound on term (6.18
Subdiffusivity of random walk on the 2D invasion percolation cluster
We derive "quenched" subdiffusive lower bounds for the exit time tau(n) from
a box of size n for the simple random walk on the planar invasion percolation
cluster. The first part of the paper is devoted to proving an almost sure
analog of H. Kesten's subdiffusivity theorem for the random walk on the
incipient infinite cluster and the invasion percolation cluster using ideas of
M. Aizenman, A. Burchard and A. Pisztora. The proof combines lower bounds on
the instrinsic distance in these graphs and general inequalities for reversible
Markov chains. In the second part of the paper, we present a sharpening of
Kesten's original argument, leading to an explicit almost sure lower bound for
tau(n) in terms of percolation arm exponents. The methods give tau(n) \geq
n^{2+epsilon_0 + kappa}, where epsilon_0>0 depends on the instrinsic distance
and (assuming the exact value of the backbone exponent) kappa can be taken to
be 17/384 on the hexagonal lattice.Comment: 35 pages, 2 figures. Presentation reorganize