52 research outputs found
Random matrices in non-confining potentials
We consider invariant matrix processes diffusing in non-confining cubic
potentials of the form . We construct the
trajectories of such processes for all time by restarting them whenever an
explosion occurs, from a new (well chosen) initial condition, insuring
continuity of the eigenvectors and of the non exploding eigenvalues. We
characterize the dynamics of the spectrum in the limit of large dimension and
analyze the stationary state of this evolution explicitly. We exhibit a sharp
phase transition for the limiting spectral density at a critical value
. If , then the potential presents a well near
deep enough to confine all the particles inside, and the spectral
density is supported on a compact interval. If however, the
steady state is in fact dynamical with a macroscopic stationary flux of
particles flowing across the system. In this regime, the eigenvalues allocate
according to a stationary density profile with full support in
, flanked with heavy tails such that as
. Our method applies to other non-confining potentials and we
further investigate a family of quartic potentials, which were already studied
in Br\'ezin et al. to count planar diagrams.Comment: 32 pages, 7 figure
Tracy-Widom at high temperature
We investigate the marginal distribution of the bottom eigenvalues of the
stochastic Airy operator when the inverse temperature tends to . We
prove that the minimal eigenvalue, whose fluctuations are governed by the
Tracy-Widom law, converges weakly, when properly centered and scaled,
to the Gumbel distribution. More generally we obtain the convergence in law of
the marginal distribution of any eigenvalue with given index . Those
convergences are obtained after a careful analysis of the explosion times
process of the Riccati diffusion associated to the stochastic Airy operator. We
show that the empirical measure of the explosion times converges weakly to a
Poisson point process using estimates proved in [L. Dumaz and B. Vir\'ag. Ann.
Inst. H. Poincar\'e Probab. Statist. 49, 4, 915-933, (2013)]. We further
compute the empirical eigenvalue density of the stochastic Airy ensemble on the
macroscopic scale when . As an application, we investigate the
maximal eigenvalues statistics of -ensembles when the repulsion
parameter when . We study the double scaling limit
and argue with heuristic and numerical arguments
that the statistics of the marginal distributions can be deduced following the
ideas of [A. Edelman and B. D. Sutton. J. Stat. Phys. 127, 6, 1121-1165 (2007)]
and [J. A. Ram\'irez, B. Rider and B. Vir\'ag. J. Amer. Math. Soc. 24, 919-944
(2011)] from our later study of the stochastic Airy operator.Comment: 5 figure
A diffusive matrix model for invariant -ensembles
We define a new diffusive matrix model converging towards the -Dyson
Brownian motion for all that provides an explicit construction
of -ensembles of random matrices that is invariant under the
orthogonal/unitary group. We also describe the eigenvector dynamics of the
limiting matrix process; we show that when and that two eigenvalues
collide, the eigenvectors of these two colliding eigenvalues fluctuate very
fast and take the uniform measure on the orthocomplement of the eigenvectors of
the remaining eigenvalues
Invariant -ensembles and the Gauss-Wigner crossover
We define a new diffusive matrix model converging towards the -Dyson
Brownian motion for all that provides an explicit construction
of -ensembles of random matrices that is invariant under the
orthogonal/unitary group. For small values of , our process allows one
to interpolate smoothly between the Gaussian distribution and the Wigner
semi-circle. The interpolating limit distributions form a one parameter family
that can be explicitly computed. This also allows us to compute the finite-size
corrections to the semi-circle.Comment: 3 figure
From Sine kernel to Poisson statistics
[no abstract available
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