241 research outputs found
Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit
We consider the double scaling limit in the random matrix ensemble with an
external source \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on Hermitian matrices, where is a diagonal matrix with two eigenvalues of equal multiplicities. The value is critical since the eigenvalues
of accumulate as on two intervals for and on one
interval for . These two cases were treated in Parts I and II, where
we showed that the local eigenvalue correlations have the universal limiting
behavior known from unitary random matrix ensembles. For the critical case
new limiting behavior occurs which is described in terms of Pearcey
integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish
this result by applying the Deift/Zhou steepest descent method to a -matrix valued Riemann-Hilbert problem which involves the construction of a
local parametrix out of Pearcey integrals. We resolve the main technical issue
of matching the local Pearcey parametrix with a global outside parametrix by
modifying an underlying Riemann surface.Comment: 36 pages, 9 figure
Discretization Dependence of Criticality in Model Fluids: a Hard-core Electrolyte
Grand canonical simulations at various levels, -20, of fine- lattice
discretization are reported for the near-critical 1:1 hard-core electrolyte or
RPM. With the aid of finite-size scaling analyses it is shown convincingly
that, contrary to recent suggestions, the universal critical behavior is
independent of (\grtsim 4); thus the continuum RPM
exhibits Ising-type (as against classical, SAW, XY, etc.) criticality. A
general consideration of lattice discretization provides effective
extrapolation of the {\em intrinsically} erratic -dependence, yielding
(\Tc^ {\ast},\rhoc^{\ast})\simeq (0.0493_{3},0.075) for the
RPM.Comment: 4 pages including 4 figure
The Julia sets and complex singularities in hierarchical Ising models
We study the analytical continuation in the complex plane of free energy of
the Ising model on diamond-like hierarchical lattices. It is known that the
singularities of free energy of this model lie on the Julia set of some
rational endomorphism related to the action of the Migdal-Kadanoff
renorm-group. We study the asymptotics of free energy when temperature goes
along hyperbolic geodesics to the boundary of an attractive basin of . We
prove that for almost all (with respect to the harmonic measure) geodesics the
complex critical exponent is common, and compute it
Reconstruction of Random Colourings
Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
Real roots of Random Polynomials: Universality close to accumulation points
We identify the scaling region of a width O(n^{-1}) in the vicinity of the
accumulation points of the real roots of a random Kac-like polynomial
of large degree n. We argue that the density of the real roots in this region
tends to a universal form shared by all polynomials with independent,
identically distributed coefficients c_i, as long as the second moment
\sigma=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to
the previously reported abrupt) and quite nontrivial suppression of the number
of real roots for coefficients with a nonzero mean value \mu_n = E(c_i) scaled
as \mu_n\sim n^{-1/2}.Comment: Some minor mistakes that crept through into publication have been
removed. 10 pages, 12 eps figures. This version contains all updates, clearer
pictures and some more thorough explanation
Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes
Random Hermitian matrices with a source term arise, for instance, in the
study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and
sample covariance matrices \cite{Baik:2005}.
We consider the case when the external source matrix has two
distinct real eigenvalues: with multiplicity and zero with multiplicity
. The source is small in the sense that is finite or , for . For a Gaussian potential, P\'ech\'e
\cite{Peche:2006} showed that for sufficiently small (the subcritical
regime) the external source has no leading-order effect on the eigenvalues,
while for sufficiently large (the supercritical regime) eigenvalues
exit the bulk of the spectrum and behave as the eigenvalues of
Gaussian unitary ensemble (GUE). We establish the universality of these results
for a general class of analytic potentials in the supercritical and subcritical
regimes.Comment: 41 pages, 4 figure
Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
We study a model of non-intersecting squared Bessel processes in the
confluent case: all paths start at time at the same positive value , remain positive, and are conditioned to end at time at . In
the limit , after appropriate rescaling, the paths fill out a
region in the -plane that we describe explicitly. In particular, the paths
initially stay away from the hard edge at , but at a certain critical
time the smallest paths hit the hard edge and from then on are stuck to
it. For we obtain the usual scaling limits from random matrix
theory, namely the sine, Airy, and Bessel kernels. A key fact is that the
positions of the paths at any time constitute a multiple orthogonal
polynomial ensemble, corresponding to a system of two modified Bessel-type
weights. As a consequence, there is a matrix valued
Riemann-Hilbert problem characterizing this model, that we analyze in the large
limit using the Deift-Zhou steepest descent method. There are some novel
ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure
Domino tilings and the six-vertex model at its free fermion point
At the free-fermion point, the six-vertex model with domain wall boundary
conditions (DWBC) can be related to the Aztec diamond, a domino tiling problem.
We study the mapping on the level of complete statistics for general domains
and boundary conditions. This is obtained by associating to both models a set
of non-intersecting lines in the Lindstroem-Gessel-Viennot (LGV) scheme. One of
the consequence for DWBC is that the boundaries of the ordered phases are
described by the Airy process in the thermodynamic limit.Comment: 14 pages, 8 figure
Convergence of random zeros on complex manifolds
We show that the zeros of random sequences of Gaussian systems of polynomials
of increasing degree almost surely converge to the expected limit distribution
under very general hypotheses. In particular, the normalized distribution of
zeros of systems of m polynomials of degree N, orthonormalized on a regular
compact subset K of C^m, almost surely converge to the equilibrium measure on K
as the degree N goes to infinity.Comment: 16 page
Free particle scattering off two oscillating disks
We investigate the two-dimensional classical dynamics of the scattering of
point particles by two periodically oscillating disks. The dynamics exhibits
regular and chaotic scattering properties, as a function of the initial
conditions and parameter values of the system. The energy is not conserved
since the particles can gain and loose energy from the collisions with the
disks. We find that for incident particles whose velocity is on the order of
the oscillating disk velocity, the energy of the exiting particles displays
non-monotonic gaps of allowed energies, and the distribution of exiting
particle velocities shows significant fluctuations in the low energy regime. We
also considered the case when the initial velocity distribution is Gaussian,
and found that for high energies the exit velocity distribution is Gaussian
with the same mean and variance. When the initial particle velocities are in
the irregular regime the exit velocity distribution is Gaussian but with a
smaller mean and variance. The latter result can be understood as an example of
stochastic cooling. In the intermediate regime the exit velocity distribution
differs significantly from Gaussian. A comparison of the results presented in
this paper to previous chaotic static scattering problems is also discussed.Comment: 9 doble sided pages 13 Postscript figures, REVTEX style. To appear in
Phys. Rev.
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