28,112 research outputs found
Probability densities and distributions for spiked and general variance Wishart -ensembles
A Wishart matrix is said to be spiked when the underlying covariance matrix
has a single eigenvalue different from unity. As increases through
, a gap forms from the largest eigenvalue to the rest of the spectrum, and
with of order the scaled largest eigenvalues form a well
defined parameter dependent state. Recent works by Bloemendal and Vir\'ag [BV],
and Mo, have quantified this parameter dependent state for real Wishart
matrices from different viewpoints, and the former authors have done similarly
for the spiked Wishart -ensemble. The latter is defined in terms of
certain random bidiagonal matrices. We use a recursive structure to give an
alternative construction of the spiked and more generally the general variance
Wishart -ensemble, and we give the exact form of the joint eigenvalue
PDF for the two matrices in the recurrence. In the case of real quaternion
Wishart matrices () the latter is recognised as having appeared in
earlier studies on symmetrized last passage percolation, allowing the exact
form of the scaled distribution of the largest eigenvalue to be given. This
extends and simplifies earlier work of Wang, and is an alternative derivation
to a result in [BV]. We also use the construction of the spiked Wishart
-ensemble from [BV] to give a simple derivation of the explicit form of
the eigenvalue PDF.Comment: 18 page
Local Central Limit Theorem for Determinantal Point Processes
We prove a local central limit theorem (LCLT) for the number of points
in a region in specified by a determinantal point process
with an Hermitian kernel. The only assumption is that the variance of
tends to infinity as . This extends a previous result giving a
weaker central limit theorem (CLT) for these systems. Our result relies on the
fact that the Lee-Yang zeros of the generating function for ---
the probabilities of there being exactly points in --- all lie on the
negative real -axis. In particular, the result applies to the scaled bulk
eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the
Ginibre ensemble. For the GUE we can also treat the properly scaled edge
eigenvalue distribution. Using identities between gap probabilities, the LCLT
can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE).
A LCLT is also established for the probability density function of the -th
largest eigenvalue at the soft edge, and of the spacing between -th neigbors
in the bulk.Comment: 12 pages; claims relating to LCLT for Pfaffian point processes of
version 1 withdrawn in version 2 and replaced by determinantal point
processes; improved presentation version
Random transition-rate matrices for the master equation
Random-matrix theory is applied to transition-rate matrices in the Pauli
master equation. We study the distribution and correlations of eigenvalues,
which govern the dynamics of complex stochastic systems. Both the cases of
identical and of independent rates of forward and backward transitions are
considered. The first case leads to symmetric transition-rate matrices, whereas
the second corresponds to general, asymmetric matrices. The resulting matrix
ensembles are different from the standard ensembles and show different
eigenvalue distributions. For example, the fraction of real eigenvalues scales
anomalously with matrix dimension in the asymmetric case.Comment: 15 pages, 12 figure
Phase transitions in self-dual generalizations of the Baxter-Wu model
We study two types of generalized Baxter-Wu models, by means of
transfer-matrix and Monte Carlo techniques. The first generalization allows for
different couplings in the up- and down triangles, and the second
generalization is to a -state spin model with three-spin interactions. Both
generalizations lead to self-dual models, so that the probable locations of the
phase transitions follow. Our numerical analysis confirms that phase
transitions occur at the self-dual points. For both generalizations of the
Baxter-Wu model, the phase transitions appear to be discontinuous.Comment: 29 pages, 13 figure
Relaxation rate of the reverse biased asymmetric exclusion process
We compute the exact relaxation rate of the partially asymmetric exclusion
process with open boundaries, with boundary rates opposing the preferred
direction of flow in the bulk. This reverse bias introduces a length scale in
the system, at which we find a crossover between exponential and algebraic
relaxation on the coexistence line. Our results follow from a careful analysis
of the Bethe ansatz root structure.Comment: 22 pages, 12 figure
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