31 research outputs found
Accuracy and range of validity of the Wigner surmise for mixed symmetry classes in random matrix theory
Schierenberg et al. [Phys. Rev. E 85, 061130 (2012)] recently applied the
Wigner surmise, i.e., substitution of \infty \times \infty matrices by their 2
\times 2 counterparts for the computation of level spacing distributions, to
random matrix ensembles in transition between two universality classes. I
examine the accuracy and the range of validity of the surmise for the crossover
between the Gaussian orthogonal and unitary ensembles by contrasting them with
the large-N results that I evaluated using the Nystrom-type method for the
Fredholm determinant. The surmised expression at the best-fitting parameter
provides a good approximation for 0 \lesssim s \lesssim 2, i.e., the validity
range of the original surmise.Comment: 3 pages in REVTeX, 10 figures. (v2) Title changed, version to appear
in Phys. Rev.
Tracy-Widom distribution as instanton sum of 2D IIA superstrings
We present an analytic expression of the nonperturbative free energy of a
double-well supersymmetric matrix model in its double scaling limit, which
corresponds to two-dimensional type IIA superstring theory on a nontrivial
Ramond-Ramond background. To this end we draw upon the wisdom of random matrix
theory developed by Tracy and Widom, which expresses the largest eigenvalue
distribution of unitary ensembles in terms of a Painleve II transcendent.
Regularity of the result at any value of the string coupling constant shows
that the third-order phase transition between a supersymmetry-preserving phase
and a supersymmetry-broken phase, previously found at the planar level, becomes
a smooth crossover in the double scaling limit. Accordingly, the supersymmetry
is always broken spontaneously as its order parameter stays nonzero for the
whole region of the coupling constant. Coincidence of the result with the
unitary one-matrix model suggests that one-dimensional type 0 string theories
partially correspond to the type IIA superstring theory. Our formulation
naturally allows for introduction of an instanton chemical potential, and
reveals the presence of a novel phase transition, possibly interpreted as
condensation of instantons.Comment: 25 pages, 4 figures. v2: new subsection 4.3 and references added.
version to be published in JHE
Comment on Dirac spectral sum rules for QCD_3
Recently Magnea hep-th/9907096 , hep-th/9912207 [Phys.Rev.D61, 056005 (2000);
Phys.Rev.D62, 016005 (2000)] claimed to have computed the first sum rules for
Dirac operators in 3D gauge theories from 0D non-linear sigma models. I point
out that these computations are incorrect, and that they contradict with the
exact results for the spectral densities unambiguously derived from random
matrix theory by Nagao and myself.Comment: REVTeX 3.1, 2 pages, no figure. (v2) redundant part removed,
conclusion unchange
Renormalization group approach to multiple-arc random matrix models
We study critical and universal behaviors of unitary invariant non-gaussian
random matrix ensembles within the framework of the large-N renormalization
group. For a simple double-well model we find an unstable fixed point and a
stable inverse-gaussian fixed point. The former is identified as the critical
point of single/double-arc phase transition with a discontinuity of the third
derivative of the free energy. The latter signifies a novel universality of
large-N correlators other than the usual single arc type. This phase structure
is consistent with the universality classification of two-level correlators for
multiple-arc models by Ambjorn and Akemann. We also establish the stability of
the gaussian fixed point in the multi-coupling model.Comment: 11 pages, 1 figure, LaTeX + a4.sty, epsf.st