709 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
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