Universal deformation rings for the symmetric group S_4

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S_4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S_4,V) for every kS_4-module V which has stable endomorphism ring k and show that R(S_4,V) is isomorphic to either k, or W[t]/(t^2,2t), or the group ring W[Z/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.Comment: 12 pages, 2 figure

Universal deformation rings and tame blocks

Let k be an algebraically closed field of positive characteristic, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group and B is a block of kG of infinite tame representation type. We find all finitely generated kG-modules V that belong to B and whose endomorphism ring is isomorphic to k and determine the universal deformation ring R(G,V) for each of these modules.Comment: 14 page

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 $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. 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 $a=1$ 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 $3 \times 3$-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

Exact solution of the six-vertex model with domain wall boundary condition. Critical line between ferroelectric and disordered phases

This is a continuation of the papers [4] of Bleher and Fokin and [5] of Bleher and Liechty, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large $n$ asymptotics of $Z_n$ on the critical line between these two phases.Comment: 22 pages, 6 figures, to appear in the Journal of Statistical Physic

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

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 $f$ 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 $f$. We prove that for almost all (with respect to the harmonic measure) geodesics the complex critical exponent is common, and compute it

Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case

In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinantal formulae defined by conformal $(2m,1)$ models. Our approach follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and uses a Lax pair representation of the conformal $(2m,1)$ models (giving Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in \cite{BleherEynard}. In particular we define Baker-Akhiezer functions associated to the Lax pair to construct a kernel which is then used to compute determinantal formulae giving the correlation functions of the double scaling limit of a matrix model near the merging of two cuts.Comment: 37 pages, 4 figures. Presentation improved, typos corrected. Published in Journal Of Statistical Mechanic

Level spacing statistics of classically integrable systems -Investigation along the line of the Berry-Robnik approach-

By extending the approach of Berry and Robnik, the limiting level spacing distribution of a system consisting of infinitely many independent components is investigated. The limiting level spacing distribution is characterized by a single monotonically increasing function $\bar{\mu}(S)$ of the level spacing $S$. Three cases are distinguished: (i) Poissonian if $\bar{\mu}(+\infty)=0$, (ii) Poissonian for large $S$, but possibly not for small $S$ if $0<\bar{\mu}(+\infty)< 1$, and (iii) sub-Poissonian if $\bar{\mu}(+\infty)=1$. This implies that, even when energy-level distributions of individual components are statistically independent, non-Poissonian level spacing distributions are possible.Comment: 19 pages, 4 figures. Accepted for publication in Phys. Rev.

All genus correlation functions for the hermitian 1-matrix model

We rewrite the loop equations of the hermitian matrix model, in a way which allows to compute all the correlation functions, to all orders in the topological $1/N^2$ expansion, as residues on an hyperelliptical curve. Those residues, can be represented diagrammaticaly as Feynmann graphs of a cubic interaction field theory on the curve.Comment: latex, 19 figure