C*-algebras associated with endomorphisms and polymorphisms of compact abelian groups

A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the $K$-theory of these algebras and use it to compute the $K$-groups in a number of interesting examples.Comment: 25 page

Infinite-dimensional diffusions as limits of random walks on partitions

The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.Comment: AMSTex, 33 pages. Version 2: minor changes, typos corrected, to appear in Prob. Theor. Rel. Field

An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution

We compute exactly the asymptotic distribution of scaled height in a (1+1)--dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in $(1+1)$ dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all share the same universal Tracy-Widom distribution for the suitably scaled height variables.Comment: 5 pages Revtex, 3 .eps figures included, new references adde

Extended Seiberg-Witten Theory and Integrable Hierarchy

The prepotential of the effective N=2 super-Yang-Mills theory perturbed in the ultraviolet by the descendents of the single-trace chiral operators is shown to be a particular tau-function of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory with 2N-2 fundamental hypermultiplets at the appropriate locus of the moduli space of vacua) or a theory on a single fractional D3 brane at the ADE singularity the hierarchy is the dispersionless Toda chain. We present its explicit solutions. Our results generalize the limit shape analysis of Logan-Schepp and Vershik-Kerov, support the prior work hep-th/0302191 which established the equivalence of these N=2 theories with the topological A string on CP^1 and clarify the origin of the Eguchi-Yang matrix integral. In the higher rank case we find an appropriate variant of the quasiclassical tau-function, show how the Seiberg-Witten curve is deformed by Toda flows, and fix the contact term ambiguity.Comment: 49 page

The Library of Babel: On the origin of gravitational thermodynamics

We show that heavy pure states of gravity can appear to be mixed states to almost all probes. For AdS_5 Schwarzschild black holes, our arguments are made using the field theory dual to string theory in such spacetimes. Our results follow from applying information theoretic notions to field theory operators capable of describing very heavy states in gravity. For half-BPS states of the theory which are incipient black holes, our account is exact: typical microstates are described in gravity by a spacetime ``foam'', the precise details of which are almost invisible to almost all probes. We show that universal low-energy effective description of a foam of given global charges is via certain singular spacetime geometries. When one of the specified charges is the number of D-branes, the effective singular geometry is the half-BPS ``superstar''. We propose this as the general mechanism by which the effective thermodynamic character of gravity emerges.Comment: LaTeX, 6 eps figures, uses young.sty and wick.sty; Version 2: typos corrected, minor rewordings and clarifications, references adde

Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

Abstract. Let A be an n by N real-valued matrix with n < N; we count the number of k-faces fk(AQ) when Q is either the standard N-dimensional hypercube IN or else the positive orthant RN +. To state results simply, consider a proportional-growth asymptotic, where for fixed δ, ρ in (0, 1), we have a sequence of matrices An,Nn and of integers kn with n/Nn → δ, kn/n → ρ as n → ∞. If each matrix An,Nn has its columns in general position, then fk(AIN)/fk(I N) tends to zero or one depending on whether ρ> min(0, 2 − δ−1) or ρ < min(0, 2 − δ−1). Also, if each An,Nn is a random draw from a distribution which is invariant under right multiplication by signed permutations, then fk(ARN +)/fk(RN +) tends almost surely to zero or one depending on whether ρ> min(0, 2 − δ−1) or ρ < min(0, 2 − δ−1). We make a variety of contrasts to related work on projections of the simplex and/or cross-polytope. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Indeed, face counting is related to conditions for uniqueness of solutions of underdetermine