Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., realizing the partition function of the free fermion on a closed Riemann surface of genus g as a linear combination of 2^{2g} Pfaffians of Dirac operators. Let G=(V,E) be a finite graph embedded in a closed Riemann surface X of genus g, x_e the collection of independent variables associated with each edge e of G (collected in one vector variable x) and S the set of all 2^{2g} Spin-structures on X. We introduce 2^{2g} rotations rot_s and (2|E| times 2|E|) matrices D(s)(x), s in S, of the transitions between the oriented edges of G determined by rotations rot_s. We show that the generating function for the even subsets of edges of G, i.e., the Ising partition function, is a linear combination of the square roots of 2^{2g} Ihara-Selberg functions I(D(s)(x)) also called Feynman functions. By a result of Foata--Zeilberger holds I(D(s)(x))= det(I-D'(s)(x)), where D'(s)(x) is obtained from D(s)(x) by replacing some entries by 0. Thus each Feynman function is computable in polynomial time. We suggest that in the case of critical embedding of a bipartite graph G, the Feynman functions provide suitable discrete analogues for the Pfaffians of discrete Dirac operators

Small eigenvalues of random 3-manifolds

We show that for every $g\geq 2$ there exists a number $c(g)>0$ such that the smallest positive eigenvalue of a random closed 3-manifold $M$ of Heegaard genus $g$ is at most $c(g)/{\rm vol}(M)^2$.Comment: 52 pages. Major revisio

Analyzing Boltzmann Samplers for Bose-Einstein Condensates with Dirichlet Generating Functions

Boltzmann sampling is commonly used to uniformly sample objects of a particular size from large combinatorial sets. For this technique to be effective, one needs to prove that (1) the sampling procedure is efficient and (2) objects of the desired size are generated with sufficiently high probability. We use this approach to give a provably efficient sampling algorithm for a class of weighted integer partitions related to Bose-Einstein condensation from statistical physics. Our sampling algorithm is a probabilistic interpretation of the ordinary generating function for these objects, derived from the symbolic method of analytic combinatorics. Using the Khintchine-Meinardus probabilistic method to bound the rejection rate of our Boltzmann sampler through singularity analysis of Dirichlet generating functions, we offer an alternative approach to analyze Boltzmann samplers for objects with multiplicative structure.Comment: 20 pages, 1 figur