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European Mathematical Society Publishing House
2024-03-29 16:40:17
2
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=AIHPD&vol=2&iss=2&update_since=2024-03-29
Annales de l’Institut Henri Poincaré D
Ann. Inst. Henri Poincaré Comb. Phys. Interact.
AIHPD
2308-5827
2308-5835
General
Combinatorics
Quantum theory
10.4171/AIHPD
http://www.ems-ph.org/doi/10.4171/AIHPD
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
2
2015
2
Kac–Ward operators, Kasteleyn operators, and s-holomorphicity on arbitrary surface graphs
David
Cimasoni
Université de Genève, GENÈVE 4, SWITZERLAND
Kac-Ward operator, Kasteleyn operator, s-holomorphic functions, Ising model
The conformal invariance and universality results of Chelkak-Smirnov on the two-dimensional Ising model hold for isoradial planar graphs with critical weights. Motivated by the problem of extending these results to a wider class of graphs, we de fine a generalized notion of s-holomorphicity for functions on arbitrary weighted surface graphs. We then give three criteria for s-holomorphicity involving the Kac–Ward, Kasteleyn, and discrete Dirac operators, respectively. Also, we show that some crucial results known to hold in the planar isoradial case extend to this general setting: in particular, spin-Ising fermionic observables are s-holomorphic, and it is possible to de fine a discrete version of the integral of the square of an s-holomorphic function. Along the way, we obtain a duality result for Kac–Ward determinants on arbitrary weighted surface graphs.
Statistical mechanics, structure of matter
Manifolds and cell complexes
113
168
10.4171/AIHPD/16
http://www.ems-ph.org/doi/10.4171/AIHPD/16
Analyticity results for the cumulants in a random matrix model
Razvan
Gurau
Ecole Polytechnique, PALAISEAU CEDEX, FRANCE
Thomas
Krajewski
Aix-Marseille Université, CNRS Luminy, MARSEILLE CEDEX 9, FRANCE
Random matrices, topological expansion
Th e generating function of the cumulants in random matrix models, as well as the cumulants themselves, can be expanded as asymptotic (divergent) series indexed by maps. While at fixed genus the sums over maps converge, the sums over genera do not. In this paper we obtain alternative expansions both for the generating function and for the cumulants that cure this problem. We provide explicit and convergent expansions for the cumulants, for the remainders of their perturbative expansion (in the size of the maps) and for the remainders of their topological expansion (in the genus of the maps). We show that any cumulant is an analytic function inside a cardioid domain in the complex plane and we prove that any cumulant is Borel summable at the origin.
Probability theory and stochastic processes
169
228
10.4171/AIHPD/17
http://www.ems-ph.org/doi/10.4171/AIHPD/17