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European Mathematical Society Publishing House
2024-03-29 02:44:35
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=AIHPD&vol=4&iss=3&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
4
2017
3
Basic properties of the infinite critical-FK random map
Linxiao
Chen
Université Paris-Sud, Orsay, France and CEA Saclay, Gif-sur-Yvette, France
Fortuin–Kasteleyn percolation, random planar maps, hamburger–cheeseburer bijection, local limits, recurrent graph, ergodicity of random graphs
In this paper we investigate the critical Fortuin–Kasteleyn (cFK) random map model. For each $q \in [0, \infty]$ and integer $n \geq 1$, this model chooses a planar map of $n$ edges with a probability proportional to the partition function of critical $q$-Potts model on that map. She eld introduced the hamburger–cheeseburer bijection which maps the cFK random maps to a family of random words, and remarked that one can construct in finite cFK random maps using this bijection. We make this idea precise by a detailed proof of the local convergence. When $q = 1$, this provides an alternative construction of the UIPQ. In addition, we show that the limit is almost surely one-ended and recurrent for the simple random walk for any $q$, and mutually singular in distribution for di fferent values of $q$.
Probability theory and stochastic processes
Combinatorics
245
271
10.4171/AIHPD/40
http://www.ems-ph.org/doi/10.4171/AIHPD/40
Conformal invariance of dimer heights on isoradial double graphs
Zhongyang
Li
University of Connecticut, Storrs, USA
Dimer model, perfect matching, conformal invariance, Gaussian free field, isoradial graph
An isoradial graph is a planar graph in which each face is inscribable into a circle of common radius. We study the 2-dimensional perfect matchings on a bipartite isoradial graph, obtained from the union of an isoradial graph and its interior dual graph. Using the isoradial graph to approximate a simply-connected domain bounded by a simple closed curve, by letting the mesh size go to zero, we prove that in the scaling limit, the distribution of height is conformally invariant and converges to a Gaussian free field.
Statistical mechanics, structure of matter
Functions of a complex variable
Probability theory and stochastic processes
273
307
10.4171/AIHPD/41
http://www.ems-ph.org/doi/10.4171/AIHPD/41
Revisiting the combinatorics of the 2D Ising model
Dmitry
Chelkak
Ecole Normale Supérieure, Paris, France
David
Cimasoni
Université de Genève, Switzerland
Adrien
Kassel
ETH Zürich, Switzerland
Ising model, Kac–Ward matrix, spin correlations, fermionic observables, discrete holomorphic functions, spin structures, double-Ising model
We provide a concise exposition with original proofs of combinatorial formulas for the 2D Ising model partition function, multi-point fermionic observables, spin and energy density correlations, for general graphs and interaction constants, using the language of Kac–Ward matrices. We also give a brief account of the relations between various alternative formalisms which have been used in the combinatorial study of the planar Ising model: dimers and Grassmann variables, spin and disorder operators, and, more recently, s-holomorphic observables. In addition, we point out that these formulas can be extended to the double-Ising model, de ned as a pointwise product of two Ising spin con gurations on the same discrete domain, coupled along the boundary.
Statistical mechanics, structure of matter
309
385
10.4171/AIHPD/42
http://www.ems-ph.org/doi/10.4171/AIHPD/42
An application of cubical cohomology to Adinkras and supersymmetry representations
Charles
Doran
University of Alberta, Edmonton, Canada
Kevin
Iga
Pepperdine University, Malibu, USA
Gregory
Landweber
Bard College, Annandale-on-Hudson, USA
Cubical cohomology, supersymmetry, Adinkras, signed graphs
An Adinkra is a class of graphs with certain signs marking its vertices and edges, which encodes off-shell representations of the super Poincaré algebra. The markings on the vertices and edges of an Adinkra are cochains for cubical cohomology. This article explores the cubical cohomology of Adinkras, treating these markings analogously to characteristic classes on smooth manifolds.
Quantum theory
Combinatorics
Algebraic topology
Information and communication, circuits
387
415
10.4171/AIHPD/43
http://www.ems-ph.org/doi/10.4171/AIHPD/43