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European Mathematical Society Publishing House
2024-03-29 12:18:54
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=AIHPD&vol=4&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
1
$q$-randomized Robinson–Schensted–Knuth correspondences and random polymers
Konstantin
Matveev
Harvard University, CAMBRIDGE, UNITED STATES
Leonid
Petrov
Russian Academy of Sciences, MOSCOW, RUSSIAN FEDERATION
Robinson–Schensted–Knuth correspondence, random polymers,$q$-TASEP, Macdonald processes, random partitions, randomized insertion algorithm, interlacing particle arrays.
We introduce and study $q$-randomized Robinson–Schensted–Knuth (RSK) correspondences which interpolate between the classical ($q=0$) and geometric $q \nearrow1$) RSK correspondences (the latter ones are sometimes also called tropical). For $0
Probability theory and stochastic processes
Combinatorics
Quantum theory
Statistical mechanics, structure of matter
1
123
10.4171/AIHPD/36
http://www.ems-ph.org/doi/10.4171/AIHPD/36
2
Comparing two statistical ensembles of quadrangulations: a continued fraction approach
Éric
Fusy
École Polytechnique, PALAISEAU CEDEX, FRANCE
Emmanuel
Guitter
CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE
Planar maps, quadrangulations, two-point functions, continued fractions, hard dimers, heaps theory
We use a continued fraction approach to compare two statistical ensembles of quadrangulations with a boundary, both controlled by two parameters. In the fi rst ensemble, the quadrangulations are bicolored and the parameters control their numbers of vertices of both colors. In the second ensemble, the parameters control instead the number of vertices which are local maxima for the distance to a given vertex, and the number of those which are not. Both ensembles may be described either by their (bivariate) generating functions at fixed boundary length or, after some standard slice decomposition, by their (bivariate) slice generating functions. We fi rst show that the fixed boundary length generating functions are in fact equal for the two ensembles. We then show that the slice generating functions, although di fferent for the two ensembles, simply correspond to two di fferent ways of encoding the same quantity as a continued fraction. This property is used to obtain explicit expressions for the slice generating functions in a constructive way.
Combinatorics
General
125
176
10.4171/AIHPD/37
http://www.ems-ph.org/doi/10.4171/AIHPD/37
The distance-dependent two-point function of triangulations: a new derivation from old results
Emmanuel
Guitter
CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE
Planar maps, triangulations, two-point function, hull
We present a new derivation of the distance-dependent two-point function of random planar triangulations. As it is well-known, this function is intimately related to the generating functions of so-called slices, which are pieces of triangulation having boundaries made of shortest paths of prescribed length. We show that the slice generating functions are fully determined by a direct recursive relation on their boundary length. Remarkably, the kernel of this recursion is some quantity introduced and computed by Tutte a long time ago in the context of a global enumeration of planar triangulations. We may thus rely on these old results to solve our new recursion relation explicitly in a constructive way.
Combinatorics
177
211
10.4171/AIHPD/38
http://www.ems-ph.org/doi/10.4171/AIHPD/38
The distance-dependent two-point function of quadrangulations: a new derivation by direct recursion
Emmanuel
Guitter
CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE
Planar maps, quadrangulations, two-point function
We give a new derivation of the distance-dependent two-point function of planar quadrangulations by solving a new direct recursion relation for the associated slice generating functions. Our approach for both the derivation and the solution of this new recursion is in all points similar to that used recently by the author in the context of planar triangulations.
Combinatorics
213
244
10.4171/AIHPD/39
http://www.ems-ph.org/doi/10.4171/AIHPD/39
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
4
Terminal chords in connected chord diagrams
Julien
Courtiel
Université Paris 13, France
Karen
Yeats
University of Waterloo, Canada
Chord diagrams, terminal chords, quantum field theory, Dyson–Schwinger equations, enumerative combinatorics, analytic combinatorics
Rooted connected chord diagrams form a nice class of combinatorial objects. Recently they were shown to index solutions to certain Dyson–Schwinger equations in quantum field theory. Key to this indexing role are certain special chords which are called terminal chords. Terminal chords provide a number of combinatorially interesting parameters on rooted connected chord diagrams which have not been studied previously. Understanding these parameters better has implications for quantum fi eld theory. Speci cally, we show that the distributions of the number of terminal chords and the number of adjacent terminal chords are asymptotically Gaussian with logarithmic means, and we prove that the average index of the first terminal chord is $2n/3$. Furthermore, we obtain a method to determine any next-to$^i$ leading log expansion of the solution to these Dyson–Schwinger equations, and have asymptotic information about the coe cients of the log expansions.
Combinatorics
Quantum theory
417
452
10.4171/AIHPD/44
http://www.ems-ph.org/doi/10.4171/AIHPD/44
12
4
2017
A combinatorial identity for the speed of growth in an anisotropic KPZ model
Sunil
Chhita
Bonn University, Germany
Patrik
Ferrari
Bonn University, Germany
Random surfaces, interacting particle systems, random tilings, limit shapes, determinantal processes, Kasteleyn matrices
The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [5], which belongs to the KPZ anisotropic universality class, was computed using multi-time correlations. The model was recently generalized by Toninelli in [38] and for this generalization the stationarymeasure is known but the time correlations are unknown. In this note, we obtain algebraic and combinatorial proofs for the expression of the speed of growth from the prescribed dynamics.
Combinatorics
Probability theory and stochastic processes
Statistical mechanics, structure of matter
453
477
10.4171/AIHPD/45
http://www.ems-ph.org/doi/10.4171/AIHPD/45
12
4
2017
Dimers on rail yard graphs
Cédric
Boutillier
Université Pierre et Marie Curie, Paris, France
Jérémie
Bouttier
Université Paris-Saclay, Gif-sur-Yvette, France
Guillaume
Chapuy
Université Paris Diderot, France
Sylvie
Corteel
Université Paris Diderot, France
Sanjay
Ramassamy
Brown University, Providence, USA
Schur process, dimer models, Aztec diamond, plane partitions, determinantal processes, free fermions, vertex operators
We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs (RYG). The transfer matrices used to compute the partition function are shown to be isomorphic to certain operators arising in the so-called boson–fermion correspondence. This allows to reformulate the RYG dimer model as a Schur process, i.e. as a random sequence of integer partitions subject to some interlacing conditions. Beyond the computation of the partition function, we provide an explicit expression for all correlation functions or, equivalently, for the inverse Kasteleyn matrix of the RYG dimer model. This expression, which is amenable to asymptotic analysis, follows from an exact combinatorial description of the operators localizing dimers in the transfer-matrix formalism, and then a suitable application of Wick’s theorem. Plane partitions, domino tilings of the Aztec diamond, pyramid partitions, and steep tilings arise as particular cases of the RYG dimermodel. For the Aztec diamond, we provide new derivations of the edge-probability generating function, of the biased creation rate, of the inverse Kasteleyn matrix and of the arctic circle theorem.
Statistical mechanics, structure of matter
Combinatorics
Probability theory and stochastic processes
479
539
10.4171/AIHPD/46
http://www.ems-ph.org/doi/10.4171/AIHPD/46
12
4
2017