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Parafermionic observables and their applications to planar statistical physics models
, 2013
"... This volume is based on the PhD thesis of the author. Through the examples of the selfavoiding walk, the randomcluster model, the Ising model and others, the book explores in details two important techniques: 1. Discrete holomorphicity and parafermionic observables, which have been used in the pa ..."
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Cited by 7 (2 self)
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This volume is based on the PhD thesis of the author. Through the examples of the selfavoiding walk, the randomcluster model, the Ising model and others, the book explores in details two important techniques: 1. Discrete holomorphicity and parafermionic observables, which have been used in the past few years to study planar models of statistical physics (in particular their conformal invariance), such as randomcluster models and loop O(n)models. 2. The RussoSeymourWelsh theory for percolationtype models with dependence. This technique was initially available for Bernoulli percolation only. Recently, it has been extended to models with dependence, thus opening the way to a deeper study of their critical regime. The book is organized as follows. The first part provides a general introduction to planar statistical physics, as well as a first example of the parafermionic observable and its application to the computation of the connective constant for the selfavoiding walk on the hexagonal lattice. The second part deals with the family of randomcluster models. It studies the RussoSeymourWelsh theory of crossing probabilities for these models. As an application, the critical point of the randomcluster model is computed on the square lattice. Then, the parafermionic observable is introduced and two of its applications are described in detail. This part contains a chapter describing basic properties of the randomcluster model. The third part is devoted to the Ising model and its randomcluster representation, the FKIsing model. After a first chapter gathering the basic properties of the Ising model, the theory of sholomorphic functions as well as Smirnov and ChelkakSmirnov’s proofs of conformal invariance (for these two models) are presented. Conformal invariance paves the way to a better understanding of the critical phase and the two next chapters are devoted to the study of the geometry of the critical phase, as well as the relation between the critical and nearcritical phases. The last part presents possible directions of future research by describing other models and several open questions.
Spectral curves of periodic Fisher graphs
"... We study the spectral curves of dimer models on periodic Fisher graphs, obtained from periodic ferromagnetic Ising models on Z2. The spectral curve is defined by the zero locus of the determinant of a modified weighted adjacency matrix. We prove that either they are disjoint from the unit torus (T2 ..."
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We study the spectral curves of dimer models on periodic Fisher graphs, obtained from periodic ferromagnetic Ising models on Z2. The spectral curve is defined by the zero locus of the determinant of a modified weighted adjacency matrix. We prove that either they are disjoint from the unit torus (T2 = {(z, w) : z  = 1, w  = 1}) or they intersect T2 at a single real point. 1
KacWard operators, Kasteleyn operators, and sholomorphicity on arbitrary surface graphs
, 1307
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Critical surface of the 12 model
, 2015
"... Abstract. The 12 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. There are three types of edge, and three corresponding parameters a, b, c. It is proved that, when a ≥ b ≥ c> 0, the surface given by √a = √b + √ ..."
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Abstract. The 12 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either 1 or 2. There are three types of edge, and three corresponding parameters a, b, c. It is proved that, when a ≥ b ≥ c> 0, the surface given by √a = √b + √c is critical. The proof hinges upon a representation of the partition function in terms of that of an Isingtype model on an enhanced graph, and thereby on the partition function of a dimer model. This dimer model may be studied via the Pfaffian representation of Fisher, Kasteleyn, and Temperley. It is proved, in addition, that the twoedge correlation function decays exponentially with distance when a2 < b2 + c2. 1. Introduction and
12 model, dimers and clusters
"... A 12 model is a probability measure on subgraphs of a hexagonal lattice, satisfying the condition that the degree of present edges at each vertex is either 1 or 2. We prove that for any translationinvariant Gibbs measure of the 12 model on the plane, almost surely there are no infinite paths. We ..."
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A 12 model is a probability measure on subgraphs of a hexagonal lattice, satisfying the condition that the degree of present edges at each vertex is either 1 or 2. We prove that for any translationinvariant Gibbs measure of the 12 model on the plane, almost surely there are no infinite paths. We discover a measurepreserving correspondence between the 12 model and the dimer model on a decorated graph, and construct an explicit translationinvariant measure P for 12 model on the infinite periodic hexagonal lattice. We prove that the behavior of infinite clusters is different for small and large local weights under the measure P, which is an evidence of the existence of a phase transition. 1 12 Model Computer Scientists M. Schwartz and J. Bruck ([12]) proposed the uniform 12 model (notallequalrelation), as a graphical model whose partition function (total number of configurations) can be computed by computing determinants via the holographic algorithm ([13]). A general version of the 12 model is explored in ([10]), as an application
LOCALITY OF CONNECTIVE CONSTANTS, I. TRANSITIVE GRAPHS
"... Abstract. The connective constant µ(G) of a quasitransitive graph G is the exponential growth rate of the number of selfavoiding walks from a given origin. We prove a locality theorem for connective constants, namely, that the connective constants of two graphs are close in value whenever the grap ..."
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Abstract. The connective constant µ(G) of a quasitransitive graph G is the exponential growth rate of the number of selfavoiding walks from a given origin. We prove a locality theorem for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. The proof exploits a generalized bridge decomposition of selfavoiding walks, which is valid subject to the assumption that the underlying graph is quasitransitive and possesses a socalled graph height function.
ON THE ASYMPTOTICS OF DIMERS ON TORI
"... Abstract. We study asymptotics of the dimer model on large toric graphs. Let L be a weighted Z2periodic planar graph, and let Z2E be a largeindex sublattice of Z2. For L bipartite we show that the dimer partition function ZE on the quotient L{pZ2Eq has the asymptotic expansion Z “ exptA f0 ` fsc ` ..."
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Abstract. We study asymptotics of the dimer model on large toric graphs. Let L be a weighted Z2periodic planar graph, and let Z2E be a largeindex sublattice of Z2. For L bipartite we show that the dimer partition function ZE on the quotient L{pZ2Eq has the asymptotic expansion Z “ exptA f0 ` fsc ` op1qu where A is the area of L{pZ2Eq, f0 is the free energy density in the bulk, and fsc is a finitesize correction term depending only on the conformal shape of the domain together with some paritytype information. Assuming a conjectural condition on the zero locus of the dimer characteristic polynomial, we show that an analogous expansion holds for L nonbipartite. The functional form of the finitesize correction differs between the two classes, but is universal within each class. Our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated doubledimer models. 1.
Principal minors Pfaffian halftree theorem
"... A halftree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a halftree theorem for the Pfaffian principal minors of a skewsymmetric matrix whose column sum is zero; introducing an explicit algorithm, we fully characterize halftrees involv ..."
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A halftree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a halftree theorem for the Pfaffian principal minors of a skewsymmetric matrix whose column sum is zero; introducing an explicit algorithm, we fully characterize halftrees involved. This question naturally arose in the context of statistical mechanics where we aimed at relating perfect matchings and trees on the same graph. As a consequence of the Pfaffian halftree theorem, we obtain a refined version of the matrixtree theorem in the case of skewsymmetric matrices, as well as a linebundle version of this result.