Crossing Probabilities of Multiple Ising Interfaces

We prove that in the scaling limit, the crossing probabilities of multiple interfaces in the critical planar Ising model with alternating boundary conditions are conformally invariant expressions given by the pure partition functions of multiple SLE(\kappa) with \kappa=3. In particular, this identifies the scaling limits with ratios of specific correlation functions of conformal field theory.Comment: 30 pages, 1 figure; v3: removed an appendix & other minor improvement

Schramm's formula for multiple loop-erased random walks

We revisit the computation of the discrete version of Schramm's formula for the loop-erased random walk derived by Kenyon. The explicit formula in terms of the Green function relies on the use of a complex connection on a graph, for which a line bundle Laplacian is defined. We give explicit results in the scaling limit for the upper half-plane, the cylinder and the Moebius strip. Schramm's formula is then extended to multiple loop-erased random walks.Comment: 59 pages, 19 figures. v2: reformulation of Section 2.3, minor correction

Sandpile probabilities on triangular and hexagonal lattices

We consider the Abelian sandpile model on triangular and hexagonal lattices. We compute several height probabilities on the full plane and on half-planes, and discuss some properties of the universality of the model.Comment: 26 pages, 12 figures. v2 and v3: minor correction

Boundary Partitions in Trees and Dimers

Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure, i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities, and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural algebraic interpretation and can be computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiple-strand SLE_2, SLE_8, and SLE_4.Comment: 46 pages, 12 figures. v4 has additional diagrams and other minor change

Exact solution of the 2d2d dimer model: Corner free energy, correlation functions and combinatorics

In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model {\it via} the so-called height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predict particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms.Comment: Final version to be published in Nuclear Physics B (53 pages and a lot of figures

Perfect matchings: Modified Aztec diamonds, covering graphs andn-matchings

In the Introduction, we present the problems we are going to study and we establish the basic definitions, concepts and results that are used throughout. We begin the first chapter with a presentation of the Aztec diamond and the behaviour of its random domino tilings. We introduce the dual-matching-problem and we explore the structure of the perfect matchings of modified Aztec diamonds. We show that some of these matchings can be extended to matchings of the dual Aztec diamond, pointing out a bijection between these types of matchings. We determine the number of perfect matchings for each of the modified graphs and the placement probabilities of the edges belonging to such a matching at a given location. We conclude with a theorem presenting the common asymptotic behaviour of the dual and the modified Aztec diamonds and we deduce a version of the Arctic Circle Theorem for these graphs. The second part is dedicated to the study of non-ramified perfect n-matchings, their decomposition into perfect matchings and 2-matchings as well as their relations to the perfect matchings of covering graphs. For the n-covering graphs we use the permutation derived graph construction. We determine the number of liftings of a given n-matching to a matching of a branched covering graph and then of a n-covering graph, together with necessary and sufficient conditions for the existence of the lifting. In particular, for the case of 2-matchings, we obtain a uniform behaviour of liftings of cycles. First, we deduce a theorem that relates the number of perfect matchings of the branched covering graph we have introduced to the number of perfect 2-matchings of the initial graph. Then we study the 2-covering graphs, their number, we determine the number of liftings of a 2-matchings (as a power of 2) and we obtain a theorem that characterizes the 2-matchings as the average of perfect matchings of 2-covering graphs. We conclude with some considerations about the maximum, minimum and the realization of this average and methods of computing it