Philippe Ruelle

Scaling fields in the two-dimensional Abelian sandpile model.

We consider the unoriented two-dimensional Abelian sandpile model from a perspective based on two-dimensional (conformal) field theory. We compute lattice correlation functions for various cluster variables (at and off criticality), from which we infer the field-theoretic description in the scaling limit. We find perfect agreement with the predictions of a c=-2 conformal field theory and its massive perturbation, thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory. The question of the height 2 variable is also addressed, with, however, no definite conclusion yet.

QUANTUM MECHANICS ON P-ADIC FIELDS

A formulation of quantum mechanics on p‐adic number fields is presented. Quantum amplitudes are taken as complex functions of p‐adic variables and it is shown how the Weyl approach to quantum mechanics can be generalized to the p‐adic case. The p‐adic analogs of simple one‐dimensional systems (free particle, compact and noncompact oscillators) are defined by a ‘‘group of motion,’’ which is an Abelian subgroup of SL (2,Qp). In each case the evolution operator is a unitary representation of the appropriate group. Its spectrum is given by characters and its eigenstates are calculated.

Height variables in the Abelian sandpile model: Scaling fields and correlations

We compute the lattice 1-site probabilities, on the upper half-plane, of the four height variables in the two-dimensional Abelian sandpile model. We find their exact scaling form when the insertion point is far from the boundary, and when the boundary is either open or closed. Comparing with the predictions of a logarithmic conformal theory with central charge c=-2, we find a full compatibility with the following field assignments: the heights 2, 3 and 4 behave like (an unusual realization of) the logarithmic partner of a primary field with scaling dimension 2, the primary field itself being associated with the height 1 variable. Finite size corrections are also computed and successfully compared with numerical simulations. Relying on these field assignments, we formulate a conjecture for the scaling form of the lattice 2-point correlations of the height variables on the plane, which remain as yet unknown. The way conformal invariance is realized in this system points to a local field theory with c=-2 which is different from the triplet theory. Comment: 68 pages, 17 figures; v2: published version (minor corrections, one comment added)

A c=-2 boundary changing operator for the Abelian sandpile model

We consider the unoriented two-dimensional Abelian sandpile model on the half-plane with open and closed boundary conditions. We show that the operator effecting the change from closed to open, or from open to closed, is a boundary primary field of weight -1/8, belonging to a c=-2 logarithmic conformal field theory. Comment: 5 pages, 1 figure, revtex4; comments added and Eq.(11) corrected, published version

Pre-logarithmic and logarithmic fields in a sandpile model

We consider the unoriented two-dimensional Abelian sandpile model on the half-plane with open and closed boundary conditions, and relate it to the boundary logarithmic conformal field theory with central charge c=-2. Building on previous results, we first perform a complementary lattice analysis of the operator effecting the change of boundary condition between open and closed, which confirms that this operator is a weight -1/8 boundary primary field, whose fusion agrees with lattice calculations. We then consider the operators corresponding to the unit height variable and to a mass insertion at an isolated site of the upper half plane and compute their one-point functions in presence of a boundary containing the two kinds of boundary conditions. We show that the scaling limit of the mass insertion operator is a weight zero logarithmic field. Comment: 18 pages, 9 figures. v2: minor corrections + added appendix

Logarithmic Conformal Field Theory and Boundary Effects in the Dimer Model

We study the finite-size corrections of the dimer model on a square lattice with two different boundary conditions: free and periodic. We find that the finite-size corrections depend in a crucial way on the parity of ; we also show that such unusual finite-size behavior can be fully explained in the framework of the logarithmic conformal field theory.

Logarithmic scaling for height variables in the Abelian sandpile model

We report on the exact computation of the scaling form of the 1-point function, on the upper-half plane, of the height 2 variable in the two-dimensional Abelian sandpile model. By comparing the open versus the closed boundary condition, we find that the scaling field associated to the height 2 is a logarithmic scalar field of scaling dimension 2, belonging to a c=-2 logarithmic conformal field theory. This identification is confirmed by numerical simulations and extended to the height 3 and 4 variables, which exhibit the same scaling form. Using the conformal setting, we make precise proposals for the bulk 2-point functions of all height variables. Comment: 7 pages, 2 figures

Logarithmic two-point correlators in the Abelian sandpile model

We present the detailed calculations of the asymptotics of two-site correlation functions for height variables in the two-dimensional Abelian sandpile model. By using combinatorial methods for the enumeration of spanning trees, we extend the well-known result for the correlation

Return probability for the loop-erased random walk and mean height in the Abelian sandpile model : a proof

\sigma_{1,1} \simeq 1/r^4

Grothendieck ring and Verlinde formula for the W-extended logarithmic minimal model WLM(1,p)

of minimal heights