298 research outputs found
Walks confined in a quadrant are not always D-finite
We consider planar lattice walks that start from a prescribed position, take
their steps in a given finite subset of Z^2, and always stay in the quadrant x
>= 0, y >= 0. We first give a criterion which guarantees that the length
generating function of these walks is D-finite, that is, satisfies a linear
differential equation with polynomial coefficients. This criterion applies,
among others, to the ordinary square lattice walks. Then, we prove that walks
that start from (1,1), take their steps in {(2,-1), (-1,2)} and stay in the
first quadrant have a non-D-finite generating function. Our proof relies on a
functional equation satisfied by this generating function, and on elementary
complex analysis.Comment: To appear in Theoret. Comput. Sci. (special issue devoted to random
generation of combinatorial objects and bijective combinatorics
The asymptotic determinant of the discrete Laplacian
We compute the asymptotic determinant of the discrete Laplacian on a
simply-connected rectilinear region in R^2. As an application of this result,
we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.Comment: 36 pages, 4 figures, to appear in Acta Mathematic
Walks on the slit plane: other approaches
Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is
a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i
belongs to S for all i, and none of the points w_i, i>0, lie on the half-line
H= {(k,0): k =< 0}.
In a recent paper, G. Schaeffer and the author computed the length generating
function S(t) of walks on the slit plane for several sets S. All the generating
functions thus obtained turned out to be algebraic: for instance, on the
ordinary square lattice,
S(t) =\frac{(1+\sqrt{1+4t})^{1/2}(1+\sqrt{1-4t})^{1/2}}{2(1-4t)^{3/4}}.
The combinatorial reasons for this algebraicity remain obscure.
In this paper, we present two new approaches for solving slit plane models.
One of them simplifies and extends the functional equation approach of the
original paper. The other one is inspired by an argument of Lawler; it is more
combinatorial, and explains the algebraicity of the product of three series
related to the model. It can also be seen as an extension of the classical
cycle lemma. Both methods work for any set of steps S.
We exhibit a large family of sets S for which the generating function of
walks on the slit plane is algebraic, and another family for which it is
neither algebraic, nor even D-finite. These examples give a hint at where the
border between algebraicity and transcendence lies, and calls for a complete
classification of the sets S.Comment: 31 page
Conformal invariance of crossing probabilities for the Ising model with free boundary conditions
We prove that crossing probabilities for the critical planar Ising model with
free boundary conditions are conformally invariant in the scaling limit, a
phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin.
We do so by establishing the convergence of certain exploration processes
towards SLE. We also construct an exploration tree
for free boundary conditions, analogous to the one introduced by Sheffield.Comment: 18 pages, 4 figures, v2: journal versio
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