7 research outputs found
Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane
A random walk in spatially homogeneous in the interior, absorbed at
the axes, starting from an arbitrary point and with step
probabilities drawn on Figure 1 is considered. The trivariate generating
function of probabilities that the random walk hits a given point at a given time is made explicit. Probabilities of absorption
at a given time and at a given axis are found, and their precise asymptotic
is derived as the time . The equivalence of two typical ways of
conditioning this random walk to never reach the axes is established. The
results are also applied to the analysis of the voter model with two candidates
and initially, in the population , four connected blocks of same opinions.
Then, a citizen changes his mind at a rate proportional to the number of its
neighbors that disagree with him. Namely, the passage from four to two blocks
of opinions is studied.Comment: 11 pages, 1 figur
On the functions counting walks with small steps in the quarter plane
Models of spatially homogeneous walks in the quarter plane
with steps taken from a subset of the set of jumps to the eight
nearest neighbors are considered. The generating function of the numbers of such walks starting at the origin and
ending at after steps is studied. For all
non-singular models of walks, the functions and are continued as multi-valued functions on having
infinitely many meromorphic branches, of which the set of poles is identified.
The nature of these functions is derived from this result: namely, for all the
51 walks which admit a certain infinite group of birational transformations of
, the interval of variation of splits into
two dense subsets such that the functions and are shown to be holonomic for any from the one of them and
non-holonomic for any from the other. This entails the non-holonomy of
, and therefore proves a conjecture of
Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure
A HUMAN PROOF OF GESSELâS LATTICE PATH CONJECTURE (PRELIMINARY VERSION)
Abstract. Gessel walks are planar walks confined to the positive quarter plane, that move by unit steps in any of the following directions: West, North-East, East and South-West. In 2001, Ira Gessel conjectured a closed-form expression for the number of Gessel walks of a given length starting and ending at the origin. In 2008, Kauers, Koutschan and Zeilberger gave a computer-aided proof of this conjecture. The same year, Bostan and Kauers showed, using again computer algebra tools, that the trivariate generating function of Gessel walks is algebraic. In this article we propose the first âhuman proofs â of these results. They are derived from a new expression for the generating function of Gessel walks. hal-00858083, version 1- 4 Sep 2013 1
ON THE NATURE OF THE GENERATING SERIES OF WALKS IN THE QUARTER PLANE
International audienceIn the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, i.e., we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients