94 research outputs found
Green functions and Martin compactification for killed random walks related to SU(3)
We consider the random walks killed at the boundary of the quarter plane,
with homogeneous non-zero jump probabilities to the eight nearest neighbors and
drift zero in the interior, and which admit a positive harmonic polynomial of
degree three. For these processes, we find the asymptotic of the Green
functions along all infinite paths of states, and from this we deduce that the
Martin compactification is the one-point compactification.Comment: 13 page
Green functions for killed random walks in the Weyl chamber of Sp(4)
We consider a family of random walks killed at the boundary of the Weyl
chamber of the dual of , which in addition satisfies the following
property: for any , there is in this family a walk associated with a
reflection group of order . Moreover, the case corresponds to a
process which appears naturally by studying quantum random walks on the dual of
. For all the processes belonging to this family, we find the exact
asymptotic of the Green functions along all infinite paths of states as well as
that of the absorption probabilities along the boundaries.Comment: 20 page
Counting walks in a quadrant: a unified approach via boundary value problems
The aim of this article is to introduce a unified method to obtain explicit
integral representations of the trivariate generating function counting the
walks with small steps which are confined to a quarter plane. For many models,
this yields for the first time an explicit expression of the counting
generating function. Moreover, the nature of the integrand of the integral
formulations is shown to be directly dependent on the finiteness of a naturally
attached group of birational transformations as well as on the sign of the
covariance of the walkComment: 28 pages; 6 figure
On the exit time from a cone for Brownian motion with drift
We investigate the tail distribution of the first exit time of Brownian
motion with drift from a cone and find its exact asymptotics for a large class
of cones. Our results show in particular that its exponential decreasing rate
is a function of the distance between the drift and the cone, whereas the
polynomial part in the asymptotics depends on the position of the drift with
respect to the cone and its polar cone, and reflects the local geometry of the
cone at the point where the drift is orthogonally projected
New steps in walks with small steps in the quarter plane
In this article we obtain new expressions for the generating functions
counting (non-singular) walks with small steps in the quarter plane. Those are
given in terms of infinite series, while in the literature, the standard
expressions use solutions to boundary value problems. We illustrate our results
with three examples (an algebraic case, a transcendental D-finite case, and an
infinite group model).Comment: 47 pages, 8 figures, to appear in Annals of Combinatoric
On the exit time from a cone for random walks with drift
We compute the exponential decay of the probability that a given
multi-dimensional random walk stays in a convex cone up to time , as
goes to infinity. We show that the latter equals the minimum, on the dual cone,
of the Laplace transform of the random walk increments. As an example, our
results find applications in the counting of walks in orthants, a classical
domain in enumerative combinatorics.Comment: 21 pages, 2 figures, to appear in Revista Matem\'atica Iberoamerican
Explicit expression for the generating function counting Gessel's walks
Gessel's walks are the planar walks that move within the positive quadrant
by unit steps in any of the following directions: West,
North-East, East and South-West. In this paper, we find an explicit expression
for the trivariate generating function counting the Gessel's walks with steps, which start at and end at a given point .Comment: 23 page
About a possible analytic approach for walks in the quarter plane with arbitrary big jumps
In this note, we consider random walks in the quarter plane with arbitrary
big jumps. We announce the extension to that class of models of the analytic
approach of [G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the
quarter plane, Springer-Verlag, Berlin (1999)], initially valid for walks with
small steps in the quarter plane. New technical challenges arise, most of them
being tackled in the framework of generalized boundary value problems on
compact Riemann surfaces.Comment: 7 pages, 3 figures, extended abstrac
Tutte's invariant approach for Brownian motion reflected in the quadrant
We consider a Brownian motion with drift in the quarter plane with orthogonal
reflection on the axes. The Laplace transform of its stationary distribution
satisfies a functional equation, which is reminiscent from equations arising in
the enumeration of (discrete) quadrant walks. We develop a Tutte's invariant
approach to this continuous setting, and we obtain an explicit formula for the
Laplace transform in terms of generalized Chebyshev polynomials.Comment: 14 pages, 3 figure
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