On a remarkable class of two‐dimensional random walks

1980 ◽  
Vol 21 (8) ◽  
pp. 2111-2113 ◽  
Author(s):  
F. W. Wiegel
Keyword(s):  
Random Walks ◽  
Two Dimensional

scholarly journals Limit theorems for isotropic random walks on triangle buildings

2002 ◽  
Vol 73 (3) ◽  
pp. 301-334 ◽  
Author(s):  
Marc Lindlbauer ◽  
Michael Voit
Keyword(s):  
Limit Theorem ◽  
Random Walks ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Two Dimensional ◽  
Littlewood Polynomials ◽  
Large Numbers ◽  
Moment Functions ◽  
Vertex Sets ◽  
Isotropic Random

AbstractThe spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

scholarly journals ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS

2019 ◽  
Vol 107 (3) ◽  
pp. 392-411 ◽  
Author(s):  
YAJUN ZHOU
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Random Walks ◽  
Euclidean Plane ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Uniformly Convergent

Let $p_{n}(x)=\int _{0}^{\infty }J_{0}(xt)[J_{0}(t)]^{n}xt\,dt$ be Kluyver’s probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in two-dimensional quantum field theory, we evaluate the third-order derivative $p_{5}^{\prime \prime \prime }(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $p_{n}(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $n\geq 5$, thus settling another conjecture of Borwein.

scholarly journals Limit theorems for one and two-dimensional random walks in random scenery

2013 ◽  
Vol 49 (2) ◽  
pp. 506-528 ◽  
Author(s):  
Fabienne Castell ◽  
Nadine Guillotin-Plantard ◽  
Françoise Pène
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Two Dimensional ◽  
Random Scenery

Random walks in two-dimensional glass-like environments

1999 ◽  
Vol 256 (4) ◽  
pp. 266-271 ◽  
Author(s):  
Beatrix Schulz ◽  
Steffen Trimper
Keyword(s):  
Random Walks ◽  
Two Dimensional
Sign in / Sign up

Export Citation Format

Share Document