36 research outputs found
Diffusion constants and martingales for senile random walks
We derive diffusion constants and martingales for senile random walks with
the help of a time-change. We provide direct computations of the diffusion
constants for the time-changed walks. Alternatively, the values of these
constants can be derived from martingales associated with the time-changed
walks. Using an inverse time-change, the diffusion constants for senile random
walks are then obtained via these martingales. When the walks are diffusive,
weak convergence to Brownian motion can be shown using a martingale functional
limit theorem.Comment: 17 pages, LaTeX; the proof of Proposition 2.3 has been simplified,
and an error in the proof of Theorem 2.4 has been correcte
Rotor-router aggregation on the layered square lattice
In rotor-router aggregation on the square lattice Z^2, particles starting at
the origin perform deterministic analogues of random walks until reaching an
unoccupied site. The limiting shape of the cluster of occupied sites is a disk.
We consider a small change to the routing mechanism for sites on the x- and
y-axes, resulting in a limiting shape which is a diamond instead of a disk. We
show that for a certain choice of initial rotors, the occupied cluster grows as
a perfect diamond.Comment: 11 pages, 3 figures
A Guide to Stochastic Loewner Evolution and its Applications
This article is meant to serve as a guide to recent developments in the study
of the scaling limit of critical models. These new developments were made
possible through the definition of the Stochastic Loewner Evolution (SLE) by
Oded Schramm. This article opens with a discussion of Loewner's method,
explaining how this method can be used to describe families of random curves.
Then we define SLE and discuss some of its properties. We also explain how the
connection can be made between SLE and the discrete models whose scaling limits
it describes, or is believed to describe. Finally, we have included a
discussion of results that were obtained from SLE computations. Some explicit
proofs are presented as typical examples of such computations. To understand
SLE sufficient knowledge of conformal mapping theory and stochastic calculus is
required. This material is covered in the appendices.Comment: 80 pages, 22 figures, LaTeX; this version has 5 minor corrections to
the text and improved hyperref suppor
Existence and uniqueness of the stationary measure in the continuous Abelian sandpile
Let \Lambda be a finite subset of Z^d. We study the following sandpile model
on \Lambda. The height at any given vertex x of \Lambda is a positive real
number, and additions are uniformly distributed on some interval [a,b], which
is a subset of [0,1]. The threshold value is 1; when the height at a given
vertex exceeds 1, it topples, that is, its height is reduced by 1, and the
heights of all its neighbours in \Lambda increase by 1/2d. We first establish
that the uniform measure \mu on the so called "allowed configurations" is
invariant under the dynamics. When a < b, we show with coupling ideas that
starting from any initial configuration of heights, the process converges in
distribution to \mu, which therefore is the unique invariant measure for the
process. When a = b, that is, when the addition amount is non-random, and a is
rational, it is still the case that \mu is the unique invariant probability
measure, but in this case we use random ergodic theory to prove this; this
proof proceeds in a very different way. Indeed, the coupling approach cannot
work in this case since we also show the somewhat surprising fact that when a =
b is rational, the process does not converge in distribution at all starting
from any initial configuration.Comment: 22 page
A short simple proof of closedness of convex cones and Farkas' lemma
Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas' lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas' lemma from it using well-known arguments
A short simple proof of closedness of convex cones and Farkas' lemma
Proving that a finitely generated convex cone is closed is often considered
the most difficult part of geometric proofs of Farkas' lemma. We provide a
short simple proof of this fact and (for completeness) derive Farkas' lemma
from it using well-known arguments.Comment: 2 pages; v2: note largely rewritten, provided more context, improved
presentation, added 5 reference
Exact Solutions for Loewner Evolutions
In this note, we solve the Loewner equation in the upper half-plane with
forcing function xi(t), for the cases in which xi(t) has a power-law dependence
on time with powers 0, 1/2 and 1. In the first case the trace of singularities
is a line perpendicular to the real axis. In the second case the trace of
singularities can do three things. If xi(t)=2*(kappa*t)^1/2, the trace is a
straight line set at an angle to the real axis. If xi(t)=2*(kappa*(1-t))^1/2,
the behavior of the trace as t approaches 1 depends on the coefficient kappa.
Our calculations give an explicit solution in which for kappa<4 the trace
spirals into a point in the upper half-plane, while for kappa>4 it intersects
the real axis. We also show that for kappa=9/2 the trace becomes a half-circle.
The third case with forcing xi(t)=t gives a trace that moves outward to
infinity, but stays within fixed distance from the real axis. We also solve
explicitly a more general version of the evolution equation, in which xi(t) is
a superposition of the values +1 and -1.Comment: 20 pages, 7 figures, LaTeX, one minor correction, and improved
hyperref
The asymptotics of group Russian roulette
We study the group Russian roulette problem, also known as the shooting
problem, defined as follows. We have armed people in a room. At each chime
of a clock, everyone shoots a random other person. The persons shot fall dead
and the survivors shoot again at the next chime. Eventually, either everyone is
dead or there is a single survivor. We prove that the probability of
having no survivors does not converge as , and becomes
asymptotically periodic and continuous on the scale, with period 1.Comment: 26 pages, 1 figure; Mathematica notebook and output file (calculated
exact bounds) are included with the source file
Reflected Brownian motion in generic triangles and wedges
Consider a generic triangle in the upper half of the complex plane with one
side on the real line. This paper presents a tailored construction of a
discrete random walk whose continuum limit is a Brownian motion in the
triangle, reflected instantaneously on the left and right sides with constant
reflection angles. Starting from the top of the triangle, it is evident from
the construction that the reflected Brownian motion lands with the uniform
distribution on the base. Combined with conformal invariance and the locality
property, this uniform exit distribution allows us to compute distribution
functions characterizing the hull generated by the reflected Brownian motion.Comment: LaTeX, 38 pages, 14 figures. This is the outcome of a complete
rewrite of the original paper. Results have been stated more clearly and the
proofs have been elucidate
Diamond Aggregation
Internal diffusion-limited aggregation is a growth model based on random walk
in Z^d. We study how the shape of the aggregate depends on the law of the
underlying walk, focusing on a family of walks in Z^2 for which the limiting
shape is a diamond. Certain of these walks -- those with a directional bias
toward the origin -- have at most logarithmic fluctuations around the limiting
shape. This contrasts with the simple random walk, where the limiting shape is
a disk and the best known bound on the fluctuations, due to Lawler, is a power
law. Our walks enjoy a uniform layering property which simplifies many of the
proofs.Comment: v2 addresses referee comments, new section on the abelian propert