263 research outputs found
The clairvoyant demon has a hard task
Consider the integer lattice L = ℤ2. For some m [ges ] 4, let us colour each column of this lattice independently and uniformly with one of m colours. We do the same for the rows, independently of the columns. A point of L will be called blocked if its row and column have the same colour. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m [ges ] 4 the configuration percolates with positive probability. This question remains open, but we prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture
Kinetics of ballistic annihilation and branching
We consider a one-dimensional model consisting of an assembly of two-velocity
particles moving freely between collisions. When two particles meet, they
instantaneously annihilate each other and disappear from the system. Moreover
each moving particle can spontaneously generate an offspring having the same
velocity as its mother with probability 1-q. This model is solved analytically
in mean-field approximation and studied by numerical simulations. It is found
that for q=1/2 the system exhibits a dynamical phase transition. For q<1/2, the
slow dynamics of the system is governed by the coarsening of clusters of
particles having the same velocities, while for q>1/2 the system relaxes
rapidly towards its stationary state characterized by a distribution of small
cluster sizes.Comment: 10 pages, 11 figures, uses multicol, epic, eepic and eepicemu. Also
avaiable at http://mykonos.unige.ch/~rey/pubt.htm
Polymer pinning in a random medium as influence percolation
In this article we discuss a set of geometric ideas which shed some light on
the question of directed polymer pinning in the presence of bulk disorder.
Differing from standard methods and techniques, we transform the problem to a
particular dependent percolative system and relate the pinning transition to a
percolation transition
Predicting the Drug Release Kinetics of Matrix Tablets
In this paper we develop two mathematical models to predict the release
kinetics of a water soluble drug from a polymer/excipient matrix tablet. The
first of our models consists of a random walk on a weighted graph, where the
vertices of the graph represent particles of drug, excipient and polymer,
respectively. The graph itself is the contact graph of a multidisperse random
sphere packing. The second model describes the dissolution and the subsequent
diffusion of the active drug out of a porous matrix using a system of partial
differential equations. The predictions of both models show good qualitative
agreement with experimental release curves. The models will provide tools for
designing better controlled release devices.Comment: 17 pages, 7 figures; Elaborated at the first Workshop on the
Application of Mathematics to Problems in Biomedicine, December 17-19, 2007
at the University of Otago in Dunedin, New Zealan
The universal Airy_1 and Airy_2 processes in the Totally Asymmetric Simple Exclusion Process
In the totally asymmetric simple exclusion process (TASEP) two processes
arise in the large time limit: the Airy_1 and Airy_2 processes. The Airy_2
process is an universal limit process occurring also in other models: in a
stochastic growth model on 1+1-dimensions, 2d last passage percolation,
equilibrium crystals, and in random matrix diffusion. The Airy_1 and Airy_2
processes are defined and discussed in the context of the TASEP. We also
explain a geometric representation of the TASEP from which the connection to
growth models and directed last passage percolation is immediate.Comment: 13 pages, 4 figures, proceeding for the conference in honor of Percy
Deift's 60th birthda
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