1,393 research outputs found
Renormalisation Group Theory of Branching Potts Interfaces
We develop a field-theoretic representation for the configurations of an
interface between two ordered phases of a q-state Potts model in two
dimensions, in the solid-on-solid approximation. The model resembles the field
theory of directed percolation and may be analysed using similar
renormalisation group methods. In the one-loop approximation these reveal a
simple mechanism for the emergence of a critical value q_c, such that for q<q_c
the interface becomes a fractal with a vanishing interfacial tension at the
critical point, while for q>q_c the interfacial width diverges at a finite
value of the tension, indicating a first-order transition. The value of the
Widom exponent for q<q_c within this approximation is in fair agreement with
known exact values. Some comments are made on the case of quenched randomness.
We also show that the q-> minus infinity limit of our model corresponds to
directed percolation and that the values for the exponents in the one-loop
approximation are in reasonable agreement with accepted values.Comment: 24 page
The coalescing-branching random walk on expanders and the dual epidemic process
Information propagation on graphs is a fundamental topic in distributed
computing. One of the simplest models of information propagation is the push
protocol in which at each round each agent independently pushes the current
knowledge to a random neighbour. In this paper we study the so-called
coalescing-branching random walk (COBRA), in which each vertex pushes the
information to randomly selected neighbours and then stops passing
information until it receives the information again. The aim of COBRA is to
propagate information fast but with a limited number of transmissions per
vertex per step. In this paper we study the cover time of the COBRA process
defined as the minimum time until each vertex has received the information at
least once. Our main result says that if is an -vertex -regular graph
whose transition matrix has second eigenvalue , then the COBRA cover
time of is , if is greater than a positive
constant, and , if . These bounds are independent of and hold for . They improve the previous bound of for expander graphs.
Our main tool in analysing the COBRA process is a novel duality relation
between this process and a discrete epidemic process, which we call a biased
infection with persistent source (BIPS). A fixed vertex is the source of an
infection and remains permanently infected. At each step each vertex other
than selects neighbours, independently and uniformly, and is
infected in this step if and only if at least one of the selected neighbours
has been infected in the previous step. We show the duality between COBRA and
BIPS which says that the time to infect the whole graph in the BIPS process is
of the same order as the cover time of the COBRA proces
Random walk on a randomly oriented honeycomb lattice
We study the recurrence behaviour of random walks on partially oriented
honeycomb lattices. The vertical edges are undirected while the orientation of
the horizontal edges is random: depending on their distribution, we prove a.s.
transience in some cases, and a.s. recurrence in other ones. The results extend
those obtained for the partially oriented square grid lattices (Campanino and
Petritis (2003), Campanino and Petritis (2014)).Comment: 21 pages, 1 figure; revised 'motivation' section and abstract; added
conclusion; minor correction
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