5,928 research outputs found
A -branching random walk with random selection
We consider an exactly solvable model of branching random walk with random
selection, which describes the evolution of a population with individuals
on the real line. At each time step, every individual reproduces independently,
and its offspring are positioned around its current locations. Among all
children, individuals are sampled at random without replacement to form the
next generation, such that an individual at position is chosen with
probability proportional to . We compute the asymptotic
speed and the genealogical behavior of the system
Roughening and inclination of competition interfaces
The competition interface between two growing ``Young clusters'' (diagrams),
in a two-dimensional random cone, is mapped to the path of a second-class
particle in the one-dimensional totally asymmetric simple exclusion process.
Using the asymptotics of the second class particle and hydrodynamic limits for
the exclusion process (Burgers equation), we show that the behavior of the
competition interface depends on the angle of the cone: for angles in [180^o,
270^o) the competition interface has a deterministic inclination, while for
angles in [90^o,180^o) the inclination is random. We relate the competition
model to a model of random directed polymers, and obtain some partial results
for the fluctuations of the competition interface.Comment: 4 pages, 3 figure
Multiple geodesics with the same direction
The directed last-passage percolation (LPP) model with independent
exponential times is considered. We complete the study of asymptotic directions
of infinite geodesics, started by Ferrari and Pimentel \cite{FP}. In
particular, using a recent result of \cite{CH2} and a local modification
argument, we prove there is no (random) direction with more than two geodesics
with probability 1.Comment: 10 pages, 1 figur
Degree Distribution of Competition-Induced Preferential Attachment Graphs
We introduce a family of one-dimensional geometric growth models, constructed
iteratively by locally optimizing the tradeoffs between two competing metrics,
and show that this family is equivalent to a family of preferential attachment
random graph models with upper cutoffs. This is the first explanation of how
preferential attachment can arise from a more basic underlying mechanism of
local competition. We rigorously determine the degree distribution for the
family of random graph models, showing that it obeys a power law up to a finite
threshold and decays exponentially above this threshold.
We also rigorously analyze a generalized version of our graph process, with
two natural parameters, one corresponding to the cutoff and the other a
``fertility'' parameter. We prove that the general model has a power-law degree
distribution up to a cutoff, and establish monotonicity of the power as a
function of the two parameters. Limiting cases of the general model include the
standard preferential attachment model without cutoff and the uniform
attachment model.Comment: 24 pages, one figure. To appear in the journal: Combinatorics,
Probability and Computing. Note, this is a long version, with complete
proofs, of the paper "Competition-Induced Preferential Attachment"
(cond-mat/0402268
Uncertain Price Competition in a Duopoly with Heterogeneous Availability
We study the price competition in a duopoly with an arbitrary number of
buyers. Each seller can offer multiple units of a commodity depending on the
availability of the commodity which is random and may be different for
different sellers. Sellers seek to select a price that will be attractive to
the buyers and also fetch adequate profits. The selection will in general
depend on the number of units available with the seller and also that of its
competitor - the seller may only know the statistics of the latter. The setting
captures a secondary spectrum access network, a non-neutral Internet, or a
microgrid network in which unused spectrum bands, resources of ISPs, and excess
power units constitute the respective commodities of sale. We analyze this
price competition as a game, and identify a set of necessary and sufficient
properties for the Nash Equilibrium (NE). The properties reveal that sellers
randomize their price using probability distributions whose support sets are
mutually disjoint and in decreasing order of the number of availability. We
prove the uniqueness of a symmetric NE in a symmetric market, and explicitly
compute the price distribution in the symmetric NE.Comment: 45 pages, Accepted for publication in IEEE Transaction on Automatic
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