187 research outputs found
Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime
In this paper we derive a technique of obtaining limit theorems for suprema
of L\'evy processes from their random walk counterparts. For each , let
be a sequence of independent and identically distributed
random variables and be a L\'evy processes such that
, and as . Let .
Then, under some mild assumptions, , for some random variable and some function
. We utilize this result to present a number of limit theorems
for suprema of L\'evy processes in the heavy-traffic regime
On the localized phase of a copolymer in an emulsion: supercritical percolation regime
In this paper we study a two-dimensional directed self-avoiding walk model of
a random copolymer in a random emulsion. The copolymer is a random
concatenation of monomers of two types, and , each occurring with
density 1/2. The emulsion is a random mixture of liquids of two types, and
, organised in large square blocks occurring with density and ,
respectively, where . The copolymer in the emulsion has an energy
that is minus times the number of -matches minus times the
number of -matches, where without loss of generality the interaction
parameters can be taken from the cone . To make the model mathematically tractable, we assume that the
copolymer is directed and can only enter and exit a pair of neighbouring blocks
at diagonally opposite corners.
In \cite{dHW06}, it was found that in the supercritical percolation regime , with the critical probability for directed bond percolation on
the square lattice, the free energy has a phase transition along a curve in the
cone that is independent of . At this critical curve, there is a transition
from a phase where the copolymer is fully delocalized into the -blocks to a
phase where it is partially localized near the -interface. In the present
paper we prove three theorems that complete the analysis of the phase diagram :
(1) the critical curve is strictly increasing; (2) the phase transition is
second order; (3) the free energy is infinitely differentiable throughout the
partially localized phase.Comment: 43 pages and 10 figure
Countable Random Sets: Uniqueness in Law and Constructiveness
The first part of this article deals with theorems on uniqueness in law for
\sigma-finite and constructive countable random sets, which in contrast to the
usual assumptions may have points of accumulation. We discuss and compare two
approaches on uniqueness theorems: First, the study of generators for
\sigma-fields used in this context and, secondly, the analysis of hitting
functions. The last section of this paper deals with the notion of
constructiveness. We will prove a measurable selection theorem and a
decomposition theorem for constructive countable random sets, and study
constructive countable random sets with independent increments.Comment: Published in Journal of Theoretical Probability
(http://www.springerlink.com/content/0894-9840/). The final publication is
available at http://www.springerlink.co
Expected length of the longest common subsequence for large alphabets
We consider the length L of the longest common subsequence of two randomly
uniformly and independently chosen n character words over a k-ary alphabet.
Subadditivity arguments yield that the expected value of L, when normalized by
n, converges to a constant C_k. We prove a conjecture of Sankoff and Mainville
from the early 80's claiming that C_k\sqrt{k} goes to 2 as k goes to infinity.Comment: 14 pages, 1 figure, LaTe
Positive temperature versions of two theorems on first-passage percolation
The estimates on the fluctuations of first-passsage percolation due to
Talagrand (a tail bound) and Benjamini--Kalai--Schramm (a sublinear variance
bound) are transcribed into the positive-temperature setting of random
Schroedinger operators.Comment: 15 pp; to appear in GAFA Seminar Note
Antiferromagnetic Potts model on the Erdos-Renyi random graph
We study the antiferromagnetic Potts model on the Poissonian Erd\"os-R\'enyi
random graph. By identifying a suitable interpolation structure and an extended
variational principle, together with a positive temperature second-moment
analysis we prove the existence of a phase transition at a positive critical
temperature. Upper and lower bounds on the temperature critical value are
obtained from the stability analysis of the replica symmetric solution
(recovered in the framework of Derrida-Ruelle probability cascades)and from a
positive entropy argument.Comment: 36 pages, revisions to improve resul
Trend-based analysis of a population model of the AKAP scaffold protein
We formalise a continuous-time Markov chain with multi-dimensional discrete state space model of the AKAP scaffold protein as a crosstalk mediator between two biochemical signalling pathways. The analysis by temporal properties of the AKAP model requires reasoning about whether the counts of individuals of the same type (species) are increasing or decreasing. For this purpose we propose the concept of stochastic trends based on formulating the probabilities of transitions that increase (resp. decrease) the counts of individuals of the same type, and express these probabilities as formulae such that the state space of the model is not altered. We define a number of stochastic trend formulae (e.g. weakly increasing, strictly increasing, weakly decreasing, etc.) and use them to extend the set of state formulae of Continuous Stochastic Logic. We show how stochastic trends can be implemented in a guarded-command style specification language for transition systems. We illustrate the application of stochastic trends with numerous small examples and then we analyse the AKAP model in order to characterise and show causality and pulsating behaviours in this biochemical system
Gaussian queues in light and heavy traffic
In this paper we investigate Gaussian queues in the light-traffic and in the
heavy-traffic regime. The setting considered is that of a centered Gaussian
process with stationary increments and variance
function , equipped with a deterministic drift ,
reflected at 0: We
study the resulting stationary workload process
in the limiting regimes (heavy
traffic) and (light traffic). The primary contribution is that we
show for both limiting regimes that, under mild regularity conditions on the
variance function, there exists a normalizing function such that
converges to a non-trivial
limit in
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