2,424 research outputs found
Generalized Green Functions and current correlations in the TASEP
We study correlation functions of the totally asymmetric simple exclusion
process (TASEP) in discrete time with backward sequential update. We prove a
determinantal formula for the generalized Green function which describes
transitions between positions of particles at different individual time
moments. In particular, the generalized Green function defines a probability
measure at staircase lines on the space-time plane. The marginals of this
measure are the TASEP correlation functions in the space-time region not
covered by the standard Green function approach. As an example, we calculate
the current correlation function that is the joint probability distribution of
times taken by selected particles to travel given distance. An asymptotic
analysis shows that current fluctuations converge to the process.Comment: 46 pages, 3 figure
Asymptotics of a discrete-time particle system near a reflecting boundary
We examine a discrete-time Markovian particle system on the quarter-plane
introduced by M. Defosseux. The vertical boundary acts as a reflecting wall.
The particle system lies in the Anisotropic Kardar-Parisi-Zhang with a wall
universality class. After projecting to a single horizontal level, we take the
longtime asymptotics and obtain the discrete Jacobi and symmetric Pearcey
kernels. This is achieved by showing that the particle system is identical to a
Markov chain arising from representations of the infinite-dimensional
orthogonal group. The fixed-time marginals of this Markov chain are known to be
determinantal point processes, allowing us to take the limit of the correlation
kernel.
We also give a simple example which shows that in the multi-level case, the
particle system and the Markov chain evolve differently.Comment: 16 pages, Version 2 improves the expositio
Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure
In [arXiv:0804.3035] we studied an interacting particle system which can be
also interpreted as a stochastic growth model. This model belongs to the
anisotropic KPZ class in 2+1 dimensions. In this paper we present the results
that are relevant from the perspective of stochastic growth models, in
particular: (a) the surface fluctuations are asymptotically Gaussian on a
sqrt(ln(t)) scale and (b) the correlation structure of the surface is
asymptotically given by the massless field.Comment: 13 pages, 4 figure
Non-intersecting squared Bessel paths: critical time and double scaling limit
We consider the double scaling limit for a model of non-intersecting
squared Bessel processes in the confluent case: all paths start at time
at the same positive value , remain positive, and are conditioned to end
at time at . After appropriate rescaling, the paths fill a region in
the --plane as that intersects the hard edge at at a
critical time . In a previous paper (arXiv:0712.1333), the scaling
limits for the positions of the paths at time were shown to be
the usual scaling limits from random matrix theory. Here, we describe the limit
as of the correlation kernel at critical time and in the
double scaling regime. We derive an integral representation for the limit
kernel which bears some connections with the Pearcey kernel. The analysis is
based on the study of a matrix valued Riemann-Hilbert problem by
the Deift-Zhou steepest descent method. The main ingredient is the construction
of a local parametrix at the origin, out of the solutions of a particular
third-order linear differential equation, and its matching with a global
parametrix.Comment: 53 pages, 15 figure
Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization
We study dynamic -approximation algorithms for the all-pairs
shortest paths problem in unweighted undirected -node -edge graphs under
edge deletions. The fastest algorithm for this problem is a randomized
algorithm with a total update time of and constant
query time by Roditty and Zwick [FOCS 2004]. The fastest deterministic
algorithm is from a 1981 paper by Even and Shiloach [JACM 1981]; it has a total
update time of and constant query time. We improve these results as
follows: (1) We present an algorithm with a total update time of and constant query time that has an additive error of
in addition to the multiplicative error. This beats the previous
time when . Note that the additive
error is unavoidable since, even in the static case, an -time
(a so-called truly subcubic) combinatorial algorithm with
multiplicative error cannot have an additive error less than ,
unless we make a major breakthrough for Boolean matrix multiplication [Dor et
al. FOCS 1996] and many other long-standing problems [Vassilevska Williams and
Williams FOCS 2010]. The algorithm can also be turned into a
-approximation algorithm (without an additive error) with the
same time guarantees, improving the recent -approximation
algorithm with running
time of Bernstein and Roditty [SODA 2011] in terms of both approximation and
time guarantees. (2) We present a deterministic algorithm with a total update
time of and a query time of . The
algorithm has a multiplicative error of and gives the first
improved deterministic algorithm since 1981. It also answers an open question
raised by Bernstein [STOC 2013].Comment: A preliminary version was presented at the 2013 IEEE 54th Annual
Symposium on Foundations of Computer Science (FOCS 2013
Statistics of layered zigzags: a two-dimensional generalization of TASEP
A novel discrete growth model in 2+1 dimensions is presented in three
equivalent formulations: i) directed motion of zigzags on a cylinder, ii)
interacting interlaced TASEP layers, and iii) growing heap over 2D substrate
with a restricted minimal local height gradient. We demonstrate that the
coarse-grained behavior of this model is described by the two-dimensional
Kardar-Parisi-Zhang equation. The coefficients of different terms in this
hydrodynamic equation can be derived from the steady state flow-density curve,
the so called `fundamental' diagram. A conjecture concerning the analytical
form of this flow-density curve is presented and is verified numerically.Comment: 5 pages, 4 figure
Gibbs Ensembles of Nonintersecting Paths
We consider a family of determinantal random point processes on the
two-dimensional lattice and prove that members of our family can be interpreted
as a kind of Gibbs ensembles of nonintersecting paths. Examples include
probability measures on lozenge and domino tilings of the plane, some of which
are non-translation-invariant.
The correlation kernels of our processes can be viewed as extensions of the
discrete sine kernel, and we show that the Gibbs property is a consequence of
simple linear relations satisfied by these kernels. The processes depend on
infinitely many parameters, which are closely related to parametrization of
totally positive Toeplitz matrices.Comment: 6 figure
Eynard-Mehta theorem, Schur process, and their pfaffian analogs
We give simple linear algebraic proofs of Eynard-Mehta theorem,
Okounkov-Reshetikhin formula for the correlation kernel of the Schur process,
and Pfaffian analogs of these results. We also discuss certain general
properties of the spaces of all determinantal and Pfaffian processes on a given
finite set.Comment: AMSTeX, 21 pages, a new section adde
On a conjecture of Widom
We prove a conjecture of H.Widom stated in [W] (math/0108008) about the
reality of eigenvalues of certain infinite matrices arising in asymptotic
analysis of large Toeplitz determinants. As a byproduct we obtain a new proof
of A.Okounkov's formula for the (determinantal) correlation functions of the
Schur measures on partitions.Comment: 9 page
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