scholarly journals Juggler's Exclusion Process

2012 ◽  
Vol 49 (01) ◽  
pp. 266-279
Author(s):  
Lasse Leskelä ◽  
Harri Varpanen
Keyword(s):  
Markov Process ◽  
Gibbs Measure ◽  
Equilibrium Distribution ◽  
Exclusion Process ◽  
Jump Height ◽  
The Family ◽  
System Of Particles ◽  
Unit Speed ◽  
Positive Integers ◽  
Special Case

Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.

scholarly journals Juggler's Exclusion Process

2012 ◽  
Vol 49 (1) ◽  
pp. 266-279 ◽  
Author(s):  
Lasse Leskelä ◽  
Harri Varpanen
Keyword(s):  
Markov Process ◽  
Gibbs Measure ◽  
Equilibrium Distribution ◽  
Exclusion Process ◽  
Jump Height ◽  
The Family ◽  
System Of Particles ◽  
Unit Speed ◽  
Positive Integers ◽  
Special Case

Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.

scholarly journals A Markov model for Selmer ranks in families of twists

2014 ◽  
Vol 150 (7) ◽  
pp. 1077-1106 ◽  
Author(s):  
Zev Klagsbrun ◽  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Markov Process ◽  
Markov Model ◽  
Elliptic Curve ◽  
Arbitrary Number ◽  
Equilibrium Distribution ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Quadratic Twists ◽  
The Family ◽  
Positive Integers

AbstractWe study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

scholarly journals Mapping TASEP back in time

2021 ◽  
Author(s):  
Leonid Petrov ◽  
Axel Saenz
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Exclusion Process ◽  
Discrete Space ◽  
Baxter Equation ◽  
Simple Exclusion Process ◽  
Open Questions ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
Hammersley Process

AbstractWe obtain a new relation between the distributions $$\upmu _t$$ μ t at different times $$t\ge 0$$ t ≥ 0 of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $$\upmu _t$$ μ t backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $$\upmu _t$$ μ t which in turn brings new identities for expectations with respect to $$\upmu _t$$ μ t . The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

Relaxed Markov processes

1983 ◽  
Vol 15 (04) ◽  
pp. 769-782 ◽  
Author(s):  
P. Whittle
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Equilibrium Distribution ◽  
Closed Model ◽  
Jackson Networks ◽  
The Creation

The concept of relaxing a Markov process is introduced; this is the creation of additional transitions between ergodic classes of the process in such a way as to conserve the existing equilibrium distribution within ergodic classes. The ‘open' version of a ‘closed' model of migration, polymerisation etc. often has this character. As further examples, generalized versions of Jackson networks and networks with clustering nodes are given.

Several weighted sum formulas of multiple zeta values

2017 ◽  
Vol 13 (09) ◽  
pp. 2253-2264 ◽  
Author(s):  
Minking Eie ◽  
Wen-Chin Liaw ◽  
Yao Lin Ong
Keyword(s):  
Real Number ◽  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Explicit Evaluation ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Number Formula ◽  
Positive Integers ◽  
Special Case

For a real number [Formula: see text] and positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we evaluate the sum of multiple zeta values [Formula: see text] explicitly in terms of [Formula: see text] and [Formula: see text]. The special case [Formula: see text] gives an evaluation of [Formula: see text]. An explicit evaluation of the multiple zeta-star value [Formula: see text] is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

A family of densities derived from the three-parameter Dirichlet process

2002 ◽  
Vol 39 (4) ◽  
pp. 764-774 ◽  
Author(s):  
Matthew A. Carlton
Keyword(s):  
State Space ◽  
Density Function ◽  
Dirichlet Process ◽  
Dirichlet Distribution ◽  
The State ◽  
Measurable Partition ◽  
Multivariate Distributions ◽  
The Family ◽  
Multivariate Density ◽  
Special Case

The traditional Dirichlet process is characterized by its distribution on a measurable partition of the state space - namely, the Dirichlet distribution. In this paper, we consider a generalization of the Dirichlet process and the family of multivariate distributions it induces, with particular attention to a special case where the multivariate density function is tractable.

Birth and death on a Brownian flow: a Feller semigroup and its generator and a martingale problem

1996 ◽  
Vol 33 (01) ◽  
pp. 71-87 ◽  
Author(s):  
Michael J. Phelan
Keyword(s):  
Markov Process ◽  
Spatial Configuration ◽  
Martingale Problem ◽  
Stochastic Flow ◽  
Transition Semigroup ◽  
Feller Semigroup ◽  
System Of Particles ◽  
Particle Process

We consider a system of particles in birth and death on a stochastic flow. The system includes a particle process tracking the spatial configuration of live particles on the flow. The particle process is a Markov process on the space of bounded counting measures. We show that its transition semigroup is a Feller semigroup and exhibit its pregenerator. The pregenerator defines a martingale problem. We show that the particle process solves the problem uniquely.

scholarly journals Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium Distribution

2015 ◽  
Vol 2015 ◽  
pp. 1-15
Author(s):  
Richard S. Ellis ◽  
Shlomo Ta’asan
Keyword(s):  
Equilibrium Distribution ◽  
Large Deviation ◽  
Random Variables ◽  
Rate Function ◽  
Number Density ◽  
Droplet Model ◽  
Empirical Measures ◽  
Deviation Analysis ◽  
Droplet Sizes ◽  
Positive Integers

In this paper we use large deviation theory to determine the equilibrium distribution of a basic droplet model that underlies a number of important models in material science and statistical mechanics. Given b∈N and c>b, K distinguishable particles are placed, each with equal probability 1/N, onto the N sites of a lattice, where K/N equals c. We focus on configurations for which each site is occupied by a minimum of b particles. The main result is the large deviation principle (LDP), in the limit K→∞ and N→∞ with K/N=c, for a sequence of random, number-density measures, which are the empirical measures of dependent random variables that count the droplet sizes. The rate function in the LDP is the relative entropy R(θ∣ρ∗), where θ is a possible asymptotic configuration of the number-density measures and ρ∗ is a Poisson distribution with mean c, restricted to the set of positive integers n satisfying n≥b. This LDP implies that ρ∗ is the equilibrium distribution of the number-density measures, which in turn implies that ρ∗ is the equilibrium distribution of the random variables that count the droplet sizes.

scholarly journals EXCLUSION PROCESSES AND BOUNDARY CONDITIONS

2004 ◽  
Vol 18 (01) ◽  
pp. 1-9 ◽  
Author(s):  
FARINAZ ROSHANI ◽  
MOHAMMAD KHORRAMI
Keyword(s):  
Boundary Conditions ◽  
Exclusion Process ◽  
Conditional Probabilities ◽  
Exclusion Processes ◽  
Simple Exclusion Process ◽  
The Family ◽  
Simple Exclusion ◽  
The One ◽  
Push Model ◽  
The Continuum

A family of boundary conditions corresponding to exclusion processes is introduced. This family is a generalization of the boundary conditions corresponding to the simple exclusion process, the drop-push model, and the one-parameter solvable family of pushing processes with certain rates on the continuum.1–3 The conditional probabilities are calculated using the Bethe ansatz, and it is shown that at large times they behave like the corresponding conditional probabilities of the family of diffusion-pushing processes introduced in Refs. 1–3.

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