Density and correlation in a random sequential adsorption model

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arXiv:2210.05627v2 [math.PR] 21 Nov 2023

Charles S. do Amarala, Diogo C. dos Santosb,∗

aDepartamento de Matem´atica, Centro Federal de Educa¸c˜ao Tecnol´ogica de Minas Gerais, Belo Horizonte,

Minas Gerais, Brazil

bInstituto de Matem´atica, Universidade Federal de Alagoas, Macei´o, Alagoas, Brazil

Abstract

We consider the random sequential adsorption process on the one-dimensional lattice with

nearest-neighbor exclusion. In this model, each site s∈Zstarts empty and we will try to

occupy it in time ts, where (ts)s∈Zis a sequence of independent random variables uniformly

distributed on the interval [0,1]. The site will be occupied if both of its neighbors are vacant.

We provide a method to calculate the density of occupied sites up to the time t, as well as

the pair correlation function.

1. Introduction

The present study aims to investigate properties of a Random Sequential Adsorption

(RSA) model. In these stochastic processes, particles are irreversibly deposited on a surface,

represented by a graph, whereby the presence of particles at a particular set of sites may

restrict deposition of other particles at the same or nearby sites. Since the 1930s, RSA

models have attracted the attention of physicists and mathematicians; for an overview, see

Evans [1]. Some recent works on particular RSA models can be found in [2, 3, 4].

We consider a type of RSA on the one-dimensional lattice Znamed RSA with nearest-

neighbor exclusion in which particles are deposited at each site and arrive if there are no

particles in their ﬁrst neighbors. At most one particle can occupy a site, thus, at each site,

there can only be one attempt for a particle to deposit.

In relation to this model, analytical results regarding the density of occupied sites and the

pair correlation function at any moment during deposition have already been obtained using

methods such as generating functions, the independence principle, or diﬀerential equations.

[5, 6, 7, 8]. In this work, we present a method to determine these functions based solely on

probabilistic arguments. Recently, applying the concept developed in this study, we derived

an analytical expression for the average densities of particles in the One-dimensional AB

random sequential adsorption with one deposition per site [9].

∗Corresponding author.

Email address: diogo.santos@im.ufal.br (Diogo C. dos Santos)

Preprint submitted to November 22, 2023

1.1. Deﬁnition and main results

Consider a sequence of i.i.d. random variables T= (ts)s∈Zwith uniform distribution on

[0,1]. The corresponding product measure will be denoted by P. We deﬁne the continuous-

time stochastic process where at time t= 0 all sites are vacant and each site s∈Zwill be

occupied in time tsif the sites s−1 and s+ 1 are vacant at time ts. The status of the sites

occupied and vacant will be represented by the numbers 1 and 0, respectively.

Let us denote by ω(T, t) the temporal-conﬁguration of occupied sites in {0,1}Zand for

ωs(T, t) the state of the site sin this conﬁguration, both at time t, i.e. ωs(t) is the product

of the indicator functions of the sets

{T∈[0,1]Z:Ts≤t},{T∈[0,1]Z:ωs−1(T, Ts) = 0},{T∈[0,1]Z:ωs+1(T, Ts) = 0}.

It is not diﬃcult to show that ω(T, t) is well-deﬁned for all t∈[0,1] and for P-almost all

T∈[0,1]Zusing the Harris construction.

By translational invariance, we have that the probability of a site sbeing occupied until

the time t,

φs(t) := PT∈[0,1]Z:ωs(T, t) = 1,

assume the same value for all s∈Z. Furthermore, the translations of Zare ergodic with

respect to P, and then by ergodic theorems, it follows that φ0(t) coincides with the density

of occupied sites up to time t, which is usually denoted by ρt. Pedersen and Hemmer [10],

as well as Dickman et al [5], independently determined the exact value of ρt, demonstrating

the following result.

Theorem 1. Consider the RSA with nearest-neighbor exclusion on the one-dimensional

lattice. For all t∈[0,1] and all s∈Z, we have that

ρt=1−e−2t

2.(1)

The particular case t= 1 had already been obtained by Widom [11]. Actually, this

result when t= 1 is equivalent to demonstrating that in the discrete R´enyi packing problem

[12], the proportion of the interval occupied by cars approaches 1 −e−2as the length of the

interval tends to inﬁnity. This result was proven by Page in 1959 [13] and had previously

been observed by Flory [14]. In 2015, Gerin [15] employed a probabilistic construction of

this model to provide a proof of the same result. Our approach shares a similarity with his,

although we consider a more general case as we determine the behavior of the density at

any instant during the deposition.

The approach employed by us to determine ρthas recently been applied to derive an

analytical expression for the density in a RSA model involving the deposition of two distinct

types of particles onto a lattice: particles Aand B, with the constraint that diﬀerent types

cannot occupy neighboring sites, and there is only one deposition attempt per site [9].

At each moment in time when a particle is deposited, it is of type Awith probability α

and of type Bwith probability β= 1 −α. We present an analytical expression for the

average densities of particles of types Aand Bat all time instances, considering all possible

deposition probabilities for each particle type.

Another quantity of interest in the model studied by us is the pair correlation. Given two

sites iand j, the correlation between these two sites is deﬁned as the covariance between the

state variables ωiand ωj. When evaluated at a given time t this quantity will be denoted

by Covt(i, j), i.e.,

Covt(i, j) = E[ωi(t)ωj(t)] −E[ωi(t)]E[ωj(t)],

where Eis the expectation operator with respect to P. Using translation invariance one

more time one can see that if s=|i−j|then Covt(i, j) = Covt(0, s). We will write

Covt(0, s) = Cs(t).

Pedersen and Hemmer [10], using generating functions and the independence principle

[6, 7, 8], determined the values of Cs(t) demonstrating the following result.

Theorem 2. Consider the RSA with nearest-neighbor exclusion on the one-dimensional

lattice. For all t∈[0,1] and all s∈Z, the following equalities are hold

Cs(t) = −1

2e−2t

∞

n=0

(−2t)2n+s+1

(2n+s+ 1)!.(2)

The value of pair correlation for t= 1 has already been provided by Monthus [16].

Remark 1. The functions obtained in Theorems 1 and 2 diﬀer from those found in [10]

only because we considered times with uniform distribution instead exponential distribution.

Performing an obvious change of variables one obtains that the results are the same.

The goal of this work is to present proofs of Theorems 1 and 2 using exclusively proba-

bilistic arguments.

The remainder of this paper is organized as follows. Section 2 is dedicated to presenting

the proof of the Theorem 1. The proof of the Theorem 2 is presented in Section 3. In both

cases, we divided the organization of the sections into subsections in order to organize the

presentation of the results.

2. Proof of Theorem 1

Deﬁne As

t:= {ωs(t) = 1}(the site swill be occupied up to time t). We will decompose

tas a disjoint union of events whose probabilities we can easily calculate.

2.1. Favorable events

It is clear that for As

tto occur it is necessary to have ts< t. For the other hand, if ts< t

and in addition ts< ts−1and ts< ts+1, then As

toccurs. Note that there are alternative

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摘要：

arXiv:2210.05627v2[math.PR]21Nov2023DensityandcorrelationinarandomsequentialadsorptionmodelCharlesS.doAmarala,DiogoC.dosSantosb,∗aDepartamentodeMatem´atica,CentroFederaldeEduca¸c˜aoTecnol´ogicadeMinasGerais,BeloHorizonte,MinasGerais,BrazilbInstitutodeMatem´atica,UniversidadeFederaldeAlagoas,Macei´o,...

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Density and correlation in a random sequential adsorption model

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