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Probabilistic models motivated by cooperative sequential adsorption

Vadim Shcherbakov

TL;DR

This survey reviews probabilistic models inspired by cooperative sequential adsorption (CSA), covering continuous-space and graph-based formulations, including RSA as a limiting case. It develops maximum likelihood estimation for CSA parameters, establishes asymptotic normality under increasing-domain limits, and links estimators to sums of locally determined functionals. It also analyzes a reversible, continuous-time growth-deposition model on graphs, providing localisation results to maximal cliques, phase transitions tied to graph spectrum, and connections to Ising-type models in finite components. Finally, it introduces CSA point processes as spatial-process objects (INP class), discusses their special cases (e.g., Strauss, hard-core), and highlights estimation challenges due to intractable normalising constants.

Abstract

This survey concerns probabilistic models motivated by cooperative sequential adsorption (CSA) models. CSA models are widely used in physics and chemistry for modelling adsorption processes in which adsorption rates depend on the spatial configuration of already adsorbed particles. Corresponding probabilistic models describe random sequential allocation of particles either in a subset of Euclidean space, or at vertices of a graph. Depending on a technical setup these probabilistic models are stated in terms of spatial or integer-valued interacting birth-and-death processes. In this survey we consider several such models that have been studied in recent years.

Probabilistic models motivated by cooperative sequential adsorption

TL;DR

This survey reviews probabilistic models inspired by cooperative sequential adsorption (CSA), covering continuous-space and graph-based formulations, including RSA as a limiting case. It develops maximum likelihood estimation for CSA parameters, establishes asymptotic normality under increasing-domain limits, and links estimators to sums of locally determined functionals. It also analyzes a reversible, continuous-time growth-deposition model on graphs, providing localisation results to maximal cliques, phase transitions tied to graph spectrum, and connections to Ising-type models in finite components. Finally, it introduces CSA point processes as spatial-process objects (INP class), discusses their special cases (e.g., Strauss, hard-core), and highlights estimation challenges due to intractable normalising constants.

Abstract

This survey concerns probabilistic models motivated by cooperative sequential adsorption (CSA) models. CSA models are widely used in physics and chemistry for modelling adsorption processes in which adsorption rates depend on the spatial configuration of already adsorbed particles. Corresponding probabilistic models describe random sequential allocation of particles either in a subset of Euclidean space, or at vertices of a graph. Depending on a technical setup these probabilistic models are stated in terms of spatial or integer-valued interacting birth-and-death processes. In this survey we consider several such models that have been studied in recent years.
Paper Structure (22 sections, 4 theorems, 59 equations, 4 figures)

This paper contains 22 sections, 4 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Continuous CSA model
  3. The model definition
  4. CSA as a model for time series of spatial locations
  5. MLE for CSA
  6. Asymptotic properties of MLE estimators
  7. MLE for CSA and the theory of locally determined functionals
  8. CSA growth model on graphs
  9. The model definition
  10. Localisation in the growth model with attractive interaction
  11. Open problem: repulsive interaction
  12. The reversible model
  13. The model definition
  14. Long term behaviour of the model
  15. Reversibility of the model
  16. ...and 7 more sections

Key Result

Theorem 4.1

Suppose that the graph $G$ is connected and $\beta\neq 0$.

Figures (4)

  • Figure 1: CSA simulations in $D=[0,1]^2$. Left: $1000$ points, $R=0.01$, $(\beta_i)_{i \geq 0} = (1,1000,10000,0,0, \ldots)$ Right: $500$ points, $R=0.03$, $(\beta_i)_{i \geq 0}=(1,0,0,\ldots)$, i.e. this is RSA model.
  • Figure 2: Deposition model on a linear graph $\{1\sim...\sim N\}$.
  • Figure 3: $G=\{1\sim 2\}$, $\alpha<0$, $\beta>0$. Left: $\alpha+\beta<0$; Right: $\alpha+\beta\geq 0$.
  • Figure 4: $G=\{1\sim 2\}$, $\alpha>0$, $\beta<0$. Left: $\alpha+\beta<0$; Right: $\alpha+\beta>0$.

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.1
  • Remark 2.4
  • Example 2.2
  • Remark 3.1
  • Example 3.1: Complete graph
  • Remark 3.2
  • Theorem 4.1
  • ...and 14 more