Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Restrictions of some reinforced processes to subgraphs

Margherita Disertori, Franz Merkl, Silke W. W. Rolles

Abstract

We prove that the restriction of the vertex-reinforced jump process to a subset of the vertex set is a mixture of vertex-reinforced jump processes. A similar statement holds for the non-linear hyperbolic supersymmetric sigma model. This is then applied to vertex-reinforced jump processes on subdivided versions of graphs of bounded degree, where every edge is replaced by a finite sequence of edges. We prove that discrete-time processes associated to suitable corresponding restrictions are mixtures of positive recurrent Markov chains. We also deduce a similar statement for edge-reinforced random walks.

Restrictions of some reinforced processes to subgraphs

Abstract

We prove that the restriction of the vertex-reinforced jump process to a subset of the vertex set is a mixture of vertex-reinforced jump processes. A similar statement holds for the non-linear hyperbolic supersymmetric sigma model. This is then applied to vertex-reinforced jump processes on subdivided versions of graphs of bounded degree, where every edge is replaced by a finite sequence of edges. We prove that discrete-time processes associated to suitable corresponding restrictions are mixtures of positive recurrent Markov chains. We also deduce a similar statement for edge-reinforced random walks.

Paper Structure

This paper contains 20 sections, 14 theorems, 99 equations, 2 figures.

Table of Contents

  1. Models and Results
  2. Motivation
  3. Reinforced processes
  4. Comparison with the restriction property observed by Davis and Volkov.
  5. Subdivisions.
  6. Consequences for linearly edge-reinforced random walk.
  7. Comparison with previous work on errw.
  8. Discussion.
  9. Non-linear hyperbolic supersymmetric sigma model
  10. How this paper is organized.
  11. Representation as a mixture
  12. Markov jump processes
  13. Notation.
  14. Application to vrjp
  15. Proofs for subdivided graphs
  16. ...and 5 more sections

Key Result

Theorem 1.2

Assume that the graph $G$ is finite without self-loops and partition its vertex set $\Lambda=I\cup J$, $I\cap J=\emptyset$, with $|J|\ge 2$. Consider the vrjp $(X,T)$ in exchangeable time scale on $\Lambda$ starting at $\rho\in J$ with weights $W$. The restrictions $(X^J,T^J)$ and $(X^{J\neq},T^{J\n respectively. They depend on a random vector $\beta_I\in{\mathbb R}^I$, where $\beta_{I\cup\{\rho\}

Figures (2)

  • Figure 1: A part of ${\mathbb Z}^2$ on the left and its 8-subdivided version on the right.
  • Figure 2: Illustration of the recursion $l\leadsto l-1$ for $l=4$ in the cases $\bar{e}=e_{3,3}$, $e'=e_{5,4}$, $e"=e_{6,4}$, $v'=v_{e,\frac{2}{8}}$, $v=v_{e,\frac{5}{16}}$, $v"=v_{e,\frac{3}{8}}$, $e^1=e_{12,4}$, $e^2=e_{13,4}$, $v^1=v_{e,\frac{11}{16}}$, $\bar{v}=v_{e,\frac{6}{8}}$, and $v^2=v_{e,\frac{13}{16}}$.

Theorems & Definitions (19)

  • Definition 1.1: Removal of self-loops and restriction to a subset: process
  • Theorem 1.2: Restriction of vrjp as a mixture of vrjps
  • Theorem 1.4: Vrjp on subdivided graphs
  • Theorem 1.5: Errw on subdivided graphs
  • Theorem 1.6: Decay of the effective weights by restriction
  • Lemma 1.7: Induction step for moments of effective weights
  • Theorem 1.8: Effective weights for restrictions to subsets
  • Definition 2.1: Removal of self-loops and restriction to a subset: parameters
  • Lemma 2.2: Laws of removal of self-loops and restriction to a subset
  • Remark 2.3: Properties of $\beta$, sabot-zeng15
  • ...and 9 more