[Paper Review] The *-Vertex-Reinforced Jump Process

Christophe Sabot, Pierre Tarrès|arXiv (Cornell University)|Feb 17, 2021

Stochastic processes and statistical mechanics23 references4 citations

TL;DR

This paper introduces the $\star$-Vertex-Reinforced Jump Process ($\star$-VRJP), a non-reversible continuous-time generalization of the VRJP that incorporates a $\star$-involution on vertices to model asymmetric reinforcement. Despite lacking partial exchangeability, the authors show that with randomized initial local times, the $\star$-VRJP becomes a mixture of Yaglom-reversible Markov jump processes, and the non-randomized version is a mixture of conditioned processes, establishing a key representation for the process and its long-time behavior.

ABSTRACT

We investigate the non-reversible generalization of the Vertex-Reinforced Jump Process (VRJP), called the *-Vertex-Reinforced Jump Process (*-VRJP) and introduced by Bacallado, Sabot and Tarr\\`es (2020). It can be seen as the continuous-time counterpart to the *-Edge-Reinforced Random Walk (*-ERRW), see Bacallado (2011) and Bacallado, Sabot and Tarr\\`es (2020), which is itself a non-reversible, and in fact Yaglom reversible, generalization of the original ERRW introduced by Coppersmith and Diaconis (1986). In contrast to the classical VRJP, the *-VRJP is not exchangeable after time-change, which leads to several difficulties and new phenomena. Firstly, we show that with some appropriate randomization of the initial local time, it becomes partially exchangeable after time-change. We provide a representation of the "randomized" *-VRJP as a mixture of Yaglom reversible Markov jump processes with an explicit mixing measure, and we prove that the non-randomized *-VRJP can be written as a mixture of conditioned Markov processes. Secondly, we give a representation of the *-VRJP in terms of a random Schr\\"odinger operator. The corresponding representation for the classical VRJP has proved to be very useful in the understanding of its asymptotic behavior. The construction is based on several new and rather remarkable identities between integrals on the space of *-symmetric and *-antisymmetric functions on vertices. We give a description of the randomized *-VRJP in terms of that random Schr\\"odinger operator, which allows us to prove the representation of the randomized *-VRJP as a mixture of Markov jump processes in a different and more analytic manner. Similarly as for the VRJP, we think that the representation by a random Schr\\"odinger operator and the associated identities are key-features of the *-VRJP.

Motivation & Objective

• To develop a non-reversible generalization of the Vertex-Reinforced Jump Process (VRJP) that interpolates between undirected and directed reinforcement mechanisms.
• To address the lack of exchangeability in the $\star$-VRJP, which complicates analysis compared to the classical VRJP.
• To establish a representation of the $\star$-VRJP as a mixture of Markov processes via randomization of initial local times.
• To prove that the non-randomized $\star$-VRJP can be expressed as a mixture of conditioned self-interacting jump processes after time change.
• To extend the framework of the VRJP and ERRW to directed graphs with $\star$-symmetry, enabling new insights into recurrence, transience, and mixing measures.

Proposed method

• Introduce the $\star$-VRJP on a directed graph $\mathcal{G}=(V,E)$ equipped with a $\star$-involution satisfying $ (i,j)\in E \iff (j^*,i^*)\in E $, with transition rates proportional to $ W_{i,j} e^{T_i(t) + T_{j^*}(t)} $.
• Apply a time change via $ s = C(t) = \sum_{i\in V} \log(1 + \frac{1}{2}(l_i^Z(s) + l_{i^*}^Z(s))) $, transforming the process into a form amenable to exchangeability analysis.
• Randomize initial local times to restore partial exchangeability, enabling the use of de Finetti-type representations.
• Prove that the randomized $\star$-VRJP is a mixture of Yaglom-reversible Markov jump processes, where reversibility is defined via $ \pi_i K_{i,j} = \pi_{j^*} K_{j^*,i^*} $ and $ \pi_i = \pi_{i^*} $.
• Use a discrete Feynman-Kac type formula and Poisson point process coupling to relate occupation times $ T_i(t) $ to jump counts $ N_{i,j}(t) $, showing $ W_{i,j} e^{T_i(t) + T_{j^*}(t)} \sim N_{i,j}(t) $ almost surely.
• Establish convergence of $ T(t) - t/N $ to a limit in $ \mathcal{U}_0^{W} $, using projection onto the space of divergence-free vectors and conditional convergence after first return to the starting point.

Experimental results

Research questions

• RQ1 Can a non-reversible generalization of the VRJP be constructed that preserves key structural properties like mixing measures and exchangeability under randomization?
• RQ2 How does the $\star$-VRJP behave under time change, and can it be represented as a mixture of Yaglom-reversible processes despite lacking exchangeability?
• RQ3 What is the asymptotic behavior of the occupation times $ T_i(t) $ in the $\star$-VRJP, and does $ T(t) - t/N $ converge almost surely?
• RQ4 Can the non-randomized $\star$-VRJP be expressed as a mixture of conditioned Markov processes after time change?
• RQ5 How does the $\star$-invariance of edge weights and the involution $\star$ affect the long-term dynamics and mixing measures of the process?

Key findings

• The $\star$-VRJP is not partially exchangeable in its original form, unlike the classical VRJP, which introduces significant analytical challenges.
• With appropriate randomization of initial local times, the $\star$-VRJP becomes partially exchangeable after time change and can be represented as a mixture of Yaglom-reversible Markov jump processes.
• The limiting local time vector $ T(t) - t/N $ converges almost surely to a point in $ \mathcal{U}_0^{W} $, the space of divergence-free vectors with respect to the conductance weights.
• The non-randomized $\star$-VRJP can be written as a mixture of conditioned self-interacting jump processes after a proper time change.
• The ratio $ N_{i,j}(t) / \sum_{(k,l)\in \tilde{E}} N_{k,l}(t) $ converges almost surely to a random variable $ x_{i,j} $, implying $ W_{i,j} e^{T_i(t) + T_{j^*}(t)} \sim N_{i,j}(t) $ a.s.
• After the first return to the starting point $ i_0 $, the process restarts with updated conductances $ W^{T(\tau)}_{i_0} $, and the convergence of $ T(t) - t/N $ holds conditionally on the return time $ \tau $.