Random walk isomorphism theorems for a new type of spin system

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• Andrew Swan (EPFL - Ecole Polytechnique Fédérale de Lausanne)
• Monday 09 December 2024, 14:00-15:00
• Seminar Room 2, Newton Institute.

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SSD - Stochastic systems for anomalous diffusion

Recently, Sabot and Tarr\`es introduced a new type of vertex reinforced jump process: the $\star$-VRJP. It is defined on a directed graph $G = (\Lambda, E)$ with a special involution $\star: G \mapsto G$, which sends each vertex $j$ to a conjugate vertex $j<sup>\star$, and each edge $\spin{ij}$ to a reversed conjugate edge $\spin{j</sup>\star i<sup>\star}$. Much like the ordinary VRJP , the $\star$-VRJP is linearly reinforced according to the local time $L_t$ of the walker $X_t$, but where the ordinary VRJP prefers to jump to where it has been $\P(X_{t+dt} = j \,| X_t = i, L_t) = \beta_{ij}L_t</sup>j$, the $\star$-VRJP prefers to jump to the \emph{conjugate} of where it has been $\P(X_{t+dt} = j \,| X_t = i, L_t) = \beta_{ij}L_t<sup>$.      Also much like the ordinary VRJP , the $\star$-VRJP possesses a variety of remarkable integral identities through its ``magic formula” and random Sch\”odinger representation. In the case of the VRJP , through its the deep connection with the $\HH</sup>{2|2}$ hyperbolic sigma model, the existence of these identities is seen to be a consequence of supersymmetric localisation: this naturally raises the question if there exists a ``$\star$-sigma model” counterpart to the $\star$-VRJP to give a similar supersymmetric explanation. In this talk, I will introduce this new hyperbolic sigma model, the $\HH<sup>_\star$-model, which is, in a sense, a complexification of the ordinary $\HH</sup>{n|2m}$-model, and will present several new isomorphism theorems which connect it to the $\star$-VRJP. Joint work with Sabot and Tarr\`es.

This talk is part of the Isaac Newton Institute Seminar Series series.

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