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SUMMARY:Random walk isomorphism theorems for a new type of spin system - A
 ndrew Swan (EPFL - Ecole Polytechnique Fédérale de Lausanne)
DTSTART:20241209T140000Z
DTEND:20241209T150000Z
UID:TALK223789@talks.cam.ac.uk
DESCRIPTION:Recently\, Sabot and Tarr\\`es introduced a new type of vertex
  reinforced jump process: the $\\star$-VRJP. It is defined on a directed g
 raph $G = (\\Lambda\, E)$ with a special involution $\\star: G \\mapsto G$
 \, which sends each vertex $j$ to a conjugate vertex $j^\\star$\, and each
  edge $\\spin{ij}$ to a reversed conjugate edge $\\spin{j^\\star i^\\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 ordi
 nary VRJP prefers to jump to where it has been $\\P(X_{t+dt} = j \\\,| X_t
  = i\, L_t) = \\beta_{ij}L_t^j$\, the $\\star$-VRJP prefers to jump to the
  \\emph{conjugate} of where it has been $\\P(X_{t+dt} = j \\\,| X_t = i\, 
 L_t) = \\beta_{ij}L_t^{j^\\star}$.&nbsp\;&nbsp\; &nbsp\;&nbsp\;&nbsp\;Also
  much like the ordinary VRJP\, the $\\star$-VRJP possesses a variety of re
 markable integral identities through its ``magic formula" and random Sch\\
 "odinger representation. In the case of the VRJP\, through its the deep co
 nnection with the $\\HH^{2|2}$ hyperbolic sigma model\, the existence of t
 hese identities is seen to be a consequence of supersymmetric localisation
 : this naturally raises the question if there exists a ``$\\star$-sigma mo
 del" counterpart to the $\\star$-VRJP to give a similar supersymmetric exp
 lanation. In this talk\, I will introduce this new hyperbolic sigma model\
 , the $\\HH^{2n+1|4m}_\\star$-model\, which is\, in a sense\, a complexifi
 cation of the ordinary $\\HH^{n|2m}$-model\, and will present several new 
 isomorphism theorems which connect it to the $\\star$-VRJP. Joint work wit
 h Sabot and Tarr\\`es.
LOCATION:Seminar Room 2\, Newton Institute
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