Markovian loop clusters (Q456617)

The features presented here complete the book Markov paths, loops and fields [Lecture Notes in Mathematics 2026. Berlin: Springer (2012; Zbl 1231.60002)] by the author.NEWLINENEWLINE Let on some graph \((X,L)\) conductances \(C_{x,y}\) be given and nonnegative weights \(K_x\), that define \(\lambda_x:= K_x+ \sum_{y\in X} C_{x,y}\), a transition matrix \(P_{x,y}:= C_{x,y}/\lambda x\), the semigroup \(\rho_t(x,y):= \exp(tP)_{x,y}/\lambda_x\), and the bridge law \(\operatorname{P}^{x,y}_t\) with mass \(\rho_t(x,y)\).NEWLINENEWLINE Consider the measure NEWLINE\[NEWLINE\mu:= \sum_{x\in X}\lambda_x \int^\infty_0 \operatorname{P}^{x,y}_t {dt\over t}NEWLINE\]NEWLINE on the loop space \(L(X)\), the loop ensemble Poisson process \(L_\alpha\) which has intensity \(\alpha\mu\) on \(L(X)\), and the partition \(C_\alpha\) of connected components of \(X\) (the clusters of loops), associated to \(L_\alpha\).NEWLINENEWLINE Then the law of \(C_\alpha\) is explicit in terms of Green functions, and \((C_\alpha)_{\alpha> 0}\) is a Markovian coalescing process, dual to a fragmentation process on clusters. The percolation clusters of \(X\) arise as a limiting case.NEWLINENEWLINE Moreover, a similar theory exists with oriented loops, with a better behaviour under reduction (by erasing links in the graph \(L\)), and with relationships to decreasing random trees of connected subgraphs.