Minimal configurations and sandpile measures
Let \(G=(V\cup \{s\},E)\) be a finite connected multigraph, with a distinguished vertex \(s,\) called the sink. A sandpile on \(G\) is a configuration of particles on \(V\), specified by a map \(\eta:V\rightarrow \{0,1,2,\cdots \},\) where \(\eta(x)\) is the number of particles at \(x.\) The wired uniform spanning forest is a random spanning subgraph of \(G\) that is obtained, through a limiting process, from uniformly random spanning trees of finite graphs. The authors ``give a new simple construction of the sandpile measure on an infinite graph \(G\), under the sole assumption that each tree in the wired uniform spanning forest on \(G\) has one end almost surely'' and ``also give determinantal formulas for minimal configurations [\dots] in terms of the transfer current matrix'' (from authors' abstract).
scientific article; zbMATH DE number 6348419
scientific article