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Tree number pairs for free and wired spanning forests. (English) Zbl 1488.05057

Summary: Let \(N\) be the set of pairs \((k, l) \in ({\mathrm{N}} \cup \{\infty\})^2\) such that on some infinite electric network, wired spanning forest has \(l\) trees almost surely, while free spanning forest has \(k\) trees with positive probability. It’s proved that \(N = \{(k, l) \in {{\mathrm{N}}^2}\mid k \le l\} \cup \{(\infty, \infty)\} \cup \{(k, \infty)\mid k \in {\mathrm{N}}\}\).

MSC:

05C05 Trees
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