Equilibrium and Energy

Equilibrium measures on trees

“…Proof Given any measure , setting f = I * and c = c f , by (10) we have If is an equilibrium measure then (4) holds, and conversely if is such that V p ( ) < 1 and solves (4) it is the equilibrium measure of some F set. But (4) holds if and only if But by Proposition 6, we know that c( ) solves (11), or equivalently, f ( ) is defined by (12). Hence, sobstituting above we get which is (13).…”

“…This gives us an algorithm to calculate the capacity of a set in T in terms of successive tents capacities. Moreover, by relation (11) we deduce that capacities can be expressed by means of branched continued fractions. For example, by (11) we obtain the expression In [15, p. 57] the same structure was observed for the total resistence R of an infinite tree without edges of degree 1.…”

“…Moreover, by relation (11) we deduce that capacities can be expressed by means of branched continued fractions. For example, by (11) we obtain the expression In [15, p. 57] the same structure was observed for the total resistence R of an infinite tree without edges of degree 1. In particular, for such a class of trees, we obtain the relation To end the section, we give a reformulation of Theorem 1 which provides a characterization of equilibrium measures by means of an equation for capacities.…”

“…Iterating we obtain, Hence, for any chosen edge which is not a leaf, it holds Now, f is the co-potential of a measure if and only if the flow condition (6) holds. Namely, using ( 12), ( 6) holds if and only if which is equivalent to (11), as can be seen solving with respect to c( ) . ◻ By the rescaling properties of equilibrium measure (Proposition 3), we have that if is the p−equilibrium measure for a set A ⊆ T , then c( ) = c p, (A ) .…”

“…Again in the linear case, Theorem 1 is related to some beautiful theorems of Kai-Lai Chung (see [10,11]), interpreting equilibrium measures in terms of last exit times for stochastic processes. The results of Chung, and of Benjamini and Schramm, have a probabilistic statement, or proof.…”