We initiate the treatment of KMS states on uniform Roe algebras $\mathrm {C}^{*}_u(X)$ for a class of naturally occurring flows on these algebras. We show that KMS states on $\mathrm {C}^{*}_u(X)$ always factor through the diagonal operators $\ell_{\infty} (X)$. We show the study of those states splits into understanding their strongly continuous KMS states and the KMS states which vanish on the ideal of compact operators. We show strongly continuous states are always unique when they exist and we give explicit formulas for them. We link the study of KMS states which vanish on the compacts to the Higson corona of X and provide lower bounds for the cardinality of the set of extreme KMS states. Lastly, we apply our theory to the n-branching tree: in this example, $\beta =\log (n)$ is a phase transition admitting $2^{2^{\aleph _0}}$ KMS states, no KMS states for smaller inverse temperatures, and a unique one for larger ones (the Gibbs state). Moreover, we show that the behavior of the KMS states around $\beta =\log (n)$ is chaotic.

We give combinatorially controlled series solutions to Dyson–Schwinger equations with multiple insertion places using tubings of rooted trees and investigate the algebraic relation between such solutions and the renormalization group equation.