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Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called typical processes) on the infinite $d$-regular tree $T_d$. This correspondence between ergodic theory on $T_d$ and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processes on $T_d$. It serves as a quantitative refinement of the notion of typicality and is tightly connected to the asymptotic free energy in statistical physics. Using entropy inequalities, we provide new sufficient conditions for typicality for edge Markov processes. We also extend these notions and results to processes on unimodular Galton-Watson random trees.
Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $Phi$. Let ${G_{n}}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $Phi$ one can construc
Chase-escape is a competitive growth process in which red particles spread to adjacent uncolored sites, while blue particles overtake adjacent red particles. We introduce the variant in which red particles die and describe the phase diagram for the r
This article discusses some recent trends in Ramsey theory on infinite structures. Trees and their Ramsey theory have been vital to these investigations. The main ideas behind the authors recent method of trees with coding nodes are presented, showin
A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random element $U$.
Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton-Watson trees. For example, let $mathcal{T}_1$ be the event that a Ga