Local limits of one-sided trees

A finite \emph{one-sided tree} of height $h$ is defined as a rooted planar tree obtained by grafting branches on one side, say the right, of a spine, i.e. a linear path of length $h$ starting at the root, such that the resulting tree has no simple path starting at the root of length greater than $h$. We consider the distribution $τ_N$ on the set of one-sided trees $T$ of fixed size $N$, such that the weight of $T$ is proportional to $e^{-μh(T)}$, where $μ$ is a real constant and $h(T)$ denotes the height of $T$. We show that, for $N$ large, $τ_N$ has a weak limit as a probability measure supported on infinite one-sided trees. The dependence of the limit measure $τ$ on $μ$ shows a transition at $μ_0=-\ln 2$ from a single spine phase for $μ\leq μ_0$ to a multi-spine phase for $μ> μ_0$. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $μ<μ_0$, to quadratic growth at $μ=μ_0$, and to qubic growth for $μ> μ_0$.

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