Non-linear conductances of Galton-Watson trees and application to the (near) critical random cluster model
Irene Ayuso Ventura, Quentin Berger
TL;DR
This work develops a rigorous link between non-linear $L^p$ resistive networks on Galton–Watson trees and concave recursions, enabling sharp, quenched asymptotics for the $p$-conductance $\mathcal{C}_p(\rho\leftrightarrow\partial T_n)$ on depth-$n$ GW trees with edge resistances $R_v=(R_n)^{-|v|}$. The analysis hinges on whether the offspring law $Z$ has a finite moment of order $q=s+1$ with $s=\frac{1}{p-1}$, yielding precise asymptotics for $\mathbf{E}[C_n]$ and convergence of normalized conductances to a branching martingale limit $W$ in the finite-moment case, and a heavy-tailed truncation theory in the infinite-moment case. These core results feed into sharp, tree-based predictions for the critical and near-critical random cluster model on quenched GW trees with cluster weight in $(0,2]$, including exact critical points $p_c$ and one-arm exponents. In the Ising regime ($q=2$), the paper derives critical decay rates for the root magnetization, showing $n^{-1/2}$-type decay under finite third moment and $n^{-1/(\\alpha-1)}$ decay in heavy-tailed offspring scenarios with tail exponent $\alpha\in(1,3)$, aligning with mean-field expectations on tree graphs.
Abstract
In this article, we study concave recursions on trees, which appear widely in information theory through algorithms such as belief propagation, and in statistical mechanics through models on tree-like graphs, including the Ising model, percolation, and more generally, the random cluster model. These tree recursions can, in fact, be compared with non-linear conductances, or $p$-conductances, between the root and the leaves of the tree. In this article, we estimate the $p$-conductances of $T_n$, a supercritical Galton--Watson tree of depth $n$, for any $p>1$, for a quenched realization of $T_n$. In particular, we find the sharp asymptotic behavior when $n$ goes to infinity, which depends on whether the offspring distribution admits a finite moment of order $q$, where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. We then apply our results to the random cluster model on $T_n$ (with cluster weight parameter in $(0,2]$ and wired boundary condition) providing sharp estimates on the probability that the root is connected to the leaves. As an example, for the Ising model on $T_n$ with plus boundary conditions on the leaves, we find that, at criticality, the quenched magnetization of the root decays like: (i) $n^{-1/2}$ times an explicit tree-dependent constant if the offspring distribution admits a finite third moment; (ii) $n^{-1/(α-1)}$ if the offspring distribution has a heavy tail with exponent $α\in (1,3)$.