We prove the analogue of Schramm's locality conjecture for long-range percolation on transitive graphs of polynomial growth with $\alpha \in (0,2)$. In this setting, we also prove the joint continuity of the percolation probability $\theta$ with respect to three parameters: the underlying graph with respect to the local topology, the connectivity kernel, and the percolation parameter $\beta$ for all values of $\beta \in \mathbf{R}_+$, including the critical parameter $\beta_c$. We also prove a number of results related to the supercritical sharpness of long-range percolation: the long-range order decay of the distribution of finite clusters, the truncation problem, the anchored isoperimetric dimension and the transience of the infinite percolation cluster, and the smoothness of the percolation characters. We obtain these results from proving the local existence-and-uniqueness of the linear-sized (giant) cluster. As an immediate corollary of the local existence-and-uniqueness of the giant we obtain the law of large numbers, which answers a special case of a question of Nekrashevych and Pete \cite[Question 1.3]{nekrashevych_scale-invariant_2011}. The main technical contribution is the construction of a renormalisation scheme combining iteratively merged Voronoi tiles with scale-invariant nets, related to the scale-invariant groups of Benjamini.
We determine the critical probability for Bernoulli bond percolation on essentially any finite graph. Namely, letting $\lambda(G)$ denote the spectral radius (maximum eigenvalue) of $G$, we prove that the critical probability is at $1/\lambda(G)$: above this probability there is typically a component of order $\Omega(\lambda(G))$, whereas below it all components are of order at most $O(\sqrt{|G|})$. These results in particular confirm a conjecture of Krivelevich and Samotij about percolation on graphs of a given average degree, and vastly extend theorems of Bollob\'as, Borgs, Chayes, and Riordan, who proved analogous results but only for dense graphs. Our theorems are optimal in many regimes, and also demonstrate that percolation has an unexpectedly subtle behaviour on graphs whose spectral radius is roughly the square root of their maximum degree.
We prove recurrence criteria for inhomogeneous long-range percolation in dimensions one and two. In dimension one, recurrence follows from a purely geometric scarcity condition: long edges eventually disappear on exponential scales. This applies to weight-dependent random connection models and related one-dimensional spatial scale-free graphs whenever the standard strong-decay long-edge estimate holds. In dimension two, we combine the linear chemical-distance estimate of L\"uchtrath with an area-order bound on the degree measure. Graph-distance layers in exponentially separated bands then give the required Nash-Williams cutsets for planar random geometric graphs satisfying the polynomial mixing and long-edge estimates [J. Theoret. Probab. 39 (2026), Paper No. 12]. As a concrete consequence, every connected component of the two-dimensional weight-dependent random connection model with interpolation kernel is recurrent throughout the strong-decay region $\delta>2$, $\gamma<1-\frac{1}{\delta}$, and $\alpha<1-\gamma$.
Johannes Bäumler, Lukas Lüchtrath, Christian Mönch· 0 citations
For critical Bernoulli bond percolation on $\Z^d$ in the high-dimensional regime, we prove that the rescaled empirical measure of the cluster of the origin converges, in a suitable $\sigma$-finite sense, to the total occupation measure of super-Brownian motion. Combined with a uniform lower mass bound for the critical cluster with respect to the extrinsic (Euclidean) metric, which we also prove and which is of independent interest, the measure convergence further yields the convergence of the rescaled cluster as a compact set, in the Hausdorff metric. As a consequence, we are able to obtain the sharp one-arm asymptotics $r^2\,\bP(0\leftrightarrow \partial B_r)\to \theta_1\in(0,\infty)$, hence refining a result of Kozma and Nachmias.
M. Cabezas, D. Croydon, A. Fribegh et al.· 0 citations
We consider the weight-dependent random connection model on a Poisson point process of intensity $\lambda$ on $\mathbb{R}\times(0,1)$ in which the vertices $(x,t)$ and $(y,s)$ are joined precisely when $(t\vee s)|x-y|\le\beta$. Points at distance $d$ are joined with probability $\min(1,\beta/d)^2$, the critical decay of one-dimensional long-range percolation, and edges sharing a vertex are dependent through the common mark. We prove that the model has a genuine phase transition: for $\lambda\beta<1$ almost surely all connected components are finite, while for $\lambda\beta\ge31$ an infinite component exists, so at intensity one the critical value satisfies $\beta_c\in[1,31]$; a numerical study included as an appendix places it near $2$. By kernel and profile comparisons the supercritical bound extends to the age-dependent random connection model on the line, which with indicator profile has a non-degenerate phase transition at every value of its parameter, closing a case of the one-dimensional phase diagram left open in earlier work. The lower bound is proved by disconnecting nested pairs of long edges ("rainbows") with cut-point certificates, an argument developed first in a discrete skeleton of the model with the vertices pinned to $\mathbb{Z}$. The skeleton is of independent interest: it has no supercritical phase at all, jumping from total fragmentation to trivial connectivity even though almost surely infinitely many edges cross every fixed site. The supercritical argument is a Peierls argument on the binary tiling of the hyperbolic half-plane.
We study first-passage percolation on the $\ell$-spread-out one-dimensional cycle of size $n$, where vertices are connected if their graph distance is at most $\ell$. We assign i.i.d.~non-negative random weights from a Weibull distribution $\omega_e \sim \mathrm{Exp}(1)^{1/\theta}$ to the edges for $\theta>0$ fixed. This paper investigates the transition in the asymptotic behavior of the passage time $T_n$ between two typical vertices and the hop-count of the optimal path as the connectivity parameter $\ell$ diverges with $n$. We identify two fundamentally distinct geometric regimes. In the mesoscopic regime ($1 \ll \ell \ll n$), the optimal path locally mimics a spatial branching random walk but remains globally constrained to a one-dimensional geometry. We establish a law of large numbers characterized by the front speed of a Crump--Mode--Jagers branching random walk, prove a central limit theorem with Gaussian fluctuations when $\ell\ll n^{1/4}$, and show that the expected hop-count grows proportionally with the spatial distance. In the macroscopic regime ($\ell \approx \lambda n$ for $\lambda \in (0,1/2)$), the graph becomes a highly connected mean-field network. We prove that the passage time collapses to a $\log n$ scale with constant order non-Gaussian fluctuations, explicitly determining the extreme-value limit driven by the collision of two independent non-spatial CMJ processes. We establish a law of large numbers for the hop-count. Finally, we rigorously trace the transition in the order of the mean of $T_n$ between these two regimes, demonstrating an order transition for the passage time across the critical connectivity threshold $\ell \asymp n/\log n$. Our results provide a comprehensive deterministic-range interpolation from spatial Gaussian fluctuations to mean-field extreme-value fluctuations.
We study self-avoiding walks on the canonical one-sided graphs of Lindstrom nested fractals. We prove that the connective constant $\mu$ exists and identify $\log\mu$ with the critical inverse temperature of a finite-dimensional boundary-state renormalization. If the boundary-state partition vectors are bounded at criticality, then the fixed-length counts $c_n$ satisfy two-sided polynomial bounds around $\mu^n$. We also prove that $h$-flexibility implies $c_{n+h}/c_n\to\mu^h$. For regular polygonal $N$-gaskets, we derive exact crossing recursions, determine the smallest flexibility step $h$, and obtain explicit algebraic connective constants for the $6$- and $9$-gaskets. The Vicsek graph has no flexibility step, and its successive ratios do not converge.
Hua Qiu, Yifan Wang· 0 citations
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