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 $\alpha<1-\gamma$.
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 loca...
In this paper, the random site percolation problem in a five-dimensional space for complex neighborhoods is studied. The efficient C++ code (with ordinary work division) of the classical Newman--Ziff algorithm is presented. The obtained speed-up of computations reduces 2170 years of single-core computations -- necessar...
We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+\sigma}$, establishing the universality diagram in the long-range (LR) regime for $\sigma\le2$. Using the event-based ensemble method, we simulate systems with linear...
Zi-Yu Liu, Tian-Ning Xiao, Zhi-Jie Fan et al.· 2 citations
One hallmark of exactly solvable KPZ random growth models is product-form invariant measures. In the setting of exponential last-passage percolation (LPP), this corresponds to the independence of Busemann increments along any down-right path. However, this independence breaks down when multiple asymptotic directions ar...
We study the spectrum of the adjacency matrix $A_n$ of directed inhomogeneous random graphs on $n$ vertices. We assume that $A_n$ has independent entries and diverging average degree scale $s_n$. This framework includes, as special cases, the directed Chung--Lu random graph and directed stochastic block models. Assumin...
In this paper, we derive quenched scaling limits for linear functionals and the empirical spin field of Ising models on inhomogeneous random graphs generated by a graphon (encompassing both dense and sparse graphs), in the high-temperature regime. We first prove a joint central limit theorem (CLT) for finite collection...
