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 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.
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 -- necessary for obtaining the results presented in this paper -- much below the typical half-decay time of the scientist. For complex neighborhoods, it is for neighborhoods composed with sites taken from several coordination zones (up to the seventh coordination zone), the 127 percolation thresholds are calculated (with 120 among them being estimated for the first time). For seven extended (compact) neighborhoods, the fractal dimensions are also calculated. The mean value of these fractal dimensions, averaged over these seven compact neighborhoods, is estimated as $\langle d_f\rangle\approx 3.5581(70)$. The percentage errors of the values obtained for the fractal dimensions for compact neighborhoods vary from 0.26\% to 1.76\% with respect to the theoretically predicted value based on scaling relations and the most recent estimates of critical exponents for five-dimensional space. The universality of the percolation threshold as dependent on the weighted coordination number $\zeta=\sum_i z_i r_i$ (where $z_i$ is the number of sites in the $i$-th coordination zone and $r_i$ is the Euclidean distance from the sites in the $i$-th coordination zone to the central site) is also verified. The latter manifests itself as the power law ($p_c\propto\zeta^{-g}$) with $g\approx 0.7913(43)$
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 sizes up to $L=16384$ and investigate three universality regimes: LR Wilson--Fisher (WF) A ($1<\sigma\le2$), LR Wilson--Fisher B ($2/3<\sigma\le1$), and LR mean-field (MF) ($0<\sigma\le2/3$). In the LR-WF-B regime, the anomalous dimension is consistent with $\eta=2-\sigma$, in agreement with mathematical results for $2/3<\sigma<1$, while the correlation-length exponent $\nu(\sigma)$ exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although $\eta$ remains close to $2-\sigma$ for smaller $\sigma$, statistically resolvable deviations $\delta\eta(\sigma)=\eta-(2-\sigma)>0$ start to appear near $\sigma\simeq3/2$ and grow toward the short-range crossover at $\sigma=2$. Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.
Ziyu Liu, Tianning Xiao, Zhijie Fan et al.· 0 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 are considered simultaneously, owing to the fact that jointly invariant measures are not jointly product-form. This paper shows that the failure of independence is one-sided: Busemann increments across arbitrary directions are negatively associated. As an application, we derive an exponential concentration inequality for sums of Busemann increments on the diffusive scale, even when the increments are not independent. While our argument relies on a Burke property that is special to exponential weights, all other proof ingredients$\unicode{x2014}$including hidden LPP monotonicities and braid relations for queueing maps$\unicode{x2014}$hold for arbitrary weights.
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. Assuming boundedness of the variance profile and that $s_n$ diverges faster than a suitable logarithmic function of $n$, we show that the rank-one Chung--Lu model satisfies a non-homogeneous version of the circular law, which in some situations allows for an explicit expression. Moreover, under mild conditions, we identify the asymptotic singular value distribution using tools from free probability. Finally, for finite-rank directed models, we prove the existence of eigenvalues outside the bulk and establish their joint Gaussian fluctuations at the scale $\sqrt{s_n/n}$, with an explicit covariance matrix. These results extend the theory of spectral outliers and their fluctuations to directed inhomogeneous random graphs.
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 collections of linear statistics of the spin configurations, where the limiting covariance is characterized by the resolvent of the associated graphon integral operator. Building on this result, we establish functional CLTs for the average magnetization and for the spin field indexed by suitable classes of regular test functions. We further prove convergence of the full empirical spin field, viewed as a random generalized function in negative Sobolev spaces. These scaling limits provide applications to both Bayesian neural networks and causal inference. Specifically, for the former, we derive infinite-width Gaussian-process limits for two-layer Bayesian neural networks with Ising-dependent output-layer signs, while for the latter, we establish the asymptotic normality of H\'{a}jek estimators for average treatment effects under network interference.
