Random walks with long-range jumps can drive superdiffusive transport, replacing ordinary diffusion with an effective long-range kinetic operator. Such superdiffusive kinetics is also central to critical phenomena, notably the self-avoiding walk with long-range jump statistics, or Lévy-SAW. This work investigates how the critical behavior is affected when the long-range connectivity itself becomes random. We study self-avoiding walks (SAWs) on a one-dimensional long-range random ring graph, where bonds are independently generated with Bernoulli probability $\sim|i-j|^{-(1+σ)}$. We term this walk Sparse-SAW. The same random bonds are responsible for both long-range superdiffusive transport and quenched disorder, with both simultaneously controlled by the single parameter $σ$, placing the problem beyond the conventional Harris and Weinrib-Halperin frameworks. Through large-scale Monte Carlo simulations and a Gaussian-truncated field theory, we show that Sparse-SAW belongs to the same universality class as the clean superdiffusive Lévy-SAW. The random bonds generate short-range uncorrelated and long-range correlated mass disorder while simultaneously producing the long-range kinetic operator. Under coarse-graining, the latter dominates, restoring the clean critical behavior. Our study suggests that the full non-Gaussian Bernoulli statistics may lead to disorder physics beyond the conventional theory of quenched disorder, while establishing random graphs as an efficient platform for extracting the critical exponents of the clean superdiffusive Lévy-SAW universality class.
Random walks with long-range jumps can drive superdiffusive transport, replacing ordinary diffusion with an effective long-range kinetic operator. Such superdiffusive kinetics is also central to critical phenomena, notably the self-avoiding walk with long-range jump statistics, or Lévy-SAW. This work investigates how the critical behavior is affected when the long-range connectivity itself becomes random. We study self-avoiding walks (SAWs) on a one-dimensional long-range random ring graph, where bonds are independently generated with Bernoulli probability ∼∣i−j∣−(1+σ). We term this walk Sparse-SAW. The same random bonds are responsible for both long-range superdiffusive transport and quenched disorder, with both simultaneously controlled by the single parameter σ, placing the problem beyond the conventional Harris and Weinrib-Halperin frameworks. Through large-scale Monte Carlo simulations and a Gaussian-truncated field theory, we show that Sparse-SAW belongs to the same universality class as the clean superdiffusive Lévy-SAW. The random bonds generate short-range uncorrelated and long-range correlated mass disorder while simultaneously producing the long-range kinetic operator. Under coarse-graining, the latter dominates, restoring the clean critical behavior. Our study suggests that the full non-Gaussian Bernoulli statistics may lead to disorder physics beyond the conventional theory of quenched disorder, while establishing random graphs as an efficient platform for extracting the critical exponents of the clean superdiffusive Lévy-SAW universality class.
This work demonstrates that sparse long-range random bonds on a one-dimensional lattice alone can generate quantum-chaotic spectral correlations and also drive a localization transition in a noninteracting single-particle Hamiltonian. The model is a one-dimensional ring in which each pair of sites is connected independently with a probability pij=dij−(1+σ). Each bond carries identical unit hopping and on-site disorder is absent. Despite the absence of on-site disorder and interaction, the model displays quantum chaotic spectra with Gaussian orthogonal ensemble (GOE) level statistics at small σ and localized eigenstates with Poisson statistics at larger σ. The transition occurs in the range $ 0.80 \lesssim σ_c \lesssim 0.85,farabovethesummabilitythresholdofthemeanhoppingprofile(σ=0$). A Gaussian field theory retaining only the mean and variance of the Bernoulli bonds instead predicts a threshold at σ=1, suggesting that higher cumulants are infrared-relevant. Our findings hint towards a universality class that is distinct from both the power-law random banded matrix model and the standard Anderson transition.
We study the one-dimensional long-range self-avoiding walk in the grand-canonical ensemble where the statistical weight of a jump of length r decays algebraically as r−(d+σ). Using large-scale Monte Carlo simulations with an efficient irreversible update scheme, we obtain high-precision estimates of the critical fugacity zc, the universal Binder ratio QNc, the correlation-length exponent ν, and the anomalous dimension η. For σ>1, the critical fugacity varies smoothly with σ, while the Binder ratio and the critical exponents remain consistent with the short-range universality class. For σ≤1, by contrast, the results clearly depart from short-range behavior, identifying σ=1 as the boundary between long-range and short-range regimes. In the long-range Wilson-Fisher regime with 1/2<σ≤1, η agrees with the long-range Gaussian-fixed-point prediction ηGFP=2−σ, whereas QNc and ν vary nontrivially with σ and exhibit discontinuous jumps at σ=1. These findings are in good agreement with the recently proposed universality diagram in the (d,σ) plane for long-range O(n) models, with the self-avoiding walk corresponding to the n→0 limit.
We introduce and numerically study the branching annihilating random walks with long-range attraction (BAWL). The long-range attraction makes hopping biased in such a manner that particle's hopping along the direction to the nearest particle has larger transition rate than hopping against the direction. Still, unlike the Lévy flight, a particle only hops to one of its nearest-neighbor sites. The strength of bias takes the form x^{-σ} with non-negative σ, where x is the distance to the nearest particle from a particle to hop. By extensive Monte Carlo simulations, we show that the critical decay exponent δ varies continuously with σ up to σ=1 and δ is the same as the critical decay exponent of the directed Ising (DI) universality class for σ≥1. Investigating the behavior of the density in the absorbing phase, we argue that σ=1 is indeed the threshold that separates the DI and non-DI critical behavior. We also show by Monte Carlo simulations that branching bias with symmetric hopping exhibits the same critical behavior as the BAWL.
The Voronoi construction is ubiquitous across the natural sciences and engineering. In statistical mechanics, however, only its dual, the Delaunay triangulation, has been considered in the investigation of critical phenomena. In this paper we set to fill this gap by studying three prominent systems of classical statistical mechanics, the equilibrium spin-1/2 Ising model, the nonequilibrium contact process, and the conserved stochastic sandpile model on two-dimensional random Voronoi graphs. Particular motivation comes from the fact that these graphs have vertices of constant coordination number, making it possible to isolate topological effects of quenched disorder from node-intrinsic coordination number disorder. Using large-scale numerical simulations and finite-size scaling techniques, we are able to demonstrate that all three systems belong to their respective clean universality classes. Therefore, quenched disorder introduced by the randomness of the lattice is irrelevant and does not influence the character of the phase transitions. We report the critical points to considerable precision and, for the Ising model, also the first correction-to-scaling exponent.
An important yet largely unsolved problem in the statistical mechanics of disordered quantum systems is to understand how quenched disorder affects quantum phase transitions in systems of itinerant fermions. In the clean limit, continuous quantum phase transitions of the symmetry-breaking type in Dirac materials such as graphene and the surfaces of topological insulators are described by relativistic (2+1)-dimensional quantum field theories of the Gross-Neveu-Yukawa (GNY) type. We study the universal critical properties of the chiral Ising, XY, and Heisenberg GNY models perturbed by quenched random-mass disorder, both uncorrelated or with long-range power-law correlations. Using the replica method combined with a controlled triple epsilon expansion below four dimensions, we find a variety of new finite-randomness critical and multicritical points with nonzero Yukawa coupling between low-energy Dirac fields and bosonic order parameter fluctuations, and compute their universal critical exponents. Analyzing bifurcations of the renormalization-group flow, we find instances of the fixed-point annihilation scenario—continuously tuned by the power-law exponent of long-range disorder correlations and associated with an exponentially large crossover length—as well as the transcritical bifurcation and the supercritical Hopf bifurcation. The latter is accompanied by the birth of a stable limit cycle on the critical hypersurface, which represents the first instance of fermionic quantum criticality with emergent discrete scale invariance.
We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities p(r)∝1/r2+σ, establishing the universality diagram in the long-range (LR) regime for σ≤2. 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<σ≤2), LR Wilson--Fisher B (2/3<σ≤1), and LR mean-field (MF) (0<σ≤2/3). In the LR-WF-B regime, the anomalous dimension is consistent with η=2−σ, in agreement with mathematical results for 2/3<σ<1, while the correlation-length exponent ν(σ) exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although η remains close to 2−σ for smaller σ, statistically resolvable deviations δη(σ)=η−(2−σ)>0 start to appear near σ≃3/2 and grow toward the short-range crossover at σ=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.