We consider a stochastic process on $\mathbb{Z}^d$ for $d \geq 1$. Given a translation invariant and ergodic starting configuration of finite clusters, each cluster $C$ performs a continuous time simple random walk with rate $|C|^{-\alpha}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. In all dimensions, we show that if $\alpha\ge 0$, there is almost surely no spontaneous creation of an infinite cluster within finite time. Moreover, for any $\alpha\le-1-2/d$ there is a finite-time blowup almost surely. In the regime $\alpha\in(-1,0)$ we show that the behavior greatly depends on the initial configuration. In addition, in dimension one, we establish the exact phase diagram.
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 the frog model on $\mathbb Z^d$ and on the discrete tori $\mathbb T_L^d$, $d\ge 2$, with a symmetric, translation-invariant, and heavy-tailed transition kernel satisfying \[ Q(x,y)\asymp |x-y|^{-(d+\alpha)}, \qquad \alpha>0. \] Starting from an i.i.d. Poisson$(\lambda)$ number of sleeping particles per site and one active particle at the origin. Active particles perform independent $Q$-random walks and activate the particles they encounter. We first determine the timescale for activating distant vertices. When $\alpha\in(0,d)$, the time required to activate all vertices within distance $L$ of the origin is, with high probability, \[ (\log L)^{\Delta+o(1)}, \qquad \Delta^{-1}:=\log_2\left(\frac{2d}{d+\alpha}\right), \] as $L\to\infty$. This polylogarithmic spreading contrasts sharply with the linear spreading of the classical frog model driven by simple random walks; see Alves, Machado, and Popov (2002) and Ram\'irez and Sidoravicius (2004). When $\alpha>d$, we recover this classical linear behavior by proving matching linear upper and lower bounds; at $\alpha=d$, we prove a linear upper bound. Finally, we consider the finite-lifespan model on $\mathbb T_L^d$, in which each particle is removed after taking $\ell$ steps. We show that the cover lifespan, defined as the smallest $\ell$ for which the torus is entirely activated, is asymptotic to the cover time of a Poisson$(\lambda L^d)$ cloud of independent stationary random walkers.
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 first-passage percolation on the $\ell$-spread-out line graph, where each vertex $i\in\{0,1,\dots,n\}$ is connected to all others at distance at most $\ell$. Here, we focus on the microscopic regime, with $\ell$ fixed as $n\to\infty$. Independent nonnegative weights are assigned to these edges. We obtain a law of large numbers and precise fluctuation results for the passage time $T_n$ from $0$ to $n$. If the weight distribution has finite variance or a heavy tail with exponent above $2/\ell_c$ where $\ell_c=\ell(\ell+1)/2$, then $T_n$ satisfies a Gaussian CLT with $\sqrt{n}$ scaling. In contrast, for heavier-tailed distributions, with index below the threshold, we show that $T_n$, appropriately centered and scaled, converges to a non-Gaussian stable law. We also prove an LLN and CLT for the number of edges in the minimizing path. The key tool is a pivot-node decomposition; the geodesic can be segmented into i.i.d. blocks, leading to a renewal structure. Our results extend the classical one-dimensional CLT to include finite-range connectivity and heavy tails, revealing a new distributional phase transition in the fluctuations of $T_n$.
We consider the voter model on the giant component of a hyperbolic random graph, which is a spatial scale-free network, in the sparse and linear-giant regime $\alpha\in(1/2,1)$. We find that the quenched expected consensus time has order $n^{2-1/\alpha}$, as the number of vertices $n\to\infty$, with probability arbitrarily close to one. This is generalised to the voter model where each vertex changes its opinion at rates $q(v)={\rm d}(v)^\varphi$, where we also establish the consensus time orders for all $\varphi\geq 0$. These orders have 3 regimes, with a phase transition at $\varphi=2-2\alpha$. For the upper bounds, our main proof idea is to connect the meeting set to some fixed target vertex of appropriate height in the product chain electrical network, to make rigorous an argument due to Durrett.
Given a subset $A \subseteq \mathbb F_2^n$, we can consider the distribution of the intersection size of $A$ with a uniformly random $d$-flat $F$. Motivated by the edge statistics problem and the hypercube statistics problem, the affine subspace statistics problem concerns the maximum of $\mathbb{P}[|F\cap A|=s]$ among $A \subseteq \mathbb F_2^n$ for any fixed $s\in\{1,\dots,2^d\}$ over a uniformly random $d$-flat $F$. We use $\lambda^*(d,s)$ to denote the limit of the maximum when $n$ goes to infinity. In this note, we prove tight bounds for $\lambda^*(d,s)$ in two different regimes. For $s=j2^k$ where $j$ is a positive odd integer, the best known lower bound construction achieving $\lambda^*(d,s)\ge 1-2^{-k}$ is due to taking $A$ as the union of $j$ parallel $(n-d+k)$-flats in $\mathbb F_2^n$. Our main result is a matching upper bound with an additive error term of $O(2^{-3k/2})$. We also study the case $s=1$, where we determine $\lambda^*(d,1)$ exactly. We show that the random construction where each point is included with probability $2^{-d}$ is optimal.
Ting-Wei Chao, Zixuan Xu, D. Zakharov· 0 citations
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