arXiv subjects

Tobias Johnson

Publications and source records attributed to Tobias Johnson.

At least 19 recordsLinked to original sources

arXiv2026

Activated random walk exhibits self-organized criticality

To explain the ubiquity of power laws and fractals in nature, Bak, Tang, and Wiesenfeld formulated simple conditions for a system to self-organize into a critical state. Dickman, Mu\~noz, Vespignani, and Zapperi postulated that the self-organized critical state matches the critical state in corresponding fixed-energy models undergoing traditional phase transitions. Although the theory has been applied broadly over the past five decades, no mathematical model has been proven to exhibit the conjectured behavior. Indeed, the originally proposed abelian sandpile model displays nonuniversal behavior stemming from its slow mixing. Marking the first result of its kind, we prove that the 1-d activated random walk model mixes quickly into a stationary state with power-law avalanches and limiting critical density that equals the critical value for the fixed-energy version.

cond-mat.stat-mech↗

arXiv2026

Explosivity in 1-d Activated Random Walk

We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability. This extends the same result recently proven by Brown, Hoffman, and Son for i.i.d. initial distributions to the setting of ergodic ones, thus completing the proof of a conjecture of Rolla's in dimension one. As a corollary we obtain that, for supercritical ergodic initial distributions with any positive density of particles initially active, the system will stay active almost surely. Our result is another piece of evidence attesting to the universality of the phase transition of Activated Random Walk on $\mathbb{Z}$.

math.PR↗

arXiv2025

The odometer in subcritical activated random walk

We consider the activated random walk particle system, a model of self-organized criticality, on $\mathbb{Z}$ with i.i.d.-Bernoulli initial configuration. We show that at subcritical density, the system's odometer function, which counts the number of actions taken at each site, has a stretched exponential tail. It follows that the expected odometer at each site is finite.

math.PR↗

arXiv2025

Cutoff for activated random walk

We prove that the mixing time of driven-dissipative activated random walk on an interval of length $n$ with uniform or central driving exhibits cutoff at $n$ times the critical density for activated random walk on the integers. The proof uses a new result for arbitrary graphs showing that the chain is mixed once activity is likely at every site.

math.PR↗

arXiv2024

The hockey-stick conjecture for activated random walk

We prove a conjecture of Levine and Silvestri that the driven-dissipative activated random walk model on an interval drives itself directly to and then sustains a critical density. This marks the first rigorous confirmation of a sandpile model behaving as in Bak, Tang, and Wiesenfeld's original vision of self-organized criticality.

math.PR↗

arXiv2024

The density conjecture for activated random walk

Bak, Tang, and Wiesenfeld developed their theory of self-organized criticality in the late 1980s to explain why many real-life processes exhibit signs of critical behavior despite the absence of a tuning parameter. A decade later, Dickman, Muñoz, Vespignani, and Zapperi explained self-organized criticality as an external force pushing a hidden parameter toward the critical value of a traditional absorbing-state phase transition. As evidence, they observed empirically that for various sandpile models, the particle density in a finite box under driven-dissipative dynamics converges to the critical density of an infinite-volume version of the model. We give the first proof of this well-known density conjecture in any setting by establishing it for activated random walk in one dimension. We prove that two other natural versions of the model have the same critical value, further establishing activated random walk as a universal model of self-organized criticality.

math.PR↗

arXiv2023

Particle density in diffusion-limited annihilating systems

Place an $A$-particle at each site of a graph independently with probability $p$ and otherwise place a $B$-particle. $A$- and $B$-particles perform independent continuous time random walks at rates $λ_A$ and $λ_B$, respectively, and annihilate upon colliding with a particle of opposite type. Bramson and Lebowitz studied the setting $λ_A = λ_B$ in the early 1990s. Despite recent progress, many basic questions remain unanswered for when $λ_A \neq λ_B$. For the critical case $p=1/2$ on low-dimensional integer lattices, we give a lower bound on the expected number of particles at the origin that matches physicists' predictions. For the process with $λ_B=0$ on the integers and the bidirected regular tree, we give sharp upper and lower bounds for the expected total occupation time of the root at and approaching criticality.

math.PR↗

arXiv2022

Diffusion-limited annihilating systems and the increasing convex order

We consider diffusion-limited annihilating systems with mobile $A$-particles and stationary $B$-particles placed throughout a graph. Mutual annihilation occurs whenever an $A$-particle meets a $B$-particle. Such systems, when ran in discrete time, are also referred to as parking processes. We show for a broad family of graphs and random walk kernels that augmenting either the size or variability of the initial placements of particles increases the total occupation time by $A$-particles of a given subset of the graph. A corollary is that the same phenomenon occurs with the total lifespan of all particles in internal diffusion-limited aggregation.

math.PR↗

arXiv2022

Concentration inequalities from monotone couplings for graphs, walks, trees and branching processes

Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Peköz, Röllin, and Ross (2016, arXiv:1309.4183 [math.PR]) exploited characterizing distributional fixed point equations to obtain uniform error bounds for generalized gamma approximations using Stein's method. Here we show how monotone couplings arising with these fixed point equations can be used to obtain sharper tail bounds that, in many cases, outperform competing moment-based bounds and the uniform bounds obtainable with Stein's method. Applications are given to concentration inequalities for preferential attachment random graphs, branching processes, random walk local time statistics and the size of random subtrees of uniformly random binary rooted plane trees.

math.PR↗

arXiv2021

Continuous phase transitions on Galton-Watson trees

Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton-Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton-Watson tree is infinite, and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a nonzero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterize the event undergoing the phase transition.

math.PR↗

arXiv2019

Infection spread for the frog model on trees

The frog model is an infection process in which dormant particles begin moving and infecting others once they become infected. We show that on the rooted $d$-ary tree with particle density $Ω(d^2)$, the set of visited sites contains a linearly expanding ball and the number of visits to the root grows linearly with high probability.

math.PR↗

arXiv2019

Cover time for the frog model on trees

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $μ$ on the full $d$-ary tree of height $n$. If $μ= Ω( d^2)$, all of the vertices are visited in time $Θ(n\log n)$ with high probability. Conversely, if $μ= O(d)$ the cover time is $\exp(Θ(\sqrt n))$ with high probability.

math.PR↗

arXiv2019

Random tree recursions: which fixed points correspond to tangible sets of trees?

Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a Galton-Watson tree falls in $\mathcal{B}$. The metaproperty makes $p$ satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is $p$, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.

math.PR↗

arXiv2019

Sensitivity of the frog model to initial conditions

The frog model is an interacting particle system on a graph. Active particles perform independent simple random walks, while sleeping particles remain inert until visited by an active particle. Some number of sleeping particles are placed at each site sampled independently from a certain distribution, and then one particle is activated to begin the process. We show that the recurrence or transience of the model is sensitive not just to the expectation but to the entire distribution. This is in contrast to closely related models like branching random walk and activated random walk.

math.PR↗

arXiv2017

Bounds to the normal for proximity region graphs

In a proximity region graph ${\cal G}$ in $\mathbb{R}^d$, two distinct points $x,y$ of a point process $μ$ are connected when the 'forbidden region' $S(x,y)$ these points determine has empty intersection with $μ$. The Gabriel graph, where $S(x,y)$ is the open disc with diameter the line segment connecting $x$ and $y$, is one canonical example. When $μ$ is a Poisson or binomial process, under broad conditions on the regions $S(x,y)$, bounds on the Kolmogorov and Wasserstein distances to the normal are produced for functionals of ${\cal G}$, including the total number of edges and the total length. Variance lower bounds, not requiring strong stabilization, are also proven to hold for a class of such functionals.

math.PR↗

arXiv2017

Stochastic orders and the frog model

The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove that certain frog model statistics are monotone in the initial configuration for two nonstandard stochastic dominance relations: the increasing concave and the probability generating function orders. This extends many canonical theorems. We connect recurrence for random initial configurations to recurrence for deterministic configurations. Also, the limiting shape of activated sites on the integer lattice respects both of these orders. Other implications include monotonicity results on transience of the frog model where the number of frogs per vertex decays away from the origin, on survival of the frog model with death, and on the time to visit a given vertex in any frog model.

math.PR↗

arXiv2017

Size biased couplings and the spectral gap for random regular graphs

Let $λ$ be the second largest eigenvalue in absolute value of a uniform random $d$-regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then $λ=2\sqrt{d-1} +o(1)$ with high probability. In the present work we show that $λ=O(\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress towards a conjecture of Vu that the bound holds for all $1\le d\le n/2$. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at $d=o(n^{1/2})$. We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on $d$-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.

math.PR↗

arXiv2017

Local limit of the fixed point forest

Consider the following partial "sorting algorithm" on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is 1. This process imposes a forest structure on the set of all permutations of size $n$, where the roots are the permutations starting with 1 and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this "fixed point forest" exhibits a rich structure. In this paper, we consider the fixed point forest in the limit $n\to \infty$ and show using Stein's method that at a random permutation the local structure weakly converges to a tree defined in terms of independent Poisson point processes. We also show that the distribution of the length of the longest path to a leaf converges to the geometric distribution with mean $e-1$, and the length of the shortest path converges to the Poisson distribution with mean 1. In addition, the higher moments are bounded and hence the expectations converge as well.

math.PR↗