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Jacob Richey

Publications and source records attributed to Jacob Richey.

15 recordsLinked to original sources
arXiv2026

Letter frequency in shifts of finite type with one forbidden word

This work considers combinatorial and statistical aspects of {\em{shifts of finite type}}, which are families of words over a finite alphabet which avoid a fixed class of {\emph{forbidden}} sub-words. The overarching question we are interested in is: how do local statistics of a uniformly random element of the shift space depend on combinatorial features of the forbidden set? We focus on the binary alphabet $\{0,1\}$, the class of shift spaces where a single pattern is forbidden, and the average frequency of $1$s (equivalently, the probability of observing $1$ at a given position). In this case, the relevant combinatorial information is encoded by a two-variable auto-correlation polynomial associated to the forbidden word, which we call the {\em{border polynomial}}. We present several results and examples characterizing the ordering of all words by their letter frequencies: for example, we describe the set of patterns which, when forbidden, cause the frequency of $1$s to increase, decrease, or stay exactly $1/2$. Our methodologies include novel explicit local injections and bijections, generating function analysis, and a connection with a probabilistic notion of letter frequency.

math.CO↗
arXiv2026

Law of large numbers for activated random walk on villages

We consider activated random walk (ARW), an interacting particle system and prototypical model of self-organized criticality in a setting which combines mean-field behavior with the geometry of an arbitrary graph, which we call the village model of ARW, or VARW for short. VARW is obtained from a fixed graph by replacing each vertex with a 'village' that consists of n replicas of that vertex. We focus on VARW where particles walk according to a strictly sub-stochastic transition kernel on a finite underlying graph, so mass is sometimes lost (which guarantees that the system eventually stabilizes almost surely). Under a subcriticality assumption on the initial state we prove a law of large numbers as n goes to infinity for the resulting stable configuration of particles and the odometer of the process, to a limit which is uniquely characterized by a system of non-linear equations.

math.PR↗
arXiv2026

Local Density of Activated Random Walk on $\mathbb{Z}$

We consider one-dimensional activated random walk (ARW) on $\mathbb{Z}$ started from a `point source' initial condition, with many particles at the origin and no other particles. We prove that, uniformly throughout a macroscopic window around the source, the probability that a site contains a sleeping particle after the configuration is stabilized is approximately the critical density. This represents a first step towards understanding the local structure of the critical stationary measure for ARW.

math.PR↗
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↗
arXiv2024

Word length, bias and bijections in Penney's ante

Fix two words over the binary alphabet $\{0,1\}$, and generate iid Bernoulli$(p)$ bits until one of the words occurs in sequence. This setup, commonly known as Penney's ante, was popularized by Conway, who found (in unpublished work) a simple formula for the probability that a given word occurs first. We study win probabilities in Penney's ante from an analytic and combinatorial perspective, building on previous results for the case $p = \frac{1}{2}$ and words of the same length. For words of arbitrary lengths, our results bound how large the win probability can be for the longer word. When $p = \frac{1}{2}$ we characterize when a longer word can be statistically favorable, and for $p \neq \frac{1}{2}$ we present a conjecture describing the optimal pairs, which is supported by computer computations. Additionally, we find that Penney's ante often exhibits symmetry under the transformation $p \to 1-p$. We construct new explicit bijections that account for these symmetries, under conditions that can be easily verified by examining auto- and cross-correlations of the words.

math.CO↗
arXiv2024

Shifts of Finite Type Obtained by Forbidding a Single Pattern

Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981, 183-208) showed that the autocorrelation polynomial $\phi_w(t)$ of $w$, which records the set of self-overlaps of $w$, explicitly determines for each $n$, the number $|B_n(w)|$ of words of length $n$ that avoid $w$. We consider this and related problems from the viewpoint of symbolic dynamics, focusing on the setting of $X_{\{w\}}$, the space of all bi-infinite sequences that avoid $w$. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981, 183-208) and other work to show that the sequence $|B_n(w)|$ is equivalent to several invariants of $X_{\{w\}}$. We then give a finite-state labeled graphical representation $L_w$ of $X_{\{w\}}$ and show that $w$ can be recovered from the graph isomorphism class of the unlabeled version of $L_w$. Using $L_w$, we apply ideas from probability and Perron-Frobenius theory to obtain results comparing features of $X_{\{w\}}$ for different $w$. Next, we give partial results on the problem of classifying the spaces $X_{\{w\}}$ up to conjugacy. Finally, we extend some of our results to spaces of multi-dimensional arrays that avoid a given finite pattern.

math.DS↗
arXiv2023

Diffusion-limited annihilating-coalescing systems

We study a family of interacting particle systems with annihilating and coalescing reactions. Two types of particles are interspersed throughout a transitive unimodular graph. Both types diffuse as simple random walks with possibly different jump rates. Upon colliding, like particles coalesce up to some cap and unlike particles annihilate. We describe a phase transition as the initial particle density is varied and provide estimates for the expected occupation time of the root. For the symmetric setting with no cap on coalescence, we prove that the limiting occupation probability of the root is asymptotic to 2/3 the occupation probability for classical coalescing random walk. This addresses an open problem from Stephenson.

math.PR↗
arXiv2023

Random walks on regular trees can not be slowed down

A random walk on a regular tree (or any non-amenable graph) has positive speed. We ask whether such a walk can be slowed down by applying carefully chosen time-dependent permutations of the vertices. We prove that on trees the random walk can not be slowed down.

math.PR↗
arXiv2020

Active Phase for Activated Random Walk on Z

We consider the Activated Random Walk model on $\mathbb{Z}$. In this model, each particle performs a continuous-time simple symmetric random walk, and falls asleep at rate $\lambda$. A sleeping particle does not move but it is reactivated in the presence of another particle. We show that for any sleep rate $\lambda < \infty$ if the density $ \zeta $ is close enough to $1$ then the system stays active.

math.PR↗
arXiv2020

Rumor source detection with multiple observations under adaptive diffusions

Recent work, motivated by anonymous messaging platforms, has introduced adaptive diffusion protocols which can obfuscate the source of a rumor: a "snapshot adversary" with access to the subgraph of "infected" nodes can do no better than randomly guessing the entity of the source node. What happens if the adversary has access to multiple independent snapshots? We study this question when the underlying graph is the infinite $d$-regular tree. We show that (1) a weak form of source obfuscation is still possible in the case of two independent snapshots, but (2) already with three observations there is a simple algorithm that finds the rumor source with constant probability, regardless of the adaptive diffusion protocol. We also characterize the tradeoff between local spreading and source obfuscation for adaptive diffusion protocols (under a single snapshot). These results raise questions about the robustness of anonymity guarantees when spreading information in social networks.

cs.SI↗
arXiv2020

Intersections of random sets

We consider a variant of a classical coverage process, the boolean model in $\mathbb{R}^d$. Previous efforts have focused on convergence of the unoccupied region containing the origin to a well studied limit $C$. We study the intersection of sets centered at points of a Poisson point process confined to the unit ball. Using a coupling between the intersection model and the original boolean model, we show that the scaled intersection converges weakly to the same limit $C$. Along the way, we present some tools for studying statistics of a class of intersection models.

math.PR↗
arXiv2018

Activated Random Walk on a cycle

We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle $\mathbb{Z}/n\mathbb{Z}$. One starts with a mass density $μ>0$ of initially active particles, each of which performs a simple symmetric random walk at rate one and falls asleep at rate $λ>0.$ Sleepy particles become active on coming in contact with other active particles. There have been several recent results concerning fixation/non-fixation of the ARW dynamics on infinite systems depending on the parameters $μ$ and $λ$. On the finite graph $\mathbb{Z}/n\mathbb{Z}$, unless there are more than $n$ particles, the process fixates (reaches an absorbing state) almost surely in finite time. We show that the number of steps the process takes to fixate is linear in $n$ (up to poly-logarithmic terms), when the density is sufficiently low compared to the sleep rate, and exponential in $n$ when the sleep rate is sufficiently small compared to the density, reflecting the fixation/non-fixation phase transition in the corresponding infinite system as established by Rolla, Sidoravicius (2012).

math.PR↗
arXiv2016

A smooth transition from Wishart to GOE

It is well known that an $n \times n$ Wishart matrix with $d$ degrees of freedom is close to the appropriately centered and scaled Gaussian Orthogonal Ensemble (GOE) if $d$ is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when $d = \Theta ( n^{3} )$. Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when $d / n^{3} \to c \in (0, \infty)$. This shows, in particular, that the phase transition from Wishart to GOE is smooth.

math.ST↗