arXiv subjects

Lionel Levine

Publications and source records attributed to Lionel Levine.

At least 37 records · Page 2Linked to original sources

arXiv2021

Random walks with local memory

We prove a quenched invariance principle for a class of random walks in random environment on $\mathbb{Z}^d$, where the walker alters its own environment. The environment consists of an outgoing edge from each vertex. The walker updates the edge $e$ at its current location to a new random edge $e'$ (whose law depends on $e$) and then steps to the other endpoint of $e'$. We show that a native environment for these walks (i.e., an environment that is stationary in time from the perspective of the walker) consists of the wired uniform spanning forest oriented toward the walker, plus an independent outgoing edge from the walker.

math.PR↗

arXiv2020

Abelian networks IV. Dynamics of nonhalting networks

An abelian network is a collection of communicating automata whose state transitions and message passing each satisfy a local commutativity condition. This paper is a continuation of the abelian networks series of Bond and Levine (2016), for which we extend the theory of abelian networks that halt on all inputs to networks that can run forever. A nonhalting abelian network can be realized as a discrete dynamical system in many different ways, depending on the update order. We show that certain features of the dynamics, such as minimal period length, have intrinsic definitions that do not require specifying an update order. We give an intrinsic definition of the \emph{torsion group} of a finite irreducible (halting or nonhalting) abelian network, and show that it coincides with the critical group of Bond and Levine (2016) if the network is halting. We show that the torsion group acts freely on the set of invertible recurrent components of the trajectory digraph, and identify when this action is transitive. This perspective leads to new results even in the classical case of sinkless rotor networks (deterministic analogues of random walks). In Holroyd et. al (2008) it was shown that the recurrent configurations of a sinkless rotor network with just one chip are precisely the unicycles (spanning subgraphs with a unique oriented cycle, with the chip on the cycle). We generalize this result to abelian mobile agent networks with any number of chips. We give formulas for generating series such as \[ \sum_{n \geq 1} r_n z^n = \det (\frac{1}{1-z}D - A ) \] where $r_n$ is the number of recurrent chip-and-rotor configurations with $n$ chips; $D$ is the diagonal matrix of outdegrees, and $A$ is the adjacency matrix. A consequence is that the sequence $(r_n)_{n \geq 1}$ completely determines the spectrum of the simple random walk on the network.

cs.FL↗

arXiv2020

Double jump phase transition in a soliton cellular automaton

In this paper, we consider the soliton cellular automaton introduced in [Takahashi 1990] with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birth-and-death chains and Galton-Watson forests. Using these ideas, we establish limit theorems showing that if the first $n$ boxes are occupied independently with probability $p\in(0,1)$, then the number of solitons is of order $n$ for all $p$, and the length of the longest soliton is of order $\log n$ for $p<1/2$, order $\sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: For each fixed $j\geq 1$, the top $j$ soliton lengths have the same order as the longest for $p\leq 1/2$, whereas all but the longest have order at most $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\text{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.

math.PR↗

arXiv2020

Anxiety Detection Leveraging Mobile Passive Sensing

Anxiety disorders are the most common class of psychiatric problems affecting both children and adults. However, tools to effectively monitor and manage anxiety are lacking, and comparatively limited research has been applied to addressing the unique challenges around anxiety. Leveraging passive and unobtrusive data collection from smartphones could be a viable alternative to classical methods, allowing for real-time mental health surveillance and disease management. This paper presents eWellness, an experimental mobile application designed to track a full-suite of sensor and user-log data off an individual's device in a continuous and passive manner. We report on an initial pilot study tracking ten people over the course of a month that showed a nearly 76% success rate at predicting daily anxiety and depression levels based solely on the passively monitored features.

cs.CY↗

arXiv2020

The devil's staircase for chip-firing on random graphs and on graphons

We study the behavior of the activity of the parallel chip-firing upon increasing the number of chips on an Erdős--Rényi random graph. We show that in various situations the resulting activity diagrams converge to a devil's staircase as we increase the number of vertices. Our method is to generalize the parallel chip-firing to graphons, and to prove a continuity result for the activity. We also show that the activity of a chip configuration on a graphon does not necessarily exist, but it does exist for every chip configuration on a large class of graphons.

math.CO↗

arXiv2018

How long does it take for Internal DLA to forget its initial profile?

Internal DLA is a discrete model of a moving interface. On the cylinder graph $\mathbb{Z}_N \times \mathbb{Z}$, a particle starts uniformly on $\mathbb{Z}_N \times \{0\}$ and performs simple random walk on the cylinder until reaching an unoccupied site in $\mathbb{Z}_N \times \mathbb{Z}_{\geq 0}$, which it occupies forever. This operation defines a Markov chain on subsets of the cylinder. We first show that a typical subset is rectangular with at most logarithmic fluctuations. We use this to prove that two Internal DLA chains started from different typical subsets can be coupled with high probability by adding order $N^2 \log N$ particles. For a lower bound, we show that at least order $N^2$ particles are required to forget which of two independent typical subsets the process started from.

math.PR↗

arXiv2018

Sandpiles on the square lattice

We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $\mathbb{Z}^2$. We also determine the asymptotic spectral gap, asymptotic mixing time and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus $\left( \mathbb{Z} / m\mathbb{Z} \right)^2$. The techniques use analysis of the space of functions on $\mathbb{Z}^2$ which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in $\ell^p(\mathbb{Z}^2)$ as linear combinations of certain discrete derivatives of Green's functions, extending a result of Schmidt and Verbitskiy.

math.PR↗

arXiv2018

Abelian logic gates

An abelian processor is an automaton whose output is independent of the order of its inputs. Bond and Levine have proved that a network of abelian processors performs the same computation regardless of processing order (subject only to a halting condition). We prove that any finite abelian processor can be emulated by a network of certain very simple abelian processors, which we call gates. The most fundamental gate is a "toppler", which absorbs input particles until their number exceeds some given threshold, at which point it topples, emitting one particle and returning to its initial state. With the exception of an adder gate, which simply combines two streams of particles, each of our gates has only one input wire. Our results can be reformulated in terms of the functions computed by processors, and one consequence is that any increasing function from N^k to N^l that is the sum of a linear function and a periodic function can be expressed in terms of (possibly nested) sums of floors of quotients by integers.

cs.DM↗

arXiv2018

Formation of large-scale random structure by competitive erosion

We study the following one-dimensional model of annihilating particles. Beginning with all sites of $\mathbb{Z}$ uncolored, a blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then, a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after $n$ blue and $n$ red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$. The resulting random color configuration, after rescaling by $n^{1/4}$ and taking $n\to \infty$, has an explicit description in terms of alternating extrema of Brownian motion (the global maximum on a certain interval, the global minimum attained after that maximum, etc.).

math.PR↗

arXiv2017

The Apollonian structure of integer superharmonic matrices

We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely characterizes the PDE which determines the continuum scaling limit of the Abelian sandpile on the lattice $\mathbb{Z}^2$.

math.AP↗

arXiv2016

The gap of the area-weighted Motzkin spin chain is exponentially small

We prove that the energy gap of the model proposed by Zhang, Ahmadain, and Klich [1] is exponentially small in the square of the system size. In [2] a class of exactly solvable quantum spin chain models was proposed that have integer spins ($s$), with a nearest neighbors Hamiltonian, and a unique ground state. The ground state can be seen as a uniform superposition of all $s-$colored Motzkin walks. The half-chain entanglement entropy provably violates the area law by a square root factor in the system's size ($\sim\sqrt{n}$) for $s>1$. For $s=1$, the violation is logarithmic [3]. Moreover in [2] it was proved that the gap vanishes polynomially and is $O(n^{-c})$ with $c\ge2$. Recently, a deformation of [2], which we call "weighted Motzkin quantum spin chain" was proposed [1]. This model has a unique ground state that is a superposition of the $s-$colored Motzkin walks weighted by $t^{\text{area\{Motzkin walk\}}}$ with $t>1$. The most surprising feature of this model is that it violates the area law by a factor of $n$. Here we prove that the gap of this model is upper bounded by $8ns\text{ }t^{-n^{2}/3}$ for $t>1$.

quant-ph↗

arXiv2016

Laplacian growth, sandpiles and scaling limits

Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of (internal) Laplacian growth, including the abelian sandpile, internal DLA, rotor aggregation, and the scaling limits of these models on the lattice Z^d as the mesh size goes to zero. These models provide a window into the tools of discrete potential theory: harmonic functions, martingales, obstacle problems, quadrature domains, Green functions, smoothing. We also present one new result: rotor aggregation in Z^d has O(log r) fluctuations around a Euclidean ball, improving a previous power-law bound. We highlight several open questions, including whether these fluctuations are O(1).

math.PR↗

arXiv2016

Interpolating between random walk and rotor walk

We introduce a family of stochastic processes on the integers, depending on a parameter $p \in [0,1]$ and interpolating between the deterministic rotor walk (p=0) and the simple random walk (p=1/2). This p-rotor walk is not a Markov chain but it has a local Markov property: for each $x \in \mathbb{Z}$ the sequence of successive exits from $x$ is a Markov chain. The main result of this paper identifies the scaling limit of the p-rotor walk with two-sided i.i.d. initial rotors. The limiting process takes the form $\sqrt{\frac{1-p}{p}} X(t)$, where $X$ is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation \begin{equation} X(t) = \mathcal{B}(t) + a \sup_{s\leq t} X(s) + b \inf_{s\leq t} X(s) \end{equation} for all $t \in [0,\infty)$. Here $\mathcal{B}(t)$ is a standard Brownian motion and $a,b<1$ are constants depending on the marginals of the initial rotors on $\mathbb{N}$ and $-\mathbb{N}$ respectively. Chaumont and Doney [CD99] have shown that the above equation has a pathwise unique solution $X(t)$, and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, $\limsup X(t) = +\infty$ and $\liminf X(t) = -\infty$ [CDH00]. This last result, together with the main result of this paper, implies that the p-rotor walk is recurrent for any two-sided i.i.d. initial rotors and any $0<p<1$.

math.PR↗

arXiv2016

Abelian networks I. Foundations and examples

In Deepak Dhar's model of abelian distributed processors, automata occupy the vertices of a graph and communicate via the edges. We show that two simple axioms ensure that the final output does not depend on the order in which the automata process their inputs. A collection of automata obeying these axioms is called an "abelian network". We prove a least action principle for abelian networks. As an application, we show how abelian networks can solve certain linear and nonlinear integer programs asynchronously. In most previously studied abelian networks, the input alphabet of each automaton consists of a single letter; in contrast, we propose two non-unary examples of abelian networks: "oil and water" and "abelian mobile agents".

cs.FL↗

arXiv2015

Mixing time and eigenvalues of the abelian sandpile Markov chain

The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph $G$. By viewing this chain as a (nonreversible) random walk on an abelian group, we give a formula for its eigenvalues and eigenvectors in terms of `multiplicative harmonic functions' on the vertices of $G$. We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest non-integer vector in the dual Laplacian lattice, while the mixing time is at most a constant times the smoothing parameter of the Laplacian lattice. We find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on $G$: If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where $G$ is the complete graph on $n$ vertices, we show that the sandpile chain exhibits cutoff at time $\frac{1}{4π^{2}}n^{3}\log n$.

math.PR↗

arXiv2015

Abelian networks III. The critical group

The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. We generalize Dhar's burning algorithm to abelian networks, and estimate the running time of an abelian network on an arbitrary input up to a constant additive error.

cs.FL↗

arXiv2015

Multi-Eulerian tours of directed graphs

Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e)=tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is bounded in terms of the Pham index, a measure of 'Eulerianness'.

math.CO↗

arXiv2015

CoEulerian graphs

We suggest a measure of "Eulerianness" of a finite directed graph and define a class of "coEulerian" graphs. These are the graphs whose Laplacian lattice is as large as possible. As an application, we address a question in chip-firing posed by Bjorner, Lovasz, and Shor in 1991, who asked for "a characterization of those digraphs and initial chip configurations that guarantee finite termination." Bjorner and Lovasz gave an exponential time algorithm in 1992. We show that this can be improved to linear time if the graph is coEulerian, and that the problem is NP-complete for general directed multigraphs.

math.CO↗