Optimization in First-Passage Resetting.

We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is nonstationary and its probability distribution exhibits rich features. In a finite domain, we define a nontrivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.

Trapping and escape in a turbid medium.

We investigate the absorption of diffusing molecules in a fluid-filled spherical beaker that contains many small reactive traps. The molecules are absorbed either by hitting a trap or by escaping via the beaker walls. In the physical situation where the number N of traps is large and their radii a are small compared to the beaker radius R, the fraction of molecules E that escape to the beaker wall and the complementary fraction T that eventually are absorbed by the traps depend only on the dimensionless parameter combination λ = Na/R. We compute E and T as a function of λ for a spherical beaker and for beakers of other three-dimensional shapes. The asymptotic behavior is found to be universal: 1 - E ∼ λ for λ → 0 and E ∼ λ-1/2 for λ → ∞.

Structural Transitions in Densifying Networks.

We introduce a minimal generative model for densifying networks in which a new node attaches to a randomly selected target node and also to each of its neighbors with probability p. The networks that emerge from this copying mechanism are sparse for p<1/2 and dense (average degree increasing with number of nodes N) for p≥1/2. The behavior in the dense regime is especially rich; for example, individual network realizations that are built by copying are disparate and not self-averaging. Further, there is an infinite sequence of structural anomalies at p=2/3, 3/4, 4/5, etc., where the N dependences of the number of triangles (3-cliques), 4-cliques, undergo phase transitions. When linking to second neighbors of the target can occur, the probability that the resulting graph is complete-all nodes are connected-is nonzero as N→∞.

Mortality, redundancy, and diversity in stochastic search.

We investigate a stochastic search process in one dimension under the competing roles of mortality, redundancy, and diversity of the searchers. This picture represents a toy model for the fertilization of an oocyte by sperm. A population of N independent and mortal diffusing searchers all start at x=L and attempt to reach the target at x=0. When mortality is irrelevant, the search time scales as τ_{D}/lnN for lnN≫1, where τ_{D}~L^{2}/D is the diffusive time scale. Conversely, when the mortality rate µ of the searchers is sufficiently large, the search time scales as sqrt[τ_{D}/µ], independent of N. When searchers have distinct and high mortalities, a subpopulation with a nontrivial optimal diffusivity is most likely to reach the target. We also discuss the effect of chemotaxis on the search time and its fluctuations.

Depletion-controlled starvation of a diffusing forager.

We study the starvation of a lattice random walker in which each site initially contains one food unit and the walker can travel S steps without food before starving. When the walker encounters food, it is completely eaten, and the walker can again travel S steps without food before starving. When the walker hits an empty site, the time until the walker starves decreases by 1. In spatial dimension d=1, the average lifetime of the walker ⟨τ⟩∝S, while for d>2, ⟨τ⟩≃exp(S^{ω}), with ω→1 as d→∞; the latter behavior suggests that the upper critical dimension is infinite. In the marginal case of d=2, ⟨τ⟩∝S^{z}, with z≈2. Long-lived walks explore a highly ramified region so they always remain close to sources of food and the distribution of distinct sites visited does not obey single-parameter scaling.

First passage on disordered intervals.

We derive unexpected first-passage properties for nearest-neighbor hopping on finite intervals with disordered hopping rates, including (a) a highly variable spatial dependence of the first-passage time, (b) huge disparities in first-passage times for different realizations of hopping rates, (c) significant discrepancies between the first moment and the square root of the second moment of the first-passage time, and (d) bimodal first-passage time distributions. Our approach relies on the backward equation, in conjunction with probability generating functions, to obtain all moments, as well as the distribution of first-passage times. Our approach is simpler than previous approaches based on the forward equation, in which computing the mth moment of the first-passage time requires all preceding moments.

Catalytic coagulation.

We introduce an autocatalytic aggregation model in which the rate at which two clusters merge is controlled by the third "catalytic" cluster, whose mass must equal the mass of one of the reaction partners. The catalyst is unaffected by the joining event and can participate in or catalyze subsequent reactions. This model is meant to mimic the self-replicating reactions that occur in models for the origin of life. We solve the kinetics of catalytic coagulation for the case of mass-independent reaction rates and show that the total cluster density decays as t^{-1/3}, while the density of clusters of fixed mass decays as t^{-2/3}. These behaviors contrast with the corresponding t^{-1} and t^{-2} scalings for classic aggregation. We extend our model to mass-dependent reaction rates, to situations where only "magic" mass clusters can catalyze reactions, and to include steady monomer input.

Universal exploration dynamics of random walks.

The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. We introduce a more fundamental quantity, the time τn required by a random walk to find a site that it never visited previously when the walk has already visited n distinct sites, which encompasses the full dynamics about the visitation statistics. To study it, we develop a theoretical approach that relies on a mapping with a trapping problem, in which the spatial distribution of traps is continuously updated by the random walk itself. Despite the geometrical complexity of the territory explored by a random walk, the distribution of the τn can be accounted for by simple analytical expressions. Processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes.

Growth inside a corner: the limiting interface shape.

We investigate the growth of a crystal that is built by depositing cubes inside a corner. The interface of this crystal approaches a deterministic growing limiting shape in the long-time limit. Building on known results for the corresponding two-dimensional system and accounting for basic three-dimensional symmetries, we conjecture a governing equation for the evolution of the interface profile. We solve this equation analytically and find excellent agreement with simulations of the growth process. We also present a generalization to arbitrary spatial dimension.

Rounding effects in record statistics.

We analyze record-breaking events in time series of continuous random variables that are subsequently discretized by rounding to integer multiples of a discretization scale Δ>0. Rounding leads to ties of an existing record, thereby reducing the number of new records. For an infinite number of random variables that are drawn from distributions with a finite upper limit, the number of discrete records is finite, while for distributions with a thinner than exponential upper tail, fewer discrete records arise compared to continuous variables. In the latter case, the record sequence becomes highly regular at long times.

Fate of 2D kinetic ferromagnets and critical percolation crossing probabilities.

We present evidence for a deep connection between the zero-temperature coarsening of both the two-dimensional time-dependent Ginzburg-Landau equation and the kinetic Ising model with critical continuum percolation. In addition to reaching the ground state, the time-dependent Ginzburg-Landau equation and kinetic Ising model can fall into a variety of topologically distinct metastable stripe states. The probability to reach a stripe state that winds a times horizontally and b times vertically on a square lattice with periodic boundary conditions equals the corresponding exactly solved critical percolation crossing probability P(a,b) for a spanning path with winding numbers a and b.

Complete visitation statistics of one-dimensional random walks.

We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that n_{1},n_{2},n_{3},... distinct sites are visited at times t_{1},t_{2},t_{3},.... From this multiple-time distribution, we show that the visitation statistics of one-dimensional random walks are temporally correlated, and we quantify the non-Markovian nature of the process. We exploit these ideas to derive unexpected results for the two-time trapping problem and to determine the visitation statistics of two important stochastic processes, the run-and-tumble particle and the biased random walk.

Divergence and consensus in majority rule.

We investigate majority rule dynamics in a population with two classes of people, each with two opinion states ±1, and with tunable interactions between people in different classes. In an update, a randomly selected group adopts the majority opinion if all group members belong to the same class; if not, majority rule is applied with rate ε. Consensus is achieved in a time that scales logarithmically with population size if ε≥ε_{c}=1/9. For ε<ε_{c}, the population can get trapped in a polarized state, with one class preferring the +1 state and the other preferring -1. The time to escape this polarized state and reach consensus scales exponentially with population size.

Facilitated asymmetric exclusion.

We introduce a class of facilitated asymmetric exclusion processes in which particles are pushed by neighbors from behind. For the simplest version in which a particle can hop to its vacant right neighbor only if its left neighbor is occupied, we determine the steady-state current and the distribution of cluster sizes on a ring. We show that an initial density downstep develops into a rarefaction wave that can have a jump discontinuity at the leading edge, while an upstep results in a shock wave. This unexpected rarefaction wave discontinuity occurs generally for facilitated exclusion processes.

Shepherd model for knot-limited polymer ejection from a capsid.

We construct a tractable model to describe the rate at which a knotted polymer is ejected from a spherical capsid via a small pore. Knots are too large to fit through the pore and must reptate to the end of the polymer for ejection to occur. The reptation of knots is described by symmetric exclusion on the line, with the internal capsid pressure represented by an additional biased particle that drives knots to the end of the chain. We compute the exact ejection speed for a finite number of knots L and find that it scales as 1/L. We establish a mapping to the solvable zero-range process. We also construct a continuum theory for many knots that matches the exact discrete theory for large L.

Topology-controlled Potts coarsening.

We uncover unusual topological features in the long-time relaxation of the q-state kinetic Potts ferromagnet on the triangular lattice that is instantaneously quenched to zero temperature from a zero-magnetization initial state. For q=3, the final state is either the ground state (frequency ≈0.75), a frozen three-hexagon state (frequency ≈0.16), a two-stripe state (frequency ≈0.09), or a three-stripe state (frequency <2×10^{-4}). Other final state topologies, such as states with more than three hexagons, occur with probability 10^{-5} or smaller, for q=3. The relaxation to the frozen three-hexagon state is governed by a time that scales as L^{2}lnL. We provide a heuristic argument for this anomalous scaling and present additional new features of Potts coarsening on the triangular lattice for q=3 and for q>3.

Nonuniversal opinion dynamics driven by opposing external influences.

We study the opinion dynamics of a generalized voter model in which N voters are additionally influenced by two opposing news sources whose effect is to promote political polarization. As the influence of these news sources is increased, the mean time to reach consensus scales nonuniversally as N^{α}. The parameter α quantifies the influence of the news sources and increases without bound as the news sources become increasingly influential. The time to reach a politically polarized state, in which roughly equal fractions of the populations are in each opinion state, is generally short, and the steady-state opinion distribution exhibits a transition from near consensus to a politically polarized state as a function of α.

Exclusion in junction geometries.

We investigate the dynamics of the asymmetric exclusion process at a junction. When two input roads are initially fully occupied and a single output road is initially empty, the ensuing rarefaction wave has a rich spatial structure. The density profile also changes dramatically as the initial densities are varied. Related phenomenology arises when one road feeds into two. Finally, we determine the phase diagram of the open system, where particles are fed into two roads at rate α for each road, the two roads merge into one, and particles are extracted from the single output road at rate ß.

Voter models on heterogeneous networks.

We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual "imports" its state from a randomly chosen neighbor. Here the average time TN to reach consensus for a network of N nodes with an uncorrelated degree distribution scales as N mu1 2/mu2, where mu k is the kth moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is "exported" to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree k to overspread the population-the fixation probability--is proportional to k for the voter model and to 1k for the invasion process.

Exciting hard spheres.

We investigate the collision cascade that is generated by a single moving particle in a static and homogeneous hard-sphere gas. We argue that the number of moving particles at time t grows as t;{xi} and the number collisions up to time t grows as t;{eta} , with xi=2d(d+2) , eta=2(d+1)(d+2) , and d the spatial dimension. These growth laws are the same as those from a hydrodynamic theory for the shock wave emanating from an explosion. Our predictions are verified by molecular dynamics simulations in d=1 and 2. For a particle incident on a static gas in a half-space, the resulting backsplatter ultimately contains almost all the initial energy.