Rightmost Particle Research Articles

Dynamically emergent correlations between particles in a switching harmonic trap.

We study a one dimensional gas of N noninteracting diffusing particles in a harmonic trap, whose stiffness switches between two values μ_{1} and μ_{2} with constant rates r_{1} and r_{2}, respectively. Despite the absence of direct interaction between the particles, we show that strong correlations between them emerge in the stationary state at long times, induced purely by the dynamics itself. We compute exactly the joint distribution of the positions of the particles in the stationary state, which allows us to compute several physical observables analytically. In particular, we show that the extreme value statistics (EVS), i.e., the distribution of the position of the rightmost particle, has a nontrivial shape in the large N limit. The scaling function characterizing this EVS has a finite support with a tunable shape (by varying the parameters). Remarkably, this scaling function turns out to be universal. First, it also describes the distribution of the position of the kth rightmost particle in a 1d trap. Moreover, the distribution of the position of the particle farthest from the center of the harmonic trap in d dimensions is also described by the same scaling function for all d≥1. Numerical simulations are in excellent agreement with our analytical predictions.

Dynamics at the edge for independent diffusing particles.

We study the dynamics of the outliers for a large number of independent Brownian particles in one dimension. We derive the multitime joint distribution of the position of the rightmost particle, by two different methods. We obtain the two-time joint distribution of the maximum and second maximum positions, and we study the counting statistics at the edge. Finally, we derive the multitime joint distribution of the running maximum, as well as the joint distribution of the arrival time of the first particle at several space points.

Gumbel Laws in the Symmetric Exclusion Process

We consider the symmetric exclusion particle system on $\mathbb{Z}$ starting from an infinite particle step configuration in which there are no particles to the right of a maximal one. We show that the scaled position $X_t/(\sigma b_t) - a_t$ of the right-most particle at time $t$ converges to a Gumbel limit law, where $b_t = \sqrt{t/\log t}$, $a_t = \log(t/(\sqrt{2\pi}\log t))$, and $\sigma$ is the standard deviation of the random walk jump probabilities. This work solves a problem left open in Arratia (1983). Moreover, to investigate the influence of the mass of particles behind the leading one, we consider initial profiles consisting of a block of $L$ particles, where $L \to \infty$ as $t \to \infty$. Gumbel limit laws, under appropriate scaling, are obtained for $X_t$ when $L$ diverges in $t$. In particular, there is a transition when $L$ is of order $b_t$, above which the displacement of $X_t$ is similar to that under a infinite particle step profile, and below which it is of order $\sqrt{t\log L}$. Proofs are based on recently developed negative dependence properties of the symmetric exclusion system. Remarks are also made on the behavior of the right-most particle starting from a step profile in asymmetric nearest-neighbor exclusion, which complement known results.

Extreme Statistics and Spacing Distribution in a Brownian Gas Correlated by Resetting.

We study a one-dimensional gas of N Brownian particles that diffuse independently, but are simultaneously reset to the origin at a constant rate r. The system approaches a nonequilibrium stationary state with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the kth rightmost particle, and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realization of this resetting gas using optical traps.

Distribution of a Second-Class Particle’s Position in the Two-Species ASEP with a Special Initial Configuration

In this paper, we consider the two-species asymmetric simple exclusion process (ASEP) consisting of $$N-1$$ first-class particles and one second-class particle. We assume that all particles are located at arbitrary positions but the second-class particle is the rightmost particle at time $$t=0$$ . We find the exact formula of the distribution of the second-class particle’s position at time t by directly using the transition probabilities of the two-species ASEP, which is a different approach from the coupling method Tracy and Widom used in [J Phys A 42:425002, 2009].

Branching random walk with infinite progeny mean: A tale of two tails

We study the extremes of branching random walks under the assumption that the underlying Galton–Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the nth generation when the tail of the displacement behaves like exp(−K(x)), where either K is a regularly varying function of index r>0, or K has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when r>1.

At the edge of a one-dimensional jellium

We consider a one-dimensional classical Wigner jellium, not necessarily charge neutral, for which the electrons are allowed to exist beyond the support of the background charge. The model can be seen as a one-dimensional Coulomb gas in which the external field is generated by a smeared background on an interval. It is a true one-dimensional Coulomb gas and not a one-dimensional log-gas. The system exists if and only if the total background charge is greater than the number of electrons minus one. For various backgrounds, we show convergence to point processes, at the edge of the support of the background. In particular, this provides asymptotic analysis of the fluctuations of the right-most particle. Our analysis reveals that these fluctuations are not universal, in the sense that depending on the background, the tails range anywhere from exponential to Gaussian-like behavior, including for instance Tracy-Widom-like behavior. We also obtain a Renyi-type probabilistic representation for the order statistics of the particle system beyond the support of the background.

Large deviations for discrete β-ensembles

We consider discrete β-ensembles as introduced by Borodin, Gorin and Guionnet (2017) [9]. Under general assumptions, we establish a large deviation principle for their rightmost particle. We apply our general results to two classes of measures that are related to Jack symmetric functions.

The stochastic Fisher-KPP Equation with seed bank and on/off branching coalescing Brownian motion

We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of “on/off”-moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an “on/off-branching Brownian motion”. This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be “switched off” for an exponential amount of time after which they get “switched on” again. On/off branching Brownian motion shows qualitatively different behaviour to classical branching Brownian motion and is an interesting object for study in itself. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed sqrt{2}, is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by sqrt{sqrt{5}-1}approx 1.111 under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology.

Fluctuations of the Rightmost Particle in the Catalytic Branching Brownian Motion

In this article we establish the magnitude of fluctuations of the extreme particle in the model of binary branching Brownian motion with a single catalytic point at the origin.

Edge fluctuations and third-order phase transition in harmonically confined long-range systems

We study the distribution of the position of the rightmost particle x max in a N-particle Riesz gas in one dimension confined in a harmonic trap. The particles interact via long-range repulsive potential, of the form r −k with −2 < k < ∞ where r is the inter-particle distance. In equilibrium at temperature O(1), the gas settles on a finite length scale L N that depends on N and k. We numerically observe that the typical fluctuation of y max = x max/L N around its mean is of . Over this length scale, the distribution of the typical fluctuations has a N independent scaling form. We show that the exponent η k obtained from the Hessian theory predicts the scale of typical fluctuations remarkably well. The distribution of atypical fluctuations to the left and right of the mean ⟨y max⟩ are governed by the left and right large deviation functions (LDFs), respectively. We compute these LDFs explicitly ∀ k > −2. We also find that these LDFs describe a pulled to pushed type phase transition as observed in Dyson’s log-gas (k → 0) and 1d one component plasma (k = −1). Remarkably, we find that the phase transition remains third order for the entire regime. Our results demonstrate the striking universality of the third order transition even in models that fall outside the paradigm of Coulomb systems and the random matrix theory. We numerically verify our analytical expressions of the LDFs via Monte Carlo simulation using an importance sampling algorithm.

The distance between the two BBM leaders

We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit d 12 of this random variable can be expressed in terms of PDEs related to the Fisher–KPP equation. We use such a representation to determine the sharp asymptotics of as a → +∞. These tail asymptotics were previously known to ‘exponential order;’ we discover an algebraic correction to this behavior.

Genealogy and spatial distribution of the N-particle branching random walk with polynomial tails

The N-particle branching random walk is a discrete time branching particle system with selection. We have N particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent’s location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behaviour of this particle system in the case where the jump distribution has regularly varying tails and the number of particles is large. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

Truncated linear statistics in the one dimensional one-component plasma

In this paper, we study the probability distribution of the observable , with 1 ⩽ N′ ⩽ N and x 1 < x 2 <⋯< x N representing the ordered positions of N particles in a 1D one-component plasma, i.e. N harmonically confined charges on a line, with pairwise repulsive 1D Coulomb interaction |x i − x j |. This observable represents an example of a truncated linear statistics—here proportional to the center of mass of the N′ = κN (with 0 < κ ⩽ 1), rightmost particles. It interpolates between the position of the rightmost particle (in the limit κ → 0) and the full center of mass (in the limit κ → 1). We show that, for large N, s fluctuates around its mean ⟨s⟩ and the typical fluctuations are Gaussian, of width O(N −3/2). The atypical large fluctuations of s, for fixed κ, are instead described by a large deviation form , where the rate function ϕ κ (s) is computed analytically. We show that ϕ κ (s) takes different functional forms in five distinct regions in the (κ, s) plane separated by phase boundaries, thus leading to a rich phase diagram in the (κ, s) plane. Across all the phase boundaries the rate function ϕ κ (s) undergoes a third-order phase transition. This rate function is also evaluated numerically using a sophisticated importance sampling method, and we find a perfect agreement with our analytical predictions.

Infinite system of random walkers: winners and losers

We study an infinite system of particles initially occupying a half-line y ⩽ 0 and undergoing random walks on the entire line. The right-most particle is called a leader. Surprisingly, every particle except the original leader may never achieve the leadership throughout the evolution. For the equidistant initial configuration, the kth particle attains the leadership with probability e−2 k −1(ln k)−1/2 when k ≫ 1. This decay law provides a quantitative measure of the correlation between earlier misfortune proportional to the label k and eternal failure. We also show that the winner defined as the first walker overtaking the initial leader has label k ≫ 1 with probability decaying as .

Rank dependent branching-selection particle systems

We consider a large family of branching-selection particle systems. The branching rate of each particle depends on its rank and is given by a function $b$ defined on the unit interval. There is also a killing measure $D$ supported on the unit interval as well. At branching times, a particle is chosen among all particles to the left of the branching one by sampling its rank according to $D$. The measure $D$ is allowed to have total mass less than one, which corresponds to a positive probability of no killing. Between branching times, particles perform independent Brownian Motions in the real line. This setting includes several well known models like Branching Brownian Motion (BBM), $N$-BBM, rank dependent BBM, and many others. We conjecture a scaling limit for this class of processes and prove such a limit for a related class of branching-selection particle system. This family is rich enough to allow us to use the behavior of solutions of the limiting equation to prove the asymptotic velocity of the rightmost particle under minimal conditions on $b$ and $D$. The behavior turns out to be universal and depends only on $b(1)$ and the total mass of $D$. If the total mass is one, the number of particles in the system $N$ is conserved and the velocities $v_N$ converge to $\sqrt{2 b(1)}$. When the total mass of $D$ is less than one, the number of particles in the system grows up in time exponentially fast and the asymptotic velocity of the rightmost one is $\sqrt{2 b(1)}$ independently of the number of initial particles.

Anomalous spreading in reducible multitype branching Brownian motion

We consider a two-type reducible branching Brownian motion, defined as a two type branching particle system on the real line, in which particles of type $1$ can give birth to particles of type $2$, but not reciprocally. This process has been shown by Biggins to exhibit an anomalous spreading behaviour under specific conditions: in that situation, the rightmost particle at type $t$ is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type $1$ or of type $2$. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer. The aim of this article is to refine the previous results and study the asymptotic behaviour of the extremal process of the two-type reducible branching Brownian motion. If the branching Brownian motion exhibits an anomalous spreading behaviour, its asymptotic differs from what it typically expected in branching Brownian motions.

A Shape Theorem for a One-Dimensional Growing Particle System with a Bounded Number of Occupants per Site

We consider a one-dimensional discrete-space birth process with a bounded number of particle per site. Under the assumptions of the finite range of interaction, translation invariance, and non-degeneracy, we prove a shape theorem. We also derive a limit estimate and an exponential estimate on the fluctuations of the position of the rightmost particle.

Particle-number distribution in large fluctuations at the tip of branching random walks.

We investigate properties of the particle distribution near the tip of one-dimensional branching random walks at large times t, focusing on unusual realizations in which the rightmost lead particle is very far ahead of its expected position, but still within a distance smaller than the diffusion radius ∼sqrt[t]. Our approach consists in a study of the generating function G_{Δx}(λ)=∑_{n}λ^{n}p_{n}(Δx) for the probabilities p_{n}(Δx) of observing n particles in an interval of given size Δx from the lead particle to its left, fixing the position of the latter. This generating function can be expressed with the help of functions solving the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation with suitable initial conditions. In the infinite-time and large-Δx limits, we find that the mean number of particles in the interval grows exponentially with Δx, and that the generating function obeys a nontrivial scaling law, depending on Δx and λ through the combined variable [Δx-f(λ)]^{3}/Δx^{2}, where f(λ)≡-ln(1-λ)-ln[-ln(1-λ)]. From this property, one may conjecture that the growth of the typical particle number with the size of the interval is slower than exponential, but, surprisingly enough, only by a subleading factor at large Δx. The scaling we argue is consistent with results from a numerical integration of the FKPP equation.

How to generate the tip of branching random walks evolved to large times

In a branching process, the number of particles increasesexponentially with time, which makes numerical simulations for large timesdifficult. In many applications, however, only the region close to the extremalparticles is relevant (the “tip”). We present a simple algorithmwhich allows to simulate a branching random walk in one dimension, keepingonly the particles that arrive within some distance of the rightmost particle ata predefined time T. The complexity of the algorithm growslinearly with T. We can furthermore choose to require that therealizations have their rightmost particle arbitrarily far on the right from itstypical position. We illustrate our algorithm by evaluating an observable forwhich no other practical method is known.