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.

Optimal Resetting Brownian Bridges via Enhanced Fluctuations.

We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time t_{f} is finite and the searcher returns to its starting point at t_{f}. This is simply a Brownian motion with a Poissonian resetting rate r to the origin which is constrained to start and end at the origin at time t_{f}. We unveil a surprising general mechanism that enhances fluctuations of a Brownian bridge, by introducing a small amount of resetting. This is verified for different observables, such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. This mechanism, valid for a Brownian bridge in arbitrary dimensions, leads to a finite optimal resetting rate that minimizes the time to search a fixed target. The physical reason behind an optimal resetting rate in this case is entirely different from that of resetting Brownian motions without the bridge constraint. We also derive an exact effective Langevin equation that generates numerically the trajectories of a resetting Brownian bridge in all dimensions via a completely rejection-free algorithm.

Universal Survival Probability for a d-Dimensional Run-and-Tumble Particle.

We consider an active run-and-tumble particle (RTP) in d dimensions and compute exactly the probability S(t) that the x component of the position of the RTP does not change sign up to time t. When the tumblings occur at a constant rate, we show that S(t) is independent of d for any finite time t (and not just for large t), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed v of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem.

Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids.

We present a numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids established by Maimbourg et al. [Phys. Rev. Lett. 116, 015902 (2016)]. For soft sphere interactions, we obtain the numerical solution by an iterative algorithm and a straightforward discretization of time. We also discuss the case of hard spheres for which we first derive analytically the dynamical mean field theory as a non-trivial limit of that of soft spheres. We present numerical results for the memory function and the mean square displacement. Our results reproduce and extend kinetic theory in the dilute or short-time limit, while they also describe dynamical arrest toward the glass phase in the dense strongly interacting regime.

Time Between the Maximum and the Minimum of a Stochastic Process.

We present an exact solution for the probability density function P(τ=t_{min}-t_{max}|T) of the time difference between the minimum and the maximum of a one-dimensional Brownian motion of duration T. We then generalize our results to a Brownian bridge, i.e., a periodic Brownian motion of period T. We demonstrate that these results can be directly applied to study the position difference between the minimal and the maximal heights of a fluctuating (1+1)-dimensional Kardar-Parisi-Zhang interface on a substrate of size L, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for Lévy flights and find that it differs from the Brownian motion result.

Exact Persistence Exponent for the 2D-Diffusion Equation and Related Kac Polynomials.

We compute the persistence for the 2D-diffusion equation with random initial condition, i.e., the probability p_{0}(t) that the diffusion field, at a given point x in the plane, has not changed sign up to time t. For large t, we show that p_{0}(t)∼t^{-θ(2)} with θ(2)=3/16. Using the connection between the 2D-diffusion equation and Kac random polynomials, we show that the probability q_{0}(n) that Kac's polynomials, of (even) degree n, have no real root decays, for large n, as q_{0}(n)∼n^{-3/4}. We obtain this result by using yet another connection with the truncated orthogonal ensemble of random matrices. This allows us to compute various properties of the zero crossings of the diffusing field, equivalently of the real roots of Kac's polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature.

Multicritical Edge Statistics for the Momenta of Fermions in Nonharmonic Traps.

We compute the joint statistics of the momenta p_{i} of N noninteracting fermions in a trap, near the Fermi edge, with a particular focus on the largest one p_{max}. For a 1D harmonic trap, momenta and positions play a symmetric role, and hence the joint statistics of momenta are identical to that of the positions. In particular, p_{max}, as x_{max}, is distributed according to the Tracy-Widom distribution. Here we show that novel "momentum edge statistics" emerge when the curvature of the potential vanishes, i.e., for "flat traps" near their minimum, with V(x)∼x^{2n} and n>1. These are based on generalizations of the Airy kernel that we obtain explicitly. The fluctuations of p_{max} are governed by new universal distributions determined from the nth member of the second Painlevé hierarchy of nonlinear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.

Kinetic Energy of a Trapped Fermi Gas at Finite Temperature.

We study the statistics of the kinetic (or, equivalently, potential) energy for N noninteracting fermions in a 1d harmonic trap of frequency ω at finite temperature T. Remarkably, we find an exact solution for the full distribution of the kinetic energy, at any temperature T and for any N, using a nontrivial mapping to an integrable Calogero-Moser-Sutherland model. As a function of temperature T and for large N, we identify (i) a quantum regime, for T∼ℏω, where quantum fluctuations dominate and (ii) a thermal regime, for T∼Nℏω, governed by thermal fluctuations. We show how the mean and the variance as well as the large deviation function associated with the distribution of the kinetic energy cross over from the quantum to the thermal regime as T increases.

Exact Extremal Statistics in the Classical 1D Coulomb Gas.

We consider a one-dimensional classical Coulomb gas of N-like charges in a harmonic potential-also known as the one-dimensional one-component plasma. We compute, analytically, the probability distribution of the position x_{max} of the rightmost charge in the limit of large N. We show that the typical fluctuations of x_{max} around its mean are described by a nontrivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of x_{max} for Dyson's log gas. We also compute the large deviation functions of x_{max} explicitly and show that the system exhibits a third-order phase transition, as in the log gas. Our theoretical predictions are verified numerically.

Exact Statistics of Record Increments of Random Walks and Lévy Flights.

We study the statistics of increments in record values in a time series {x_{0}=0,x_{1},x_{2},…,x_{n}} generated by the positions of a random walk (discrete time, continuous space) of duration n steps. For arbitrary jump length distribution, including Lévy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of n for large n, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability Q(n) that the record increments decrease monotonically up to step n. Remarkably, Q(n) is universal (i.e., independent of the jump distribution) for each n, decaying as Q(n)∼A/sqrt[n] for large n, with a universal amplitude A=e/sqrt[π]=1.53362….

Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature.

We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry. We show that for short time t, the probability distribution P(H,t) of the height H at a given point x takes the scaling form P(H,t)∼exp[-Φ_{drop}(H)/sqrt[t]] where the rate function Φ_{drop}(H) is computed exactly for all H. While it is Gaussian in the center, i.e., for small H, the probability distribution function has highly asymmetric non-Gaussian tails that we characterize in detail. This function Φ_{drop}(H) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Thanks to a recently discovered connection between the KPZ equation and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full counting statistics at the edge.

Finite-temperature free fermions and the Kardar-Parisi-Zhang equation at finite time.

We consider the system of N one-dimensional free fermions confined by a harmonic well V(x)=mω(2)x(2)/2 at finite inverse temperature ß=1/T. The average density of fermions ρ(N)(x,T) at position x is derived. For N≫1 and ß∼O(1/N), ρ(N)(x,T) is given by a scaling function interpolating between a Gaussian at high temperature, for ß≪1/N, and the Wigner semicircle law at low temperature, for ß≫N(-1). In the latter regime, we unveil a scaling limit, for ßℏω=bN(-1/3), where the fluctuations close to the edge of the support, at x∼±âˆš[2ℏN/(mω)], are described by a limiting kernel K(b)(ff)(s,s') that depends continuously on b and is a generalization of the Airy kernel, found in the Gaussian unitary ensemble of random matrices. Remarkably, exactly the same kernel K(b)(ff)(s,s') arises in the exact solution of the Kardar-Parisi-Zhang equation in 1+1 dimensions at finite time t, with the correspondence t=b(3).

Fluctuating interfaces subject to stochastic resetting.

We study one-dimensional fluctuating interfaces of length L, where the interface stochastically resets to a fixed initial profile at a constant rate r. For finite r in the limit L→∞, the system settles into a nonequilibrium stationary state with non-Gaussian interface fluctuations, which we characterize analytically for the Kardar-Parisi-Zhang and Edwards-Wilkinson universality class. Our results are corroborated by numerical simulations. We also discuss the generality of our results for a fluctuating interface in a generic universality class.

Phase transitions and edge scaling of number variance in Gaussian random matrices.

We consider N × N Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over [-√2],√2]. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P(N,L)(N(L)) that N(L) eigenvalues lie within the box [-L,L]. This probability scales as P(N,L)(N(L) = κ(L)N) ≈ exp(-ßN(2)ψ(L)(κ(L))), where ß is the Dyson index of the ensemble and ψ(L)(κ(L)) is a ß-independent rate function that we compute exactly. We identify three regimes as L is varied: (i) N(-1)≪L < √2 (bulk), (ii) L∼√2 on a scale of O(N(-2/3)) (edge), and (iii) L > sqrt[2] (tail). We find a dramatic nonmonotonic behavior of the number variance V(N)(L) as a function of L: after a logarithmic growth ∝ln(NL) in the bulk (when L∼O(1/N)), V(N)(L) decreases abruptly as L approaches the edge of the semicircle before it decays as a stretched exponential for L > sqrt[2]. This "dropoff" of V(N)(L) at the edge is described by a scaling function V(ß) that smoothly interpolates between the bulk (i) and the tail (iii). For ß = 2 we compute V(2) explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for ß = 2 the full statistics of particle-number fluctuations at zero temperature of 1D spinless fermions in a harmonic trap.

First Order Transition for the Optimal Search Time of Lévy Flights with Resetting.

We study analytically an intermittent search process in one dimension. There is an immobile target at the origin and a searcher undergoes a discrete time jump process starting at x_{0}≥0, where successive jumps are drawn independently from an arbitrary jump distribution f(η). In addition, with a probability 0≤r<1, the position of the searcher is reset to its initial position x_{0}. The efficiency of the search strategy is characterized by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump distribution f(η), initial position x_{0} and resetting probability r, we compute analytically the MFPT. For the heavy-tailed Lévy stable jump distribution characterized by the Lévy index 0<µ<2, we show that, for any given x_{0}, the MFPT has a global minimum in the (µ,r) plane at (µ^{*}(x_{0}),r^{*}(x_{0})). We find a remarkable first-order phase transition as x_{0} crosses a critical value x_{0}^{*} at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.

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.

Fluctuations in the active Dyson Brownian motion and the overdamped Calogero-Moser model.

Recently we introduced the active Dyson Brownian motion model (DBM), in which N run-and-tumble particles interact via a logarithmic repulsive potential in the presence of a harmonic well. We found that in a broad range of parameters the density of particles converges at large N to the Wigner semicircle law as in the passive case. In this paper we provide an analytical support for this numerical observation by studying the fluctuations of the positions of the particles in the nonequilibrium stationary state of the active DBM in the regime of weak noise and large persistence time. In this limit we obtain an analytical expression for the covariance between the particle positions for any N from the exact inversion of the Hessian matrix of the system. We show that, when the number of particles is large N≫1, the covariance matrix takes scaling forms that we compute explicitly both in the bulk and at the edge of the support of the semicircle. In the bulk the covariance scales as N^{-1}, while at the edge it scales as N^{-2/3}. Remarkably we find that these results can be transposed directly to an equilibrium model, the overdamped Calogero-Moser model in the low-temperature limit, providing an analytical confirmation of the numerical results obtained by Agarwal et al. [J. Stat. Phys. 176, 1463 (2019)0022-471510.1007/s10955-019-02349-6]. For this model our method also allows us to obtain the equilibrium two-time correlations and their dynamical scaling forms both in the bulk and at the edge. Our predictions at the edge are reminiscent of a recent result in the mathematics literature in Gorin and Kleptsyn [arXiv:2009.02006 (2023)] on the (passive) DBM. That result can be recovered by the present methods and also, as we show, using the stochastic Airy operator. Finally, our analytical predictions are confirmed by precise numerical simulations in a wide range of parameters.

Exact extreme, order, and sum statistics in a class of strongly correlated systems.

Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations. We consider a set of N independent and identically distributed random variables {X_{1},X_{2},...,X_{N}} whose common distribution has a parameter Y (or a set of parameters) which itself is random with its own distribution. For a fixed value of this parameter Y, the X_{i} variables are independent and we call them conditionally independent and identically distributed. However, once integrated over the distribution of the parameter Y, the X_{i} variables get strongly correlated yet retain a solvable structure for various observables, such as for the sum and the extremes of X_{i}^{'}s. This provides a simple procedure to generate a class of solvable strongly correlated systems. We illustrate how this procedure works via three physical examples where N particles on a line perform independent (i) Brownian motions, (ii) ballistic motions with random initial velocities, and (iii) Lévy flights, but they get strongly correlated via simultaneous resetting to the origin. Our results are verified in numerical simulations. This procedure can be used to generate an endless variety of solvable strongly correlated systems.

Optimizing the random search of a finite-lived target by a Lévy flight.

In many random search processes of interest in chemistry, biology, or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk, or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a nontrivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a nonconventional tricritical point. These results pave the way to the optimization of search processes under time constraints.

Anomalous fluctuations of currents in Sinai-type random chains with strongly correlated disorder.

We study properties of a random walk in a generalized Sinai model, in which a quenched random potential is a trajectory of a fractional Brownian motion with arbitrary Hurst parameter H, 0