In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.

In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

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.

We study a system of N non-interacting spin-less fermions trapped in a confining potential, in arbitrary dimensions d and arbitrary temperature T . The presence of the confining trap breaks the translational invariance and introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so called "bulk regime"), where the fermions do not feel the curvature of the trap, physical properties of the fermions have traditionally been understood using the Local Density (or Thomas Fermi) Approximation. However, these approximations drastically fail near the edge where the density vanishes and thermal and quantum fluctuations are thus enhanced. The main goal of this paper is to show that, even near the edge, novel universal properties emerge, independently of the details of the shape of the confining potential. We present a unified framework to investigate both the bulk and the edge properties of the fermions. We show that for large N , these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any n-point correlation function, including the average density profile, can be expressed as an n × n determinant whose entry is called the kernel, a central object for such processes. Near the edge, we derive the large N scaling form of the kernels, parametrized by d and T . In d = 1 and T = 0, this reduces to the so called Airy kernel, that appears in the Gaussian Unitary Ensemble (GUE) of random matrix theory. In d = 1 and T > 0 we show a remarkable connection between our kernel and the one appearing in the 1 + 1dimensional Kardar-Parisi-Zhang equation at finite time. Consequently our result provides a finite T generalization of the Tracy-Widom distribution, that describes the fluctuations of the rightmost fermion at T = 0. In d > 1 and T ≥ 0, while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the 1d Airy kernel at T = 0 obtained in random matrix theory. Some of our finite temperature results should be testable in present-day cold atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge. arXiv:1609.04366v1 [cond-mat.stat-mech] 14 Sep 2016 B. Large time/low temperature expansions for KPZ/fermions around the Tracy Widom distribution 52 References 53

We study the fluctuations of the largest eigenvalue λ max of N ×N random matrices in the limit of large N . The main focus is on Gaussian β-ensembles, including in particular the Gaussian orthogonal (β = 1), unitary (β = 2) and symplectic (β = 4) ensembles. The probability density function (PDF) of λ max consists, for large N , of a central part described by Tracy-Widom distributions flanked, on both sides, by two large deviations tails. While the central part characterizes the typical fluctuations of λ max -of order O(N −2/3 ) -, the large deviations tails are instead associated to extremely rare fluctuations -of order O(1). Here we review some recent developments in the theory of these extremely rare events using a Coulomb gas approach. We discuss in particular the third-order phase transition which separates the left tail from the right tail, a transition akin to the so-called Gross-Witten-Wadia phase transition found in 2-d lattice quantum chromodynamics. We also discuss the occurrence of similar third-order transitions in various physical problems, including non-intersecting Brownian motions, conductance fluctuations in mesoscopic physics and entanglement in a bipartite system.

A stochastic process, when subject to resetting to its initial condition at a constant rate, generically reaches a nonequilibrium steady state. We study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems. We show that as time progresses an inner core region around the resetting point reaches the steady state, while the region outside the core is still transient. The boundaries of the core region grow with time as power laws at late times with new exponents. Alternatively, at a fixed spatial point, the system undergoes a dynamical transition from the transient to the steady state at a characteristic space-dependent timescale t(*)(x). We calculate analytically in several examples the large deviation function associated with this spatiotemporal fluctuation and show that, generically, it has a second-order discontinuity at a pair of critical points characterizing the edges of the inner core. These singularities act as separatrices between typical and atypical trajectories. Our results are verified in the numerical simulations of several models, such as simple diffusion and fluctuating one-dimensional interfaces.

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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.PACS numbers: 05.70.Ln, Fluctuating interfaces are paradigmatic nonequilibrium systems commonly encountered in diverse physical situations, e.g., propagation of flame fronts in paper sheets, fluid flow in porous media, vortex lines in disordered superconductors, liquid-crystal turbulence, and many others. Study of such interfaces has many practical applications in the field of molecular beam epitaxy, crystal growth, fluctuating steps on metals, growing bacterial colonies or tumor, etc [1][2][3]. A well-studied model of fluctuating interfaces is the Kardar-Parisi-Zhang (KPZ) equation [4], which is believed to describe a wide class of such out-of-equilibrium growth processes.Earlier studies of the KPZ equation focused on the universal behavior of the interface roughness, a property which, for instance, in 1 + 1 space-time dimensions is characterized by the interface width W (L, t) at time t for an interface growing over a substrate of linear size L. It is then known that W (L, t) grows algebraically with time as t β for times t ≪ L z where z is the dynamic exponent, and saturates for times t ≫ L z to a L-dependent value ∼ L α . Here, α is the roughness exponent, while β = α/z is the growth exponent. For the KPZ universality class in 1+1 dimensions, one has α = 1/2 and z = 3/2 [1-3]. More recently, in this case, significant theoretical progress has shown that in the growing regime (i.e., for times t ≪ L z ), the notion of universality extends beyond the interface width and holds even for the full interface height distribution at late times [5][6][7][8][9]. For example, the scaled cumulative distribution of the interface height fluctuations in a curved (respectively, flat) geometry is described by the so-called Tracy-Widom (TW) distribution