arXiv subjects

Satya N. Majumdar

Publications and source records attributed to Satya N. Majumdar.

At least 253 records · Page 14Linked to original sources

arXiv2013

Near-extreme statistics of Brownian motion

We study the statistics of near-extreme events of Brownian motion (BM) on the time interval [0,t]. We focus on the density of states (DOS) near the maximum ρ(r,t) which is the amount of time spent by the process at a distance r from the maximum. We develop a path integral approach to study functionals of the maximum of BM, which allows us to study the full probability density function (PDF) of ρ(r,t) and obtain an explicit expression for the moments, \langle [ρ(r,t)]^k \rangle, for arbitrary integer k. We also study near-extremes of constrained BM, like the Brownian bridge. Finally we also present numerical simulations to check our analytical results.

cond-mat.stat-mech↗

arXiv2013

Exact record and order statistics of random walks via first-passage ideas

While records and order statistics of independent and identically distributed (i.i.d.) random variables X_1, ..., X_N are fully understood, much less is known for strongly correlated random variables, which is often the situation encountered in statistical physics. Recently, it was shown, in a series of works, that one-dimensional random walk (RW) is an interesting laboratory where the influence of strong correlations on records and order statistics can be studied in detail. We review here recent exact results which have been obtained for these questions about RW, using techniques borrowed from the study of first-passage problems. We also present a brief review of the well known (and not so well known) results for records and order statistics of i.i.d. variables.

cond-mat.stat-mech↗

arXiv2013

Persistence and First-Passage Properties in Non-equilibrium Systems

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.

cond-mat.stat-mech↗

arXiv2013

Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks

We investigate the statistics of the gap, G_n, between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration, L_n, which separates the occurrence of these two extremal positions. The distribution of the jumps η_i's of the RW, f(η), is symmetric and its Fourier transform has the small k behavior 1-\hat{f}(k)\sim| k|^μwith 0 < μ\leq 2. We compute the joint probability density function (pdf) P_n(g,l) of G_n and L_n and show that, when n \to \infty, it approaches a limiting pdf p(g,l). The corresponding marginal pdf of the gap, p_{\rm gap}(g), is found to behave like p_{\rm gap}(g) \sim g^{-1 - μ} for g \gg 1 and 0<μ< 2. We show that the limiting marginal distribution of L_n, p_{\rm time}(l), has an algebraic tail p_{\rm time}(l) \sim l^{-γ(μ)} for l \gg 1 with γ(1<μ\leq 2) = 1 + 1/μ, and γ(0<μ<1) = 2. For l, g \gg 1 with fixed l g^{-μ}, p(g,l) takes the scaling form p(g,l) \sim g^{-1-2μ} \tilde p_μ(l g^{-μ}) where \tilde p_μ(y) is a (μ-dependent) scaling function. We also present numerical simulations which verify our analytic results.

cond-mat.stat-mech↗

arXiv2013

Spatial extent of an outbreak in animal epidemics

Characterizing the spatial extent of epidemics at the outbreak stage is key to controlling the evolution of the disease. At the outbreak, the number of infected individuals is typically small, so that fluctuations around their average are important: then, it is commonly assumed that the susceptible-infected-recovered (SIR) mechanism can be described by a stochastic birth-death process of Galton-Watson type. The displacements of the infected individuals can be modelled by resorting to Brownian motion, which is applicable when long-range movements and complex network interactions can be safely neglected, as in case of animal epidemics. In this context, the spatial extent of an epidemic can be assessed by computing the convex hull enclosing the infected individuals at a given time. We derive the exact evolution equations for the mean perimeter and the mean area of the convex hull, and compare them with Monte Carlo simulations.

q-bio.PE↗

arXiv2013

Effect of Partial Absorption on Diffusion with Resetting

The effect of partial absorption on a diffusive particle which stochastically resets its position with a finite rate $r$ is considered. The particle is absorbed by a target at the origin with absorption `velocity' $a$; as the velocity $a$ approaches $\infty$ the absorption property of the target approaches that of a perfectly-absorbing target. The effect of partial absorption on first-passage time problems is studied, in particular, it is shown that the mean time to absorption (MTA) is increased by an additive term proportional to $1/a$. The results are extended to multiparticle systems where independent searchers, initially uniformly distributed with a given density, look for a single immobile target. It is found that the average survival probability $P^{av}$ is modified by a multiplicative factor which is a function of $1/a$, whereas the decay rate of the typical survival probability $P^{typ}$ is decreased by an additive term proportional to $1/a$.

cond-mat.stat-mech↗

arXiv2013

Exact distributions of the number of distinct and common sites visited by N independent random walkers

We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\sqrt{t} \propto 2 \sqrt{\log N} + \widetilde{s}/(2 \sqrt{\log N}) and W_N(t)/\sqrt{t} \propto \widetilde{w}/N where \widetilde{s} and \widetilde{w} are random variables whose probability density functions (pdfs) are computed exactly and are found to be non trivial. We verify our results through direct numerical simulations.

cond-mat.stat-mech↗

arXiv2013

Record-breaking statistics for random walks in the presence of measurement error and noise

We address the question of distance record-setting by a random walker in the presence of measurement error, $δ$, and additive noise, $γ$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $< R_n> \sim n^{1/2}$ for large $n$ for all jump distributions, including Lévy flights, and for all $δ$ and $γ$. In contrast to the universal growth exponent of 1/2, the pace of record setting, measured by the pre-factor of $n^{1/2}$, depends on $δ$ and $γ$. In the absence of noise ($γ=0$), the pre-factor $S(δ)$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $δ$ whereas, in case of perfect measurement $(δ=0)$, the corresponding pre-factor $T(γ)$ increases with $γ$. Our analytical results are supported by extensive numerical simulations and qualitatively similar results are found in two and three dimensions.

physics.data-an↗

arXiv2012

Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics

We study first-passage time problems for a diffusive particle with stochastic resetting with a finite rate $r$. The optimal search time is compared quantitatively with that of an effective equilibrium Langevin process with the same stationary distribution. It is shown that the intermittent, nonequilibrium strategy with non-vanishing resetting rate is more efficient than the equilibrium dynamics. Our results are extended to multiparticle systems where a team of independent searchers, initially uniformly distributed with a given density, looks for a single immobile target. Both the average and the typical survival probability of the target are smaller in the case of nonequilibrium dynamics.

cond-mat.stat-mech↗

arXiv2012

Large deviations of the top eigenvalue of large Cauchy random matrices

We compute analytically the probability density function (pdf) of the largest eigenvalue $λ_{\max}$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($β= 2$), orthogonal ($β=1$) and symplectic ($β=4$) ensembles of such heavy-tailed random matrices. We show that a central non-Gaussian regime for $λ_{\max} \sim \mathcal{O}(N)$ is flanked by large deviation tails on both sides which we compute here exactly for any value of $β$. By matching these tails with the central regime, we obtain the exact leading asymptotic behaviors of the pdf in the central regime, which generalizes the Tracy-Widom distribution known for Gaussian ensembles, both at small and large arguments and for any $β$. Our analytical results are confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2012

Reunion probability of N vicious walkers: typical and large fluctuations for large N

We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyse the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for L\ll O(\sqrt{N}) and L\geq O(\sqrt{N}). A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value L=L_c(N)\sim \sqrt{N}. This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for L \sim L_c(N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [L \gg L_c(N)] and the left [L \ll L_c(N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L \sim L_c(N) and atypical fluctuations in the right tail L \gg L_c(N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices.

cond-mat.stat-mech↗

arXiv2012

Invariant $β$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law

We construct a diffusive matrix model for the $β$-Wishart (or Laguerre) ensemble for general continuous $β\in [0,2]$, which preserves invariance under the orthogonal/unitary group transformation. Scaling the Dyson index $β$ with the largest size $M$ of the data matrix as $β=2c/M$ (with $c$ a fixed positive constant), we obtain a family of spectral densities parametrized by $c$. As $c$ is varied, this density interpolates continuously between the Mar\vcenko-Pastur ($c\to \infty$ limit) and the Gamma law ($c\to 0$ limit). Analyzing the full Stieltjes transform (resolvent) equation, we obtain as a byproduct the correction to the Mar\vcenko-Pastur density in the bulk up to order 1/M for all $β$ and up to order $1/M^2$ for the particular cases $β=1,2$.

cond-mat.stat-mech↗

arXiv2012

Record statistics and persistence for a random walk with a drift

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length ηdrawn from a continuous distribution f(η) which is symmetric around a constant drift c. We focus in particular on the case were f(η) is a symmetric stable law with a Lévy index 0 < μ\leq 2. The record statistics depends crucially on the persistence probability which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter μ. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and μ) of the jump distribution. We compute the asymptotic mean record number after n steps as well as its full distribution P(R,n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < μ\leq 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.

cond-mat.stat-mech↗

arXiv2012

Number of Common Sites Visited by N Random Walkers

We compute analytically the mean number of common sites, W_N(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N-d) plane, there are three distinct regimes for the asymptotic large t growth of W_N(t). These three regimes are separated by two critical lines d=2 and d=d_c(N)=2N/(N-1) in the (N-d) plane. For d<2, W_N(t)\sim t^{d/2} for large t (the N dependence is only in the prefactor). For 2 d_c(N), W_N(t) approaches a constant as t\to \infty. Exactly at the critical dimensions there are logaritmic corrections: for d=2, we get W_N(t)\sim t/[\ln t]^N, while for d=d_c(N), W_N(t)\sim \ln t for large t. Our analytical predictions are verified in numerical simulations.

cond-mat.stat-mech↗

arXiv2012

Survival probability of an immobile target surrounded by mobile traps

We study analytically, in one dimension, the survival probability $P_{s}(t)$ up to time $t$ of an immobile target surrounded by mutually noninteracting traps each performing a continuous-time random walk (CTRW) in continuous space. We consider a general CTRW with symmetric and continuous (but otherwise arbitrary) jump length distribution $f(η)$ and arbitrary waiting time distribution $ψ(τ)$. The traps are initially distributed uniformly in space with density $ρ$. We prove an exact relation, valid for all time $t$, between $P_s(t)$ and the expected maximum $E[M(t)]$ of the trap process up to time $t$, for rather general stochastic motion $x_{\rm trap}(t)$ of each trap. When $x_{\rm trap}(t)$ represents a general CTRW with arbitrary $f(η)$ and $ψ(τ)$, we are able to compute exactly the first two leading terms in the asymptotic behavior of $E[M(t)]$ for large $t$. This allows us subsequently to compute the precise asymptotic behavior, $P_s(t)\sim a\, \exp[-b\, t^θ]$, for large $t$, with exact expressions for the stretching exponent $θ$ and the constants $a$ and $b$ for arbitrary CTRW. By choosing appropriate $f(η)$ and $ψ(τ)$, we recover the previously known results for diffusive and subdiffusive traps. However, our result is more general and includes, in particular, the superdiffusive traps as well as totally anomalous traps.

cond-mat.stat-mech↗

arXiv2012

Can a Lamb Reach a Haven Before Being Eaten by Diffusing Lions?

We study the survival of a single diffusing lamb on the positive half line in the presence of N diffusing lions that all start at the same position L to the right of the lamb and a haven at x=0. If the lamb reaches this haven before meeting any lion, the lamb survives. We investigate the survival probability of the lamb, S_N(x,L), as a function of N and the respective initial positions of the lamb and the lions, x and L. We determine S_N(x,L) analytically for the special cases of N=1 and N--->oo. For large but finite N, we determine the unusual asymptotic form whose leading behavior is S_N(z)\simN^{-z^2}, with z=x/L. Simulations of the capture process very slowly converge to this asymptotic prediction as N reaches 10^{500}.

physics.data-an↗

arXiv2012

Record Statistics for Multiple Random Walks

We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance σ^2 of the jump distribution is finite and (II) when σ^2 is divergent as in the case of Lévy flights with index 0 < μ< 2. In both cases we find that the mean record number grows universally as \sim α_N \sqrt{n} for large n, but with a very different behavior of the amplitude α_N for N > 1 in the two cases. We find that for large N, α_N \approx 2 \sqrt{\log N} independently of σ^2 in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, α_N \approx 4/\sqrtπ, independently of 0<μ<2. For finite σ^2 we argue, and this is confirmed by our numerical simulations, that the full distribution of (R_{n,N}/\sqrt{n} - 2 \sqrt{\log N}) \sqrt{\log N} converges to a Gumbel law as n \to \infty and N \to \infty. In case II, our numerical simulations indicate that the distribution of R_{n,N}/\sqrt{n} converges, for n \to \infty and N \to \infty, to a universal nontrivial distribution, independently of μ. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poors 500 index.

cond-mat.stat-mech↗

arXiv2012

Bose-Einstein Condensation of a Gaussian Random Field in the Thermodynamic Limit

We derive the criterion for the Bose-Einstein condensation (BEC) of a Gaussian field $ϕ$ (real or complex) in the thermodynamic limit. The field is characterized by its covariance function and the control parameter is the intensity $u=\|ϕ\|_2^2/V$, where $V$ is the volume of the box containing the field. We show that for any dimension $d$ (including $d=1$), there is a class of covariance functions for which $ϕ$ exhibits a BEC as $u$ is increased through a critical value $u_c$. In this case, we investigate the probability distribution of the part of $u$ contained in the condensate. We show that depending on the parameters characterizing the covariance function and the dimension $d$, there can be two distinct types of condensate: a Gaussian distributed "normal" condensate with fluctuations scaling as $1/\sqrt{V}$, and a non Gaussian distributed "anomalous" condensate. A detailed analysis of the anomalous condensate is performed for a one-dimensional system ($d=1$). Extending this one-dimensional analysis to exactly the point of transition between normal and anomalous condensations, we find that the condensate at the transition point is still Gaussian distributed but with anomalously large fluctuations scaling as $\sqrt{\ln(L)/L}$, where $L$ is the system length. The conditional spectral density of $ϕ$, knowing $u$, is given for all the regimes (with and without BEC).

cond-mat.stat-mech↗