arXiv subjects

Satya N. Majumdar

Publications and source records attributed to Satya N. Majumdar.

At least 217 records · Page 12Linked to original sources

arXiv2016

Record statistics for random walk bridges

We investigate the statistics of records in a random sequence $\{x_B(0)=0,x_B(1),\cdots, x_B(n)=x_B(0)=0\}$ of $n$ time steps. The sequence $x_B(k)$'s represents the position at step $k$ of a random walk `bridge' of $n$ steps that starts and ends at the origin. At each step, the increment of the position is a random jump drawn from a specified symmetric distribution. We study the statistics of records and record ages for such a bridge sequence, for different jump distributions. In absence of the bridge condition, i.e., for a free random walk sequence, the statistics of the number and ages of records exhibits a `strong' universality for all $n$, i.e., they are completely independent of the jump distribution as long as the distribution is continuous. We show that the presence of the bridge constraint destroys this strong `all $n$' universality. Nevertheless a `weaker' universality still remains for large $n$, where we show that the record statistics depends on the jump distributions only through a single parameter $0<μ\le 2$, known as the Lévy index of the walk, but are insensitive to the other details of the jump distribution. We derive the most general results (for arbitrary jump distributions) wherever possible and also present two exactly solvable cases. We present numerical simulations that verify our analytical results.

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arXiv2016

Universal ground state properties of free fermions in a $d$-dimensional trap

The ground state properties of $N$ spinless free fermions in a $d$-dimensional confining potential are studied. We find that any $n$-point correlation function has a simple determinantal structure that allows us to compute several properties exactly for large $N$. We show that the average density has a finite support with an edge, and near this edge the density exhibits a universal (valid for a wide class of potentials) scaling behavior for large $N$. The associated edge scaling function is computed exactly and generalizes to any $d$ the edge electron gas result of Kohn and Mattsson in $d=3$ [Phys. Rev. Lett. 81, 3487 (1998)]. In addition, we calculate the kernel (that characterizes any $n$-point correlation function) for large $N$ and show that, when appropriately scaled, it depends only on dimension $d$, but has otherwise universal scaling forms, at the edges. The edge kernel, for higher $d$, generalizes the Airy kernel in one dimension, well known from random matrix theory.

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arXiv2015

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 $β= 1/T$. The average density of fermions $ρ_N(x,T)$ at position $x$ is derived. For $N \gg 1$ and $β\sim {\cal O}(1/N)$, $ρ_N(x,T)$ is given by a scaling function interpolating between a Gaussian at high temperature, for $β\ll 1/N$, and the Wigner semi-circle law at low temperature, for $β\gg N^{-1}$. In the latter regime, we unveil a scaling limit, for $β{\hbar ω}= b N^{-1/3}$, where the fluctuations close to the edge of the support, at $x \sim \pm \sqrt{2\hbar N/(mω)}$, are described by a limiting kernel $K^{\rm ff}_b(s,s')$ 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 $K^{\rm ff}_b(s,s')$ arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time $t$, with the correspondence $t= b^3$.

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arXiv2015

Exact gap statistics for the random average process on a ring with a tracer

We study statistics of the gaps in Random Average Process (RAP) on a ring with particles hopping symmetrically, except one tracer particle which could be driven. These particles hop either to the left or to the right by a random fraction $η$ of the space available till next particle in the respective directions. The random fraction $η\in [0,~1)$ is chosen from a distribution $R(η)$. For non-driven tracer, when $R(η)$ satisfies a necessary and sufficient condition, the stationary joint distribution of the gaps between successive particles takes an universal form that is factorized except for a global constraint. Some interesting explicit forms of $R(η)$ are found which satisfy this condition. In case of driven tracer, the system reaches a current-carrying steady state where such factorization does not hold. Analytical progress has been made in the thermodynamic limit, where we computed the single site mass distribution inside the bulk. We have also computed the two point gap-gap correlation exactly in that limit. Numerical simulations support our analytical results.

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arXiv2015

Random walk with random resetting to the maximum

We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability $r$, and with probability $(1-r)$, it undergoes symmetric random walk, i.e., it hops to one of its neighboring sites, with equal probability $(1-r)/2$. For $r=0$, it reduces to a standard random walk whose typical distance grows as $\sqrt{n}$ for large $n$. In presence of a nonzero resetting rate $0<r\le 1$, we find that both the average maximum and the average position grow ballistically for large $n$, with a common speed $v(r)$. Moreover, the fluctuations around their respective averages grow diffusively, again with the same diffusion coefficient $D(r)$. We compute $v(r)$ and $D(r)$ explicitly. We also show that the probability distribution of the difference between the maximum and the location of the walker, becomes stationary as $n\to \infty$. However, the approach to this stationary distribution is accompanied by a dynamical phase transition, characterized by a weakly singular large deviation function. We also show that $r=0$ is a special `critical' point, for which the growth laws are different from the $r\to 0$ case and we calculate the exact crossover functions that interpolate between the critical $(r=0)$ and the off-critical $(r\to 0)$ behavior for finite but large $n$.

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arXiv2015

On the Gap and Time Interval between the First Two Maxima of Long Continuous Time Random Walks

We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval $T$ where at each time step the walker waits a random time $τ$, before performing a jump drawn from a symmetric continuous probability distribution function (PDF) $f(η)$, of Lévy index $0 < μ\leq 2$. Our study includes the case where the waiting time PDF $Ψ(τ)$ has a power law tail, $Ψ(τ) \propto τ^{-1 - γ}$, with $0< γ< 1$, such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if $γ< μ/2$ (and super-diffusive if $γ> μ/2$). We investigate the joint PDF of the gap $g$ between the first two highest positions of the CTRW and the time $t$ separating these two maxima. We show that this PDF reaches a stationary limiting joint distribution $p(g,t)$ in the limit of long CTRW, $T \to \infty$. Our exact analytical results show a very rich behavior of this joint PDF in the $(γ, μ)$ plane, which we study in great detail. Our main results are verified by numerical simulations. This work provides a non trivial extension to CTRWs of the recent study in the discrete time setting by Majumdar et al. (J. Stat. Mech. P09013, 2014).

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arXiv2015

Top eigenvalue of a random matrix: large deviations and third order phase transition

We study the fluctuations of the largest eigenvalue $λ_{\max}$ of $N \times N$ 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 ${\cal O}(N^{-2/3})$ --, the large deviations tails are instead associated to extremely rare fluctuations -- of order ${\cal O}(1)$. 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.

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arXiv2015

Dynamical transition in the temporal relaxation of stochastic processes under resetting

A stochastic process, when subject to resetting to its initial condition at a constant rate, generically reaches a non-equilibrium 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. 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 spatio-temporal fluctuation and show that generically it has a second order discontinuity at a pair of critical points characterizing the edges of the inner core. Our results are verified in the numerical simulations of several models, such as simple diffusion and fluctuating one-dimensional interfaces.

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arXiv2015

Extreme value statistics of correlated random variables

Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of uncorrelated variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this note, we will first review the classical EVS for uncorrelated variables and discuss few examples of correlated variables where analytical progress can be made.

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arXiv2015

Effective Langevin equations for constrained stochastic processes

We propose a novel stochastic method to exactly generate Brownian paths conditioned to start at an initial point and end at a given final point during a fixed time $t_{f}$. These paths are weighted with a probability given by the overdamped Langevin dynamics. We show how these paths can be exactly generated by a local stochastic differential equation. The method is illustrated on the generation of Brownian bridges, Brownian meanders, Brownian excursions and constrained Ornstein-Uehlenbeck processes. In addition, we show how to solve this equation in the case of a general force acting on the particle. As an example, we show how to generate constrained path joining the two minima of a double-well. Our method allows to generate statistically independent paths, and is computationally very efficient.

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arXiv2015

The average number of distinct sites visited by a random walker on random graphs

We study the linear large $n$ behavior of the average number of distinct sites $S(n)$ visited by a random walker after $n$ steps on a large random graph. An expression for the graph topology dependent prefactor $B$ in $S(n) = Bn$ is proposed. We use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message-passing equations to calculate its value. Numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs. Scaling with system size and average graph connectivity are also analysed.

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arXiv2015

Spatial Extent of Branching Brownian Motion

We study the one dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost ($X_{\max}\geq 0$) and leftmost ($X_{\min} \leq 0$) visited sites up to time $t$. At each time step the existing particles in the system either diffuse (with diffusion constant $D$), die (with rate $a$) or split into two particles (with rate $b$). We focus on the regime $b \leq a$ where these two extreme values $X_{\max}$ and $X_{\min}$ are strongly correlated. We show that at large time $t$, the joint probability distribution function (PDF) of the two extreme points becomes stationary $P(X,Y,t \to \infty) \to p(X,Y)$. Our exact results for $p(X,Y)$ demonstrate that the correlation between $X_{\max}$ and $X_{\min}$ is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF $p(ζ)$ of the (dimensionless) span $ζ= {(X_{\max} - X_{\min})}/{\sqrt{D/b}}$, which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior ${p}(ζ) \sim \frac{1}{2} \left(1 + Δ\right) ζ$ for small spans, with $Δ= \left(\frac{a}{b} -1\right)$. In the critical case ($Δ= 0$) this distribution has a non-trivial power law tail ${p}(ζ) \sim 8 π\sqrt{3} /ζ^3$ for large spans. On the other hand, in the subcritical case ($Δ> 0$), we show that the span distribution decays exponentially as ${p}(ζ) \sim (A^2/2) ζ\exp \left(- \sqrtΔ~ζ\right)$ for large spans, where $A$ is a non-trivial function of $Δ$ which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between $X_{\max}$ and $X_{\min}$. Finally we verify our results via direct Monte Carlo simulations.

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arXiv2015

On certain functionals of the maximum of Brownian motion and their applications

We consider a Brownian motion (BM) $x(τ)$ and its maximal value $x_{\max} = \max_{0 \leq τ\leq t} x(τ)$ on a fixed time interval $[0,t]$. We study functionals of the maximum of the BM, of the form ${\cal O}_{\max}(t)=\int_0^t\, V(x_{\max} - x(τ)) {\rm d} τ$ where $V(x)$ can be any arbitrary function and develop various analytical tools to compute their statistical properties. These tools rely in particular on (i) a "counting paths" method and (ii) a path-integral approach. In particular, we focus on the case where $V(x) = δ(x-r)$, with $r$ a real parameter, which is relevant to study the density of near-extreme values of the BM (the so called density of states), $ρ(r,t)$, which is the local time of the BM spent at given distance $r$ from the maximum. We also provide a thorough analysis of the family of functionals ${T}_α(t)=\int_0^t (x_{\max} - x(τ))^α\, {\rm d}τ$, corresponding to $V(x) = x^α$, with $α$ real. As $α$ is varied, $T_α(t)$ interpolates between different interesting observables. For instance, for $α=1$, $T_{α= 1}(t)$ is a random variable of the "area", or "Airy", type while for $α=-1/2$ it corresponds to the maximum time spent by a ballistic particle through a Brownian random potential. On the other hand, for $α= -1$, it corresponds to the cost of the optimal algorithm to find the maximum of a discrete random walk, proposed by Odlyzko. We revisit here, using tools of theoretical physics, the statistical properties of this algorithm which had been studied before using probabilistic methods. Finally, we extend our methods to constrained BM, including in particular the Brownian bridge, i.e., the Brownian motion starting and ending at the origin.

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arXiv2015

Statistics of the longest interval in renewal processes

We consider renewal processes where events, which can for instance be the zero crossings of a stochastic process, occur at random epochs of time. The intervals of time between events, $τ_{1},τ_{2},...$, are independent and identically distributed (i.i.d.) random variables with a common density $ρ(τ)$. Fixing the total observation time to $t$ induces a global constraint on the sum of these random intervals, which accordingly become interdependent. Here we focus on the largest interval among such a sequence on the fixed time interval $(0,t)$. Depending on how the last interval is treated, we consider three different situations, indexed by $α=$ I, II and III. We investigate the distribution of the longest interval $\ell^α_{\max}(t)$ and the probability $Q^α(t)$ that the last interval is the longest one. We show that if $ρ(τ)$ decays faster than $1/τ^2$ for large $τ$, then the full statistics of $\ell^α_{\max}(t)$ is given, in the large $t$ limit, by the standard theory of extreme value statistics for i.i.d. random variables, showing in particular that the global constraint on the intervals $τ_i$ does not play any role at large times in this case. However, if $ρ(τ)$ exhibits heavy tails, $ρ(τ)\simτ^{-1-θ}$ for large $τ$, with index $0 <θ<1$, we show that the fluctuations of $\ell^α_{\max}(t)/t$ are governed, in the large $t$ limit, by a stationary universal distribution which depends on both $θ$ and $α$, which we compute exactly. On the other hand, $Q^α(t)$ is generically different from its counterpart for i.i.d. variables (both for narrow or heavy tailed distributions $ρ(τ)$). In particular, in the case $0<θ<1$, the large $t$ behaviour of $Q^α(t)$ gives rise to universal constants (depending also on both $θ$ and $α$) which we compute exactly.

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arXiv2015

Survival probability of a Brownian motion in a planar wedge of arbitrary angle

We study the survival probability and the first-passage time distribution for a Brownian motion in a planar wedge with infinite absorbing edges. We generalize existing results obtained for wedge angles of the form $π/n$ with $n$ a positive integer to arbitrary angles, which in particular cover the case of obtuse angles. We give explicit and simple expressions of the survival probability and the first-passage time distribution in which the difference between an arbitrary angle and a submultiple of $π$ is contained in three additional terms. As an application, we obtain the short time development of the survival probability in a wedge of arbitrary angle.

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arXiv2015

Convex Hulls of Random Walks: Large-Deviation Properties

We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter and the mean area have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P(A) and P(L). In this paper, we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P(A) and P(L) are as small as 10^{-300}. We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T, a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/\sqrt{T}, respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

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arXiv2014

Random matrices and entanglement entropy of trapped Fermi gases

We exploit and clarify the use of random matrix theory for the calculation of the entanglement entropy of free Fermi gases. We apply this method to obtain analytic predictions for Renyi entanglement entropies of a one-dimensional gas trapped by a harmonic potential in all the relevant scaling regimes. We confirm our findings with accurate numerical calculations obtained by means of an ingenious discretisation of the reduced correlation matrix.

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arXiv2014

Winding statistics of a Brownian particle on a ring

We consider a Brownian particle moving on a ring. We study the probability distributions of the total number of turns and the net number of counter-clockwise turns the particle makes till time t. Using a method based on the renewal properties of Brownian walker, we find exact analytical expressions of these distributions. This method serves as an alternative to the standard path integral techniques which are not always easily adaptable for certain observables. For large t, we show that these distributions have Gaussian scaling forms. We also compute large deviation functions associated to these distributions characterizing atypically large fluctuations. We provide numerical simulations in support of our analytical results.

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