arXiv subjects

Satya N. Majumdar

Publications and source records attributed to Satya N. Majumdar.

At least 289 records · Page 16Linked to original sources

arXiv2010

Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x_0=0, x_1,x_2.... x_n} up to n steps where x_i represents the position at step i of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Levy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek-Spitzer formula and the associated Sparre Andersen theorem.

cond-mat.stat-mech↗

arXiv2010

Probability distributions of Linear Statistics in Chaotic Cavities and associated phase transitions

We establish large deviation formulas for linear statistics on the $N$ transmission eigenvalues $\{T_i\}$ of a chaotic cavity, in the framework of Random Matrix Theory. Given any linear statistics of interest $A=\sum_{i=1}^N a(T_i)$, the probability distribution $\mathcal{P}_A(A,N)$ of $A$ generically satisfies the large deviation formula $\lim_{N\to\infty}[-2\log\mathcal{P}_A(Nx,N)/βN^2]=Ψ_A(x)$, where $Ψ_A(x)$ is a rate function that we compute explicitly in many cases (conductance, shot noise, moments) and $β$ corresponds to different symmetry classes. Using these large deviation expressions, it is possible to recover easily known results and to produce new formulas, such as a closed form expression for $v(n)=\lim_{N\to\infty}\mathrm{var}(\mathcal{T}_n)$ (where $\mathcal{T}_n=\sum_{i}T_i^n$) for arbitrary integer $n$. The universal limit $v^\star=\lim_{n\to\infty} v(n)=1/2πβ$ is also computed exactly. The distributions display a central Gaussian region flanked on both sides by non-Gaussian tails. At the junction of the two regimes, weakly non-analytical points appear, a direct consequence of phase transitions in an associated Coulomb gas problem. Numerical checks are also provided, which are in full agreement with our asymptotic results in both real and Laplace space even for moderately small $N$. Part of the results have been announced in [P. Vivo, S.N. Majumdar and O. Bohigas, {\it Phys. Rev. Lett.} {\bf 101}, 216809 (2008)].

cond-mat.mes-hall↗

arXiv2009

Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].

cond-mat.stat-mech↗

arXiv2009

Hitting probability for anomalous diffusion processes

We present the universal features of the hitting probability $Q(x,L)$, the probability that a generic stochastic process starting at $x$ and evolving in a box $[0,L]$ hits the upper boundary $L$ before hitting the lower boundary at 0. For a generic self-affine process (describing, for instance, the polymer translocation through a nanopore) we show that $Q(x,L)=Q(x/L)$ and the scaling function $Q(z)\sim z^ϕ$ as $z\to 0$ with $ϕ=θ/H$ where $H$ and $θ$ are respectively the Hurst exponent and the persistence exponent of the process. This result is verified in several exact calculations including when the process represents the position of a particle diffusing in a disordered potential. We also provide numerical supports for our analytical results.

cond-mat.stat-mech↗

arXiv2009

Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State

Using a Coulomb gas method, we compute analytically the probability distribution of the Renyi entropies (a standard measure of entanglement) for a random pure state of a large bipartite quantum system. We show that, for any order q>1 of the Renyi entropy, there are two critical values at which the entropy's probability distribution changes shape. These critical points correspond to two different transitions in the corresponding charge density of the Coulomb gas: the disappearance of an integrable singularity at the origin and the detachement of a single-charge drop from the continuum sea of all the other charges. These transitions respectively control the left and right tails of the entropy's probability distribution, as verified also by Monte Carlo numerical simulations of the Coulomb gas equilibrium dynamics.

cond-mat.stat-mech↗

arXiv2009

The Index Distribution of Gaussian Random Matrices

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N_{+}) of a random NxN matrix belonging to Gaussian orthogonal (β=1), unitary (β=2) or symplectic (β=4) ensembles. The distribution of the fraction of positive eigenvalues c=N_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-βN^2 Φ(c)] where the rate function Φ(c), symmetric around c=1/2 and universal (independent of $β$), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.

cond-mat.stat-mech↗

arXiv2009

Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology

We compute exactly the mean perimeter and area of the convex hull of N independent planar Brownian paths each of duration T, both for open and closed paths. We show that the mean perimeter < L_N > = α_N, \sqrt{T} and the mean area = β_N T for all T. The prefactors α_N and β_N, computed exactly for all N, increase very slowly (logarithmically) with increasing N. This slow growth is a consequence of extreme value statistics and has interesting implication in ecological context in estimating the home range of a herd of animals with population size N.

cond-mat.stat-mech↗

arXiv2009

Non-intersecting Brownian Interfaces and Wishart Random Matrices

We study a system of $N$ non-intersecting $(1+1)$-dimensional fluctuating elastic interfaces (`vicious bridges') at thermal equilibrium, each subject to periodic boundary condition in the longitudinal direction and in presence of a substrate that induces an external confining potential for each interface. We show that, for a large system and with an appropriate choice of the external confining potential, the joint distribution of the heights of the $N$ non-intersecting interfaces at a fixed point on the substrate can be mapped to the joint distribution of the eigenvalues of a Wishart matrix of size $N$ with complex entries (Dyson index $β=2$), thus providing a physical realization of the Wishart matrix. Exploiting this analogy to random matrix, we calculate analytically (i) the average density of states of the interfaces (ii) the height distribution of the uppermost and lowermost interfaces (extrema) and (iii) the asymptotic (large $N$) distribution of the center of mass of the interfaces. In the last case, we show that the probability density of the center of mass has an essential singularity around its peak which is shown to be a direct consequence of a phase transition in an associated Coulomb gas problem.

cond-mat.stat-mech↗

arXiv2009

Capture of particles undergoing discrete random walks

It is shown that particles undergoing discrete-time jumps in 3D, starting at a distance r0 from the center of an adsorbing sphere of radius R, are captured with probability (R - c sigma)/r0 for r0 much greater than R, where c is related to the Fourier transform of the scaled jump distribution and sigma is the distribution's root-mean square jump length. For particles starting on the surface of the sphere, the asymptotic survival probability is non-zero (in contrast to the case of Brownian diffusion) and has a universal behavior sigma/(R sqrt(6)) depending only upon sigma/R. These results have applications to computer simulations of reaction and aggregation.

cond-mat.dis-nn↗

arXiv2009

Real-space Condensation in Stochastic Mass Transport Models

The phenomenon of real-space condensation is encountered in a variety of situations such as aggregation and fragmentation processes, granular clustering, phase separation, traffic and networks. Unlike traditional Bose-Einstein condensation in the momentum space, a condensate in these systems forms in real space, e.g., upon increasing the density beyond a critical value a macroscopically large mass/cluster may form at a single site on a lattice. In this brief review, I discuss some recent developments in understanding the physical and mathematical mechanism behind this real-space condensation in a class of simple stochastic mass transport models.

cond-mat.stat-mech↗

arXiv2009

The longest excursion of stochastic processes in nonequilibrium systems

We consider the excursions, i.e. the intervals between consecutive zeros, of stochastic processes that arise in a variety of nonequilibrium systems and study the temporal growth of the longest one l_{\max}(t) up to time t. For smooth processes, we find a universal linear growth < l_{\max}(t) > \simeq Q_{\infty} t with a model dependent amplitude Q_\infty. In contrast, for non-smooth processes with a persistence exponent θ, we show that < l_{\max}(t) > has a linear growth if θ< θ_c while < l_{\max}(t) > \sim t^{1-ψ} if θ> θ_c. The amplitude Q_{\infty} and the exponent ψare novel quantities associated to nonequilibrium dynamics. These behaviors are obtained by exact analytical calculations for renewal and multiplicative processes and numerical simulations for other systems such as the coarsening dynamics in Ising model as well as the diffusion equation with random initial conditions.

cond-mat.stat-mech↗

arXiv2009

Asymptotic behavior of self-affine processes in semi-infinite domains

We propose to model the stochastic dynamics of a polymer passing through a pore (translocation) by means of a fractional Brownian motion, and study its behavior in presence of an absorbing boundary. Based on scaling arguments and numerical simulations, we present a conjecture that provides a link between the persistence exponent $θ$ and the Hurst exponent $H$ of the process, thus sheding light on the spatial and temporal features of translocation. Furthermore, we show that this conjecture applies more generally to a broad class of self affine processes undergoing anomalous diffusion in bounded domains, and we discuss some significant examples.

cond-mat.stat-mech↗

arXiv2009

Condensation of the roots of real random polynomials on the real axis

We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance = e^{-k^α}, indexed by a real α\geq 0. We compute exactly the mean number of real roots for large n. As αis varied, one finds three different phases. First, for 0 \leq α< 1, one finds that \sim (\frac{2}π) \log{n}. For 1 < α< 2, there is an intermediate phase where < N_n > grows algebraically with a continuously varying exponent, < N_n > \sim \frac{2}π \sqrt{\frac{α-1}α} n^{α/2}. And finally for α> 2, one finds a third phase where \sim n. This family of real random polynomials thus exhibits a condensation of their roots on the real line in the sense that, for large n, a finite fraction of their roots /n are real. This condensation occurs via a localization of the real roots around the values \pm \exp{[\fracα{2}(k+{1/2})^{α-1} ]}, 1 \ll k \leq n.

math-ph↗

arXiv2008

Bethe Ansatz in the Bernoulli Matching Model of Random Sequence Alignment

For the Bernoulli Matching model of sequence alignment problem we apply the Bethe ansatz technique via an exact mapping to the 5--vertex model on a square lattice. Considering the terrace--like representation of the sequence alignment problem, we reproduce by the Bethe ansatz the results for the averaged length of the Longest Common Subsequence in Bernoulli approximation. In addition, we compute the average number of nucleation centers of the terraces.

cond-mat.stat-mech↗

arXiv2008

Exact distribution of the maximal height of p vicious walkers

Using path integral techniques, we compute exactly the distribution of the maximal height H_p of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions (p-watermelons with a wall) and bridges (p-watermelons without a wall), for all integer p\ge 1. For large p, we show that < H_p > \sim \sqrt{2p} (excursions) whereas < H_p > \sim \sqrt{p} (bridges). Our exact results prove that previous numerical experiments only measured the pre-asymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.

cond-mat.stat-mech↗

arXiv2008

Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices

We present a simple Coulomb gas method to calculate analytically the probability of rare events where the maximum eigenvalue of a random matrix is much larger than its typical value. The large deviation function that characterizes this probability is computed explicitly for Wishart and Gaussian ensembles. The method is quite general and applies to other related problems, e.g. the joint large deviation function for large fluctuations of top eigenvalues. Our results are relevant to widely employed data compression techniques, namely the principal components analysis. Analytical predictions are verified by extensive numerical simulations.

cond-mat.stat-mech↗

arXiv2008

Statistical Properties of the Final State in One-dimensional Ballistic Aggregation

We investigate the long time behaviour of the one-dimensional ballistic aggregation model that represents a sticky gas of N particles with random initial positions and velocities, moving deterministically, and forming aggregates when they collide. We obtain a closed formula for the stationary measure of the system which allows us to analyze some remarkable features of the final `fan' state. In particular, we identify universal properties which are independent of the initial position and velocity distributions of the particles. We study cluster distributions and derive exact results for extreme value statistics (because of correlations these distributions do not belong to the Gumbel-Frechet-Weibull universality classes). We also derive the energy distribution in the final state. This model generates dynamically many different scales and can be viewed as one of the simplest exactly solvable model of N-body dissipative dynamics.

cond-mat.stat-mech↗

arXiv2008

Optimal Time to Sell a Stock in Black-Scholes Model: Comment on "Thou shall buy and hold", by A. Shiryaev, Z. Xu and X.Y. Zhou

We reconsider the problem of optimal time to sell a stock studied recently by Shiryaev, Xu and Zhou using path integral methods. This method allows us to confirm the results obtained by these authors and extend them to a parameter region inaccessible to the method used by Shiryaev et. al. We also obtain the full distribution of the time t_m at which the maximum of the price is reached for arbitrary values of the drift.

q-fin.PR↗