arXiv subjects

Satya N. Majumdar

Publications and source records attributed to Satya N. Majumdar.

At least 271 records · Page 15Linked to original sources

arXiv2011

Number of relevant directions in Principal Component Analysis and Wishart random matrices

We compute analytically, for large $N$, the probability $\mathcal{P}(N_+,N)$ that a $N\times N$ Wishart random matrix has $N_+$ eigenvalues exceeding a threshold $Nζ$, including its large deviation tails. This probability plays a benchmark role when performing the Principal Component Analysis of a large empirical dataset. We find that $\mathcal{P}(N_+,N)\approx\exp(-βN^2 ψ_ζ(N_+/N))$, where $β$ is the Dyson index of the ensemble and $ψ_ζ(κ)$ is a rate function that we compute explicitly in the full range $0\leq κ\leq 1$ and for any $ζ$. The rate function $ψ_ζ(κ)$ displays a quadratic behavior modulated by a logarithmic singularity close to its minimum $κ^\star(ζ)$. This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance $Δ(N)$ of the number of relevant components is also shown to grow universally (independent of $ζ)$ as $Δ(N)\sim (βπ^2)^{-1}\ln N$ for large $N$.

cond-mat.stat-mech↗

arXiv2011

Universal Order Statistics of Random Walks

We study analytically the order statistics of a time series generated by the successive positions of a symmetric random walk of n steps with step lengths of finite variance σ^2. We show that the statistics of the gap d_{k,n}=M_{k,n} -M_{k+1,n} between the k-th and the (k+1)-th maximum of the time series becomes stationary, i.e, independent of n as n\to \infty and exhibits a rich, universal behavior. The mean stationary gap (in units of σ) exhibits a universal algebraic decay for large k, /σ\sim 1/\sqrt{2πk}, independent of the details of the jump distribution. Moreover, the probability density (pdf) of the stationary gap exhibits scaling, Proba.(d_{k,\infty}=δ)\simeq (\sqrt{k}/σ) P(δ\sqrt{k}/σ), in the scaling regime when δ\sim \simeq σ/\sqrt{2πk}. The scaling function P(x) is universal and has an unexpected power law tail, P(x) \sim x^{-4} for large x. For δ\gg the scaling breaks down and the pdf gets cut-off in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual multi-scaling behavior.

cond-mat.stat-mech↗

arXiv2011

Phase transitions in the distribution of the Andreev conductance of superconductor-metal junctions with multiple transverse modes

We compute analytically the full distribution of Andreev conductance $G_{\mathrm{NS}}$ of a metal-superconductor interface with a large number $N_c$ of transverse modes, using a random matrix approach. The probability distribution $\mathcal{P}(G_{\mathrm{NS}},N_c)$ in the limit of large $N_c$ displays a Gaussian behavior near the average value $ = (2-\sqrt{2}) N_c$ and asymmetric power-law tails in the two limits of very small and very large $G_{\mathrm{NS}}$. In addition, we find a novel third regime sandwiched between the central Gaussian peak and the power law tail for large $G_{\mathrm{NS}}$. Weakly non-analytic points separate these four regimes---these are shown to be consequences of three phase transitions in an associated Coulomb gas problem.

cond-mat.mes-hall↗

arXiv2011

Large deviations of the maximal eigenvalue of random matrices

We present detailed computations of the 'at least finite' terms (three dominant orders) of the free energy in a one-cut matrix model with a hard edge a, in beta-ensembles, with any polynomial potential. beta is a positive number, so not restricted to the standard values beta = 1 (hermitian matrices), beta = 1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This model allows to study the statistic of the maximum eigenvalue of random matrices. We compute the large deviation function to the left of the expected maximum. We specialize our results to the gaussian beta-ensembles and check them numerically. Our method is based on general results and procedures already developed in the literature to solve the Pastur equations (also called "loop equations"). It allows to compute the left tail of the analog of Tracy-Widom laws for any beta, including the constant term.

math-ph↗

arXiv2011

The convex hull for a random acceleration process in two dimensions

We compute exactly the mean perimeter and the mean area of the convex hull of a random acceleration process of duration T in two dimensions. We use an exact mapping that relates, via Cauchy's formulae, the computation of the perimeter and the area of the convex hull of an arbitrary two dimensional stochastic process [x(t); y(t)] to the computation of the extreme value statistics of the associated one dimensional component process x(t). The latter can be computed exactly for the one dimensional random acceleration process even though the process in non-Markovian. Physically, our results are relevant in describing theaverage shape of a semi-flexible ideal polymer chain in two dimensions.

cond-mat.stat-mech↗

arXiv2011

Diffusion with Optimal Resetting

We consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate $r$. We consider several generalisations of the model of M. R. Evans and S. N. Majumdar (2011), Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601: (i) a space dependent resetting rate $r(x)$ ii) resetting to a random position $z$ drawn from a resetting distribution ${\cal P}(z)$ iii) a spatial distribution for the absorbing target $P_T(x)$. As an example of (i) we show that the introduction of a non-resetting window around the initial position can reduce the mean time to absorption provided that the initial position is sufficiently far from the target. We address the problem of optimal resetting, that is, minimising the mean time to absorption for a given target distribution. For an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows.

cond-mat.stat-mech↗

arXiv2011

Long-range steady state density profiles induced by localized drive

We show that the presence of a localized drive in an otherwise diffusive system results in steady-state density and current profiles that decay algebraically to their global average value, away from the drive in two or higher dimensions. An analogy to an electrostatic problem is established, whereby the density profile induced by a driving bond maps onto the electrostatic potential due to an electric dipole located along the bond. The dipole strength is proportional to the drive, and is determined self-consistently by solving the electrostatic problem. The profile resulting from a localized configuration of more than one driving bond can be straightforwardly determined by the superposition principle of electrostatics. This picture is shown to hold even in the presence of exclusion interaction between particles.

cond-mat.stat-mech↗

arXiv2011

Non-intersecting Brownian walkers and Yang-Mills theory on the sphere

We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L=L_c(N)\sim \sqrt{N} in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, is identical to the Tracy-Widom distribution describing the probability distribution of the largest eigenvalue of a random matrix. For the periodic case we obtain the Tracy-Widom distribution corresponding to the GUE random matrices, while for the absorbing and reflecting cases we get the Tracy-Widom distribution corresponding to GOE random matrices. In the absorbing case, the reunion probability is also identified as the maximal height of N non-intersecting Brownian excursions ("watermelons" with a wall) whose distribution in the asymptotic scaling limit is then described by GOE Tracy-Widom law. In addition, large deviation formulas for the maximum height are also computed.

math-ph↗

arXiv2011

Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations ~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with 0 0 (near the absorbing boundary), while R(y) ~ y^gamma exp(-y^2/2) as y -> oo, with phi = 1 - 4 epsilon + O(epsilon^2) and gamma = 1 - 2 epsilon + O(epsilon^2). Our epsilon-expansion result confirms the scaling relation phi = (1-H)/H proposed in Ref. [28]. We verify our findings via numerical simulations for H = 2/3. The tools developed here are versatile, powerful, and adaptable to different situations.

cond-mat.stat-mech↗

arXiv2011

A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix

In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably adapting a method of orthogonal polynomials developed by Gross and Matytsin in the context of Yang-Mills theory in two dimensions, we provide a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. As an added bonus, this method also allows us to compute the precise subleading terms describing the right large deviation tail of the maximal eigenvalue distribution. In the Yang-Mills language, these subleading terms correspond to non-perturbative (in $1/N$ expansion) corrections to the two-dimensional partition function in the so called `weak' coupling regime.

cond-mat.stat-mech↗

arXiv2011

Diffusion with Stochastic Resetting

We study simple diffusion where a particle stochastically resets to its initial position at a constant rate r. A finite resetting rate leads to a nonequilibrium stationary state with non-Gaussian fluctuations for the particle position. We also show that the mean time to find a stationary target by a diffusive searcher is finite and has a minimum value at an optimal resetting rate r^*. Resetting also alters fundamentally the late time decay of the survival probability of a stationary target when there are multiple searchers: while the typical survival probability decays exponentially with time, the average decays as a power law with an exponent depending continuously on the density of searchers.

cond-mat.stat-mech↗

arXiv2010

How many eigenvalues of a Gaussian random matrix are positive?

We study the probability distribution of the index ${\mathcal N}_+$, i.e., the number of positive eigenvalues of an $N\times N$ Gaussian random matrix. We show analytically that, for large $N$ and large $\mathcal{N}_+$ with the fraction $0\le c=\mathcal{N}_+/N\le 1$ of positive eigenvalues fixed, the index distribution $\mathcal{P}({\mathcal N}_+=cN,N)\sim\exp[-βN^2 Φ(c)]$ where $β$ is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function $Φ(c)$ is computed explicitly for all $0\leq c \leq 1$. It is independent of $β$ and displays a quadratic form modulated by a logarithmic singularity around $c=1/2$. As a consequence, the distribution of the index has a Gaussian form near the peak, but with a variance $Δ(N)$ of index fluctuations growing as $Δ(N)\sim \log N/βπ^2$ for large $N$. For $β=2$, this result is independently confirmed against an exact finite $N$ formula, yielding $Δ(N)= \log N/2π^2 +C+\mathcal{O}(N^{-1})$ for large $N$, where the constant $C$ has the nontrivial value $C=(γ+1+3\log 2)/2π^2\simeq 0.185248...$ and $γ=0.5772...$ is the Euler constant. We also determine for large $N$ the probability that the interval $[ζ_1,ζ_2]$ is free of eigenvalues. Part of these results have been announced in a recent letter [\textit{Phys. Rev. Lett.} {\bf 103}, 220603 (2009)].

cond-mat.stat-mech↗

arXiv2010

Statistics of layered zigzags: a two-dimensional generalization of TASEP

A novel discrete growth model in 2+1 dimensions is presented in three equivalent formulations: i) directed motion of zigzags on a cylinder, ii) interacting interlaced TASEP layers, and iii) growing heap over 2D substrate with a restricted minimal local height gradient. We demonstrate that the coarse-grained behavior of this model is described by the two-dimensional Kardar-Parisi-Zhang equation. The coefficients of different terms in this hydrodynamic equation can be derived from the steady state flow-density curve, the so called `fundamental' diagram. A conjecture concerning the analytical form of this flow-density curve is presented and is verified numerically.

cond-mat.stat-mech↗

arXiv2010

Maximum Distance Between the Leader and the Laggard for Three Brownian Walkers

We consider three independent Brownian walkers moving on a line. The process terminates when the left-most walker (the `Leader') meets either of the other two walkers. For arbitrary values of the diffusion constants D_1 (the Leader), D_2 and D_3 of the three walkers, we compute the probability distribution P(m|y_2,y_3) of the maximum distance m between the Leader and the current right-most particle (the `Laggard') during the process, where y_2 and y_3 are the initial distances between the leader and the other two walkers. The result has, for large m, the form P(m|y_2,y_3) \sim A(y_2,y_3) m^{-δ}, where δ= (2π-θ)/(π-θ) and θ= cos^{-1}(D_1/\sqrt{(D_1+D_2)(D_1+D_3)}. The amplitude A(y_2,y_3) is also determined exactly.

cond-mat.stat-mech↗

arXiv2010

Maximum of N Independent Brownian Walkers till the First Exit From the Half Space

We consider the one-dimensional target search process that involves an immobile target located at the origin and $N$ searchers performing independent Brownian motions starting at the initial positions $\vec x = (x_1,x_2,..., x_N)$ all on the positive half space. The process stops when the target is first found by one of the searchers. We compute the probability distribution of the maximum distance $m$ visited by the searchers till the stopping time and show that it has a power law tail: $P_N(m|\vec x)\sim B_N (x_1x_2... x_N)/m^{N+1}$ for large $m$. Thus all moments of $m$ up to the order $(N-1)$ are finite, while the higher moments diverge. The prefactor $B_N$ increases with $N$ faster than exponentially. Our solution gives the exit probability of a set of $N$ particles from a box $[0,L]$ through the left boundary. Incidentally, it also provides an exact solution of the Laplace's equation in an $N$-dimensional hypercube with some prescribed boundary conditions. The analytical results are in excellent agreement with Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2010

Extreme Eigenvalues of Wishart Matrices: Application to Entangled Bipartite System

We discuss an application of the random matrix theory in the context of estimating the bipartite entanglement of a quantum system. We discuss how the Wishart ensemble (the earliest studied random matrix ensemble) appears in this quantum problem. The eigenvalues of the reduced density matrix of one of the subsystems have similar statistical properties as those of the Wishart matrices, except that their {\em trace is constrained to be unity}. We focus here on the smallest eigenvalue which serves as an important measure of entanglement between the two subsystems. In the hard edge case (when the two subsystems have equal sizes) one can fully characterize the probability distribution of the minimum eigenvalue for real, complex and quaternion matrices of all sizes. In particular, we discuss the important finite size effect due to the {\em fixed trace constraint}.

cond-mat.stat-mech↗

arXiv2010

Area distribution and the average shape of a Lévy bridge

We consider a one dimensional Lévy bridge x_B of length n and index 0 < α< 2, i.e. a Lévy random walk constrained to start and end at the origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution P_B(A,n) of the area A = \sum_{m=1}^n x_B(m) under such a Lévy bridge and show that, for large n, it has the scaling form P_B(A,n) \sim n^{-1-1/α} F_α(A/n^{1+1/α}), with the asymptotic behavior F_α(Y) \sim Y^{-2(1+α)} for large Y. For α=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions. We also compute the average profile < \tilde x_B (m) > at time m of a Lévy bridge with fixed area A. For large n and large m and A, one finds the scaling form < \tilde x_B(m) > = n^{1/α} H_α({m}/{n},{A}/{n^{1+1/α}}), where at variance with Brownian bridge, H_α(X,Y) is a non trivial function of the rescaled time m/n and rescaled area Y = A/n^{1+1/α}. Our analytical results are verified by numerical simulations.

cond-mat.stat-mech↗

arXiv2010

Switching and growth for microbial populations in catastrophic responsive environments

Phase variation, or stochastic switching between alternative states of gene expression, is common among microbes, and may be important in coping with changing environments. We use a theoretical model to assess whether such switching is a good strategy for growth in environments with occasional catastrophic events. We find that switching can be advantageous, but only when the environment is responsive to the microbial population. In our model, microbes switch randomly between two phenotypic states, with different growth rates. The environment undergoes sudden "catastrophes", the probability of which depends on the composition of the population. We derive a simple analytical result for the population growth rate. For a responsive environment, two alternative strategies emerge. In the "no switching" strategy, the population maximises its instantaneous growth rate, regardless of catastrophes. In the "switching" strategy, the microbial switching rate is tuned to minimise the environmental response. Which of these strategies is most favourable depends on the parameters of the model. Previous studies have shown that microbial switching can be favourable when the environment changes in an unresponsive fashion between several states. Here, we demonstrate an alternative role for phase variation in allowing microbes to maximise their growth in catastrophic responsive environments.

q-bio.PE↗