Satya N. Majumdar

Force fluctuations in bead packs

Experimental observations and numerical simulations of the large force inhomogeneities present in stationary bead packs are presented. Forces much larger than the mean occurred but were exponentially rare. An exactly soluble model reproduced many aspects of the experiments and simulations. In this model, the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead in the pile.

Model for force fluctuations in bead packs

We study theoretically the complex network of forces that is responsible for the static structure and properties of granular materials. We present detailed calculations for a model in which the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead of the pile. We compare our results for force distribution function for this model, including exact results for certain contact angle probability distributions, with numerical simulations of force distributions in random sphere packings. This model reproduces many aspects of the force distribution observed both in experiment and in numerical simulations of sphere packings.

Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model

We establish an equivalence between the undirected Abelian sandpile model and the q→0 limit of the q-state Potts model. The equivalence is valid for arbitrary finite graphs. Two-dimensional Abelian sandpile models, thus, correspond to a conformal field theory with central charge c = −2. The equivalence also gives a Monte Carlo algorithm to generate random spanning trees. We study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the Abelian sandpile model. The average number of sites burnt upto time t varies at ta. In two dimensions our numerically determined value of a agrees with the theoretical prediction a = 85. We relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes.

Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices

We compute exact asymptotic results for the probability of the occurrence of large deviations of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (NxN) random matrix are positive (negative) decreases for large N as approximately exp [-beta theta(0)N2] where the Dyson index beta characterizes the ensemble and the exponent theta(0)=(ln 3)/4=0.274653... is universal. We compute the probability that the eigenvalues lie in the interval [zeta1,zeta2] which allows us to calculate the joint probability distribution of the minimum and the maximum eigenvalue. As a by-product, we also obtain exactly the average density of states in Gaussian ensembles whose eigenvalues are restricted to lie in the interval [zeta1,zeta2] , thus generalizing the celebrated Wigner semi-circle law to these restricted ensembles. It is found that the density of states generically exhibits an inverse square-root singularity at the location of the barriers. These results are confirmed by numerical simulations. Some of the results presented in detail here were announced in a previous paper [D. S. Dean and S. N. Majumdar, Phys. Rev. Lett. 97, 160201 (2006)].

Large Deviations of Extreme Eigenvalues of Random Matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N x N) random matrix are positive (negative) decreases for large N as approximately exp[-betatheta(0)N2] where the parameter beta characterizes the ensemble and the exponent theta(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at zeta.

Persistence and first-passage properties in nonequilibrium 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 spin 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 generalizations 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.

Abelian sandpile model on the Bethe lattice

The authors study Bak, Tang and Wiesenfelds Abelian sandpile model (1987) of self-organised criticality on the Bethe lattice. Exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained. They also determine the probability distribution of the number of distinct sites that topple at least once, the number of toplings at the origin and the total number of toplings in an avalanche. The probability that an avalanche consists of more than n toplings varies as n-1/2 for large n. The probability that its duration exceeds T decreases as 1/T for large T. These exponents are the same as for the critical percolation clusters in mean field theory.

Height correlations in the Abelian sandpile model

The authors study the distribution of heights in the self-organized critical state of the Abelian sandpile model on a d-dimensional hypercubic lattice. They calculate analytically the concentration of sites having minimum allowed value in the critical state. They also calculate, in the critical state, the probability that the heights, at two sites separated by a distance r, would both have minimum values and show that the lowest-order r-dependent term in it varies as r-2d for large r.

Large deviations of the maximum eigenvalue in Wishart random matrices

We analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N × N) Wishart matrix W = XTX (where X is a rectangular M × N matrix with independent Gaussian entries) are smaller than the mean value λ = N/c decreases for large N as , where β = 1, 2 corresponds respectively to real and complex Wishart matrices, c = N/M ≤ 1 and Φ−(x; c) is a rate (sometimes also called large deviation) function that we compute explicitly. The result for the anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of Wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier. Numerical simulations are in excellent agreement with the analytical predictions.

Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces

The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution function also appears in a rather well studied physical system, namely the fluctuating interfaces. We present an exact solution for the distribution P(hm,L) of the maximal height hm (measured with respect to the average spatial height) in the steady state of a fluctuating interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(hm,L)=L−1/2f(hmL−1/2) for all L>0 where the function f(x) is the Airy distribution function. This result is valid for both the Edwards–Wilkinson (EW) and the Kardar–Parisi–Zhang interfaces. For the free boundary case, the same scaling holds P(hm,L)=L−1/2F(hmL−1/2), but the scaling function F(x) is different from that of the periodic case. We compute this scaling function explicitly for the EW interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [S.N. Majumdar and A. Comtet, Phys. Rev. Lett. 92: 225501 (2004)].