Sample-to-sample fluctuations in the conductivity of a disordered medium

We investigate the sample-to-sample fluctuations in the conductivity of a random resistor network—equivalently, in the diffusivity of a disordered medium with symmetric hopping rates. We argue that whenever the effective conductivity ^* is strictly positive, then the fluctuations are normal, i.e., proportional to (volume)^–1/2. If the local conductivities are allowed to be zero, then ^* vanishes when approaching the percolation thresholdp _c. Close top _c the fluctuations are anomalous. From the renormalization group on hierarchical lattices we find that atp _c fluctuations and mean scale in the same fashion, i.e., there is no independent scaling exponent for the fluctuations.     

Conserved Mass Models and Particle Systems in One Dimension

In this paper we study analytically a simple one-dimensional model of mass transport. We introduce a parameter p that interpolates between continuous-time dynamics (p0 limit) and discrete parallel update dynamics (p=1). For each p, we study the model with (i) both continuous and discrete masses and (ii) both symmetric and asymmetric transport of masses. In the asymmetric continuous mass model, the two limits p=1 and p0 reduce respectively to the q-model of force fluctuations in bead packs [S. N. Coppersmith et al., Phys. Rev. E 53:4673 (1996)] and the recently studied asymmetric random average process [J. Krug and J. Garcia, cond-mat/9909034]. We calculate the steady-state mass distribution function P(m) assuming product measure and show that it has an algebraic tail for small m, P(m)m ^– , where the exponent depends continuously on p. For the asymmetric case we find (p)=(1–p)/(2–p) for 0p<1 and (1)=–1, and for the symmetric case, (p)=(2–p)²/(8–5p+p ²) for all 0p1. We discuss the conditions under which the product measure ansatz is exact. We also calculate exactly the steady-state mass–mass correlation function and show that while it decouples in the asymmetric model, in the symmetric case it has a nontrivial spatial oscillation with an amplitude decaying exponentially with distance.     

Symmetric random walks in random environments

We consider a random walk on thed-dimensional lattice ^d where the transition probabilitiesp(x,y) are symmetric,p(x,y)=p(y,x), different from zero only ify–x belongs to a finite symmetric set including the origin and are random. We prove the convergence of the finite-dimensional probability distributions of normalized random paths to the finite-dimensional probability distributions of a Wiener process and find our an explicit expression for the diffusion matrix.     

Asymptotic and Numerical Analyses for Mechanical Models of Heat Baths

A mechanical model of a particle immersed in a heat bath is studied, in which a distinguished particle interacts via linear springs with a collection of n particles with variable masses and random initial conditions; the jth particle oscillates with frequency j ^p , where p is a parameter. For p>1/2 the sequence of random processes that describe the trajectory of the distinguished particle tends almost surely, as n, to the solution of an integro-differential equation with a random driving term; the mean convergence rate is 1/n ^p–1/2. We further investigate whether the motion of the distinguished particle can be well approximated by an integration scheme—the symplectic Euler scheme—when the product of time step h and highest frequency n ^p is of order 1, that is, when high frequencies are underresolved. For 1/2<p<1 the numerical solution is found to converge to the exact solution at a reduced rate of |log h| h ^2–1/p . These results shed light on existing numerical data.     

On the phase transition to sheet percolation in random Cantor sets

Thed-dimensional random Cantor set is a generalization of the classical middle-thirds Cantor set. Starting with the unit cube [0, 1] ^d , at every stage of the construction we divide each cube remaining intoM ^d equal subcubes, and select each of these at random with probabilityp. The resulting limit set is a random fractal, which may be crossed by paths or (d–1)-dimensional sheets. We examine the critical probabilityp _s(M, d) marking the existence of these sheet crossings, and show that p_s(M,d)1–p_c(M ^d) asM, where p_c(M ^d) is the critical probability of site percolation on the lattice (M ^d) obtained by adding the diagonal edges to the hypercubic lattice ^d. This result is then used to show that, at least for sufficiently large values ofM, the phases corresponding to the existence of path and sheet crossings are distinct.     

The thermodynamics of the Curie-Weiss model with random couplings

We study the Curie-Weiss version of an Ising spin system with random, positively biased couplings. In particular, the case where the couplings _ij take the values one with probabilityp and zero with probability 1 –p, which describes the Ising model on a random graph, is considered. We prove that ifp is allowed to decrease with the system sizeN in such a way thatNp(N) asN , then the free energy converges (after trivial rescaling) to that of the standard Curie-Weiss model, almost surely. Similarly, the induced measures on the mean magnetizations converge to those of the Curie-Weiss model. Generalizations of this result to a wide class of distributions are detailed.     

Parking Cars with Spin but no Length

The car parking problem is a one-dimensional model of random packing. Cars arrive to park on a block of length x, sequentially. Each car has, independently, spin up or spin down, w.p. 0 < p 1, for spin up and q = 1 – p for spin down, respectively. Each car tries to park at a uniformly distributed random point t [0, x]. If t is within distance 1 of the location of a previously parked car of the same spin, or within distance a of the location of a previously parked car of the opposite spin, then the new car leaves without parking and the next car arrives, until saturation. We study the problem analytically as well as numerically. The expected number of up spins c(p, a) per unit length for sufficiently large x is neither monotonic in p for fixed a, nor is it monotone in a for fixed p, in general. An intuitive explanation is given for this nonmonotonicity.     

On the large-coupling-constant behavior of the Liapunov exponent in a binary alloy

We consider the usual one-dimensional tight-binding Anderson model with the random potential taking only two values, 0 and, with probabilityp and 1–p, 0<p<1. We show that the Liapunov exponent (E), E R. diverges as uniformly in the energyE. Using a result of Carmona, Klein, and Martinelli, this proves that for large enough, the integrated density of states is singular continuous. We also compute explicitly the exact asymptotics for a dense set of energies and we compare the results with numerical simulations.     

A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices

Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X*X(X ^t X) converges to the Tracy–Widom law as n,p (the dimensions of X) tend to in some ratio n/p>0. We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n–p=O(p ^1/3). We also prove that _max(n ^1/2+p ^1/2)²+O(p ^1/2 log(p)) (a.e.) for general >0.     

Two-dimensional mixed spin Ising models with bond dilution and random ±J interactions

Using the finite cluster approximation we study a mixed spin model (spins =1/2 andS=1) on a square lattice with nearest-neighbour and crystal field interactions. The nearest-neighbour couplingsK _ij are assumed to be independent random variables with distribution,P(K _ij )=p(K _ij –K)+(1–p)(K _ij –K), whereK>0. We investigate the cases =0 and =–1 corresponding to bond dilution and to random ±J interactions, respectively. In certain ranges ofp the phase diagrams exhibit tricritical behaviour and reentrance.Supported by the agreement of cooperation between the DFG and the CNR-MarocOn leave from Faculté des Sciences I, Université Hassan II, Casablanca, Morocco; and Laboratoire de Magnétisme Université de Rabat, Morocco     

Stochastic One-Dimensional Lorentz Gas on a Lattice

We study a one-dimensional stochastic Lorentz gas where a light particle moves in a fixed array of nonidentical random scatterers arranged in a lattice. Each scatterer is characterized by a random transmission/reflection coefficient. We consider the case when the transmission coefficients of the scatterers are independent identically distributed random variables. A symbolic program is presented which generates the exact velocity autocorrelation function (VACF) in terms of the moments of the transmission coefficients. The VACF is found for different types of disorder for times up to 20 collision times. We then consider a specific type of disorder: a two-state Lorentz gas in which two types of scatterers are arranged randomly in a lattice. Then a lattice point is occupied by a scatterer whose transmission coefficient is with probability p or + with probability 1–p. A perturbation expansion with respect to is derived. The ² term in this expansion shows that the VACF oscillates with time, the period of oscillation being twice the time of flight from one scatterer to its nearest neighbor. The coarse-grained VACF decays for long times like t ^–3/2, which is similar to the decay of the VACF of the random Lorentz gas with a single type of scatterer. The perturbation results and the exact ones (found up to 20 collision times) show good agreement.     

Scale Invariance of the PNG Droplet and the Airy Process

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single time (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y ^–2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.     

Stochastic analyses of the dynamics of generalized Little-Hopfield-Hemmen type neural networks

Stochastic analyses are conducted of model neural networks of the generalized Little-Hopfield-Hemmen type, in which the synaptic connections with linearly embeddedp sets of patterns are free of symmetric ones, and a Glauber dynamics of a Markovian type is assumed. Two kinds of approaches are taken to study the stochastic dynamical behavior of the network system. First, by developing the method of the nonlinear master equation in the thermodynamic limitN, an exact self-consistent equation is derived for the time evolultion of the pattern overlaps which play the role of the order parameters of the system. The self-consistent equation is shown to describe almost completely the macroscopic dynamical behavior of the network system. Second, conducting the system-size expansion of the master equation for theN-body probability distribution of the Glauber dynamics makes it possible to analyze the fluctuations. In the course of the analysis, the self-consistent equation for the pattern overlaps is derived again. The main result of the rigorous fluctuation analysis is that as far as the fluctuations are concerned, the time course of the pattern overlap fluctuations behaves independently of the fluctuations in the remaining modes of the system's macrovariables, in accordance with the self-determining property of the macroscopic motion of the pattern overlaps for neural networks with linear synaptic couplings.     

Reentrant behaviour in spin glasses

We present the phase diagram of thed-dimensional random bond Ising model as a representative system for spin glasses. We consider nearest neighbour ferromagnetic couplingsJ with concentration 1-p and impurity couplingsaJ (|a|1) with concentrationp. It is shown that for antiferromagnetic couplings, –1<a<0, the system quite generally exhibits reentrant behaviour, i.e. two phase transitions at finite temperatures, in certain ranges of the concentrationp. It is further argued that this behaviour is a quite common feature for spin glass systems characterized by competing interactions.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

First Order Asymptotics of Matrix Integrals; A Rigorous Approach Towards the Understanding of Matrix Models

We investigate the large N limit of spectral measures of matrices which relate to the Gibbs measures of a number of statistical mechanical systems on random graphs. These include the Ising and Potts models on random graphs. For most of these models, we prove that the spectral measures converge almost surely and describe their limit via solutions to an Euler equation for isentropic flow with negative pressure p()=–3^–123.     

One-dimensional random walk with self-interaction

A standard random walk on a one-dimensional integer lattice is considered where the probability ofk self-intersections of a path =(0, (1),..., (n) is proportional toe ^–k . It is proven that for <0,n ^–1/3(n) converges to a certain continuous random variable. For >0 the formulas are given for the asymptotic Westerwater velocity of a generic path and for the variance of the fluctuations about the asymptotic motion.     

Review: Gas of Wormholes: A Possible Ground State of Quantum Gravity

In order to gain insight into the possible Ground State of Quantized Einstein's Gravity, we have derived a variational calculation of the energy of the quantum gravitational field in an open space, as measured by an asymptotic observer living in an asymptotically flat space-time. We find that for Quantum Gravity (QG) it is energetically favourable to perform its quantum fluctuations not upon flat space-time but around a "gas" of wormholes of mass m _p, the Planck mass (m _p 10¹⁹ GeV) and average distance l _p, the Planck length a _p(a _p 10^–33 cm). As a result, assuming such configuration to be a good approximation to the true Ground State of Quantum Gravity, space-time, the arena of physical reality, turns out to be well described by Wheeler's quantum foam and adequately modeled by a space-time lattice with lattice constant l _p, the Planck lattice.     

Real-space renormalization group investigation of pure and disordered mixed spin Ising models ond-dimensional lattices

The mixed spin Ising model (spins =1/2 andS=1) ond-dimensional hypercubic lattices with nearest-neighbour exchange interactions is studied via a renormalization group transformation in position space. The phase diagrams in (L, K) space, i.e. in dependence of the bilinear (K) and the biquadratic (L) interaction coefficients, are qualitatively different ford=2 andd>2. For any dimensiond however it is found that all transitions are of second order. At zero-temperature (K=,L=), the ferromagnetic order disappears at (L/K)₀=2, which does not depend ond. Using an extension of this real-space renormalization group analysis we study the two-dimensional random disordered version of the above model.L is kept homogeneous and the bilinear interactionsK _ij are assumed to be independent random variables with distributionP(K _ij )=p(K _ij –K)+(1–p)(K _ij –K); whereK>0. The phase diagrams for different values ofp are obtained. At zero temperature, it is found that in the bond diluted model (=0) the value (L/K)₀ depends continuously onp, whereas in the random ±K interactions (=–1) (L/K)₀ is unique and does not depend onp.Supported by the agreement of cooperation between the DFGW. Germany and the CNR-Maroc     

Shock fluctuations in the two-dimensional asymmetric simple exclusion process

We study via computer simulations (using various serial and parallel updating techniques) the time evolution of shocks, particularly the shock width(t), in several versions of the two-dimensional asymmetric simple exclusion process (ASEP). The basic dynamics of this process consists of particles jumping independently to empty neighboring lattice sites with ratesp _up=p _down=p andp _left<p _right. If the system is initially divided into two regions with densities _left< _right, the boundary between the two regions corresponds to a shock front. Macroscopically the shock remains sharp and moves with a constant velocityv _shock=(p _right– _left)(1–p _left–p _right). We find that microscopic fluctuations cause to grow ast , 1/4. This is consistent with theoretical expectations. We also study the nonequilibrium stationary states of the ASEP on a periodic lattice, where we break translation invariance by reducing the jump rates across the bonds between two neighboring columns of the system by a factorr. We find that for fixed overall density _avg and reduction factorr sufficiently small (depending on _avg and the jump rates) the system segregates into two regions with densities ₁ and ₂=1– ₁, where these densities do not depend on the overall density _avg. The boundary between the two regions is again macroscopically sharp. We examine the shock width and the variance in the shock position in the stationary state, paying particular attention to the scaling of these quantities with system size. This scaling behavior shows many of the same features as the time-dependent scaling discussed above, providing an alternate determination of the result1/4.