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.     

Extension of the Arrowsmith–Essam Formula to the Domany–Kinzel Model

Arrowsmith and Essam gave an expansion formula for point-to-point connectedness functions of the mixed site-bond percolation model on oriented lattices, in which each term is characterized by a graph. We extend this formula to general k-point correlation functions, which are point-to-set (with k points) connectivities in the context of percolation, of the two-neighbor discrete-time Markov process (stochastic cellular automata with two parameters) in one dimension called the Domany–Kinzel model, which includes the mixed site-bond oriented percolation model on a square lattice as a special case. Our proof of the formula is elementary and based on induction with respect to time-step, which is different from the original graph-theoretical one given by Arrowsmith and Essam. We introduce a system of m interacting random walkers called m friendly walkers (m FW) with two parameters. Following the argument of Cardy and Colaiori, it is shown that our formula is useful to derive a theorem that the correlation functions of the Domany–Kinzel model are obtained as an m0 limit of the generating functions of the m FW.     

Large-n limit of the Heisenberg model: The decorated lattice and the disordered chain

The critical temperature of the generalized spherical model (large-component limit of the classical Heisenberg model) on a cubic lattice, whose every bond is decorated byL spins, is found. WhenL, the asymptotics of the temperature isT _c aL ^–1. The reduction of the number of spherical constraints for the model is found to be fairly large. The free energy of the one-dimensional generalized spherical model with random nearest neighbor interaction is calculated.     

Algebraic Decay in Hierarchical Graphs

We study the algebraic decay of the survival probability in open hierarchical graphs. We present a model of a persistent random walk on a hierarchical graph and study the spectral properties of the Frobenius–Perron operator. Using a perturbative scheme, we derive the exponent of the classical algebraic decay in terms of two parameters of the model. One parameter defines the geometrical relation between the length scales on the graph, and the other relates to the probabilities for the random walker to go from one level of the hierarchy to another. The scattering resonances of the corresponding hierarchical quantum graphs are also studied. The width distribution shows the scaling behavior P()1/.     

Absence of continuous symmetry breaking in a one-dimensional n−2 model

For a one-dimensional array ofS ^N–1 spins (N 2) with isotropic pair interactions (and more general systems) with J(j–i) obeying sup_n[n^–1 ₁ ⁿ j ²|J(j)|]<, we prove that every equilibrium state is invariant under the natural action ofSO(N). In particular, there is no long-range order of the conventional type. Included is the caseJ(n)=n ^–2.Research partially supported by U.S.N.S.F. Grant No. MCS-78-01885.S. Fairchild Scholar at Caltech. On leave from Departments of Mathematics and Physics, Princeton University, Princeton, New Jersey 08544.     

From dilute to dense self-avoiding walks on hypercubic lattices

A field-theoretic representation is presented to count the number of configurations of a single self-avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions. We evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain. The meanfield approximation is exact in the limit of infinite dimensions, and corrections to it in powers ofd ^–1 can be systematically evaluated. The connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions. We also find that the entropy per site develops a maximum atf1–(2d)^–1. Ford=2 (d=3), this maximum occurs atf~0.80 (f~0.86) and its value is about 50% (30%) higher than the entropy per site of a Hamiltonian walk (f=1).     

Superconductivity as a Kosterlitz–Thouless transition in the two-dimensional Hubbard model

We investigate a superconducting Kosterlitz–Thouless transition in the two-dimensional (2D) Hubbard model using auxiliary quantum Monte Carlo method for the ground state. The pair susceptibility is computed for both the attractive and repulsive Hubbard model. The numerical results show that the s-wave pair susceptibility scales as χ  L² for the attractive case, in agreement with previous quantum Monte Carlo studies. The scaling χ  L² also holds for the d-wave pair susceptibility for the repulsive Hubbard model if we adjust the band parameter t′.     

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.     

Trapping of random walks on the line

Several features of the trapping of random walks on a one-dimensional lattice are analyzed. The results of this investigation are as follows: (1) The correction term to the known asymptotic form for the survival probability ton steps is O(( ₂n)^–1/3), where =–ln(1–c), andc is the trap concentration. (2) The short time form for the survival probability is found to be exp[–a(c)n ^1/2], wherea(c) is given in Eq. (21). (3) The mean-square displacement of a surviving random walker is found to go liken ^2/3for largen. (4) When the distribution of trap-free regions is changed so that very large regions are much rarer than for ideally random trap placement the asymptotic survival probability changes its dependence onn. One such model is studied.     

A Self-Consistent Ornstein–Zernike Approximation for the Random Field Ising Model

We extend the self-consistent Ornstein–Zernike approximation (SCOZA), first formulated in the context of liquid-state theory, to the study of the random field Ising model. Within the replica formalism, we treat the quenched random field just as another spin variable, thereby avoiding the usual average over the random field distribution. This allows us to study the influence of the distribution on the phase diagram in finite dimensions. The thermodynamics and the correlation functions are obtained as solutions of a set a coupled partial differential equations with magnetization, temperature, and disorder strength as independent variables. A preliminary analysis based on high-temperature and 1/d series expansions shows that the theory can predict accurately the dependence of the critical temperature on disorder strength (no sharp transition, however, occurs for d4). For the bimodal distribution, we find a tricritical point which moves to weaker fields as the dimension is reduced. For the Gaussian distribution, a tricritical point may appear for d around 4.     

On the Classification of q-Algebras

The problem is the classification of the ideals of free differential algebras, or the associated quotient algebras, the q-algebras; being finitely generated, unital C-algebras with homogeneous relations and a q-differential structure. This family of algebras includes the quantum groups, or at least those that are based on simple (super) Lie or Kac–Moody algebras. Their classification would encompass the so far incompleted classification of quantized (super) Kac–Moody algebras and of the (super) Kac–Moody algebras themselves. These can be defined as singular limits of q-algebras, and it is evident that to deal with the q-algebras in their full generality is more rational than the examination of each singular limit separately. This is not just because quantization unifies algebras and superalgebras, but also because the points q=1 and q=–1 are the most singular points in parameter space. In this Letter, one of two major hurdles in this classification program has been overcome. Fix a set of integers n ₁,...,n _k, and consider the space of homogeneous polynomials of degree n ₁ in the generator e ₁, and so on. Assume that there are no constants among the polynomials of lower degree, in any one of the generators; in this case all constants in the space have been classified. The task that remains, the more formidable one, is to remove the stipulation that there are no constants of lower degree.     

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.     

On the chiral hubbard model and the chiral Kondo lattice model

The integrability of the one-dimensional chiral Hubbard model is discussed in the limit of strong interaction,U=. The system is shown to be integrable in the sense of the existence of an infinite number of constants of motion. The system is related to a chiral Kondo lattice model at strong interactionJ=+.     

Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty

We consider a weakly self-avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp[–|t–s| ^–p ] wheret ands are the times at which the common site is visited andp is a parameter. We prove that ifp<1 and is sufficiently large, then the walk behaves ballistically, i.e., the distance to the end of the walk grows linearly with the number of steps in the walk. We also give a heuristic argument that ifp>3/2, then the walk should have diffusive behavior. The proof and the heuristic argument make use of a real-space renormalization group transformation.     

Reaction,trapping, and multifractality in one-dimensional systems

In the first part of this paper, we present two variants of the A+AA and A+AP reaction in one dimension that can be investigated analytically. In the first model, pairs of neighboring particles disappear reactively at a rate which is independent of their relative distance. It is shown that the probability density (x) for a nearest neighbor distance equal tox approaches the scaling form(x) c exp(–cx/2)/(cx)^1/2 in the long-time limit, withc being the concentration of particles. The second model is a ballistic analogue of the coagulation reaction A+A A. The model is solved by reducing it to a first-passagetime problem. The anomalous relaxation dynamics can be linked in a direct way to the fractal time properties of random walks. In the second part of this paper, we discuss the complications that arise in systems with disorder. We present a new approach that relates first-passage-time characteristics in a one-dimensional random walk to properties of random maps. In particular, we show that Sinai disorder is a borderline case for the appearance of multifractal properties. Finally, we apply a previously introduced renormalization technique to calculate the survival probability of particles moving on the line in the presence of a background of imperfect traps.     

New Upper Bounds for the Connective Constants of Self-Avoiding Walks

Using a novel implementation of the Goulden–Jackson method, we compute new rigorous upper bounds for the connective constants of self-avoiding walks, breaking Alm's previous records for rectangular (hypercubic) lattices. We also give the explicit generating functions for memory 8. We then incorporate a numerical limit which gives bounds that are even better.     

Fermionic solution of the Andrews-Baxter-Forrester model. I. Unification of TBA and CTM methods

The problem of computing the one-dimensional configuration sums of the ABF model in regime III is mapped onto the problem of evaluating the grandcanonical partition function of a gas of charged particles obeying certain fermionic exclusion rules. We thus obtain a newfermionic method to compute the local height probabilities of the model. Combined with the originalbosonic approach of Andrews, Baxter, and Forrester, we obtain a new proof of (some of) Melzer's polynomial identities. In the infinite limit these identities yield Rogers-Ramanujan type identities for the Virasoro characters _l,1 ^(r–l,r) (q) as conjectured by the Stony Brook group. As a result of our work the corner transfer matrix and thermodynamic Bethe Ansatz approaches to solvable lattice models are unified.     

On the magnetic ordering implemented by the s–d exchange interaction

The RKKY perturbation scheme for the spin s–d exchange model is reexamined in the case of a finite number of electrons per impurity. The distribution of impurities is assumed to be random. The free electron gas representing the unperturbed system in the RKKY approach is replaced by a mean-field system h_r asymptotically equivalent to the reduced s–d model H_r which includes part of the s–d exchange. Below the transition temperature T_c of h_r the lowest energy levels of the s–d exchange Hamiltonian H_K resulting from second-order perturbation theory prove to be ordered in the same manner as those of h_r in the limit of weak s–d exchange, favouring the same definite parallel alignment of any pair of impurity spins. It follows therefore that in the low temperature, weak coupling regime, the s–d model exhibits ferromagnetic ordering of impurities. At temperatures above T_c there is no ordering and impurity–impurity exchange has the RKKY form. These observations are consistent with the presence of a low-temperature ferromagnetic phase in numerous compounds with s–d exchange interaction.     

The Swendsen–Wang Process Does Not Always Mix Rapidly

The Swendsen–Wang process provides one possible dynamics for the q-state Potts model. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The legitimacy of such simulations depends on the rate of convergence of the process to equilibrium, as measured by the mixing time. Empirical observations suggest that the mixing time of the Swendsen–Wang process is short in many instances of practical interest, although proofs of this desirable behavior are known only for some very special cases. Nevertheless, we show that there are occasions when the mixing time of the Swendsen–Wang process is exponential in the size of the system. This undesirable behavior is related to the phenomenon of first-order phase transitions in Potts systems with q > 2 states.     

Bödeker’s effective theory: From Langevin dynamics to Dyson–Schwinger equations

The dynamics of weakly coupled, non-abelian gauge fields at high temperature is non-perturbative if the characteristic momentum scale is of order |k|g²T. Such a situation is typical for the processes of electroweak baryon number violation in the early Universe. Bödeker has derived an effective theory that describes the dynamics of the soft field modes by means of a Langevin equation. This effective theory has been used for lattice calculations so far [G.D. Moore, Nucl. Phys. B568 (2000) 367. Available from: <hep-ph/9810313>; G.D. Moore, Phys. Rev. D62 (2000) 085011. Available from: <hep-ph/0001216>]. In this work we provide a complementary, more analytic approach based on Dyson–Schwinger equations. Using methods known from stochastic quantitation, we recast Bödeker’s Langevin equation in the form of a field theoretic path integral. We introduce gauge ghosts in order to help control possible gauge artefacts that might appear after truncation, and which leads to a BRST symmetric formulation and to corresponding Ward identities. A second set of Ward identities, reflecting the origin of the theory in a stochastic differential equation, is also obtained. Finally, Dyson–Schwinger equations are derived.