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We present and analyze a minimalist model for the vertical transport of people in a tall building by elevators. We focus on start-of-day operation in which people arrive at the ground floor of the building at a fixed rate. When an elevator arrives on the ground floor, passengers enter until the elevator capacity is reached, and then they are transported to their destination floors. We determine the distribution of times that each person waits until an elevator arrives, the number of people waiting for elevators, and transition to synchrony for multiple elevators when the arrival rate of people is sufficiently large. We validate many of our predictions by event-driven simulations., Comment: 19 pages in IOP format, 11 figures. Version 2 is slightly expanded and contains some additional results and a new figure

We introduce a socially motivated extension of the voter model in which individual voters are also influenced by two opposing, fixed-opinion news sources. These sources forestall consensus and instead drive the population to a politically polarized state, with roughly half the population in each opinion state. Two types social networks for the voters are studied: (a) the complete graph of $N$ voters and, more realistically, (b) the two-clique graph with $N$ voters in each clique. For the complete graph, many dynamical properties are soluble within an annealed-link approximation, in which a link between a news source and a voter is replaced by an average link density. In this approximation, we show that the average consensus time grows as $N^\alpha$, with $\alpha = p\ell/(1-p)$. Here $p$ is the probability that a voter consults a news source rather than a neighboring voter, and $\ell$ is the link density between a news source and voters, so that $\alpha$ can be greater than 1. The polarization time, namely, the time to reach a politically polarized state from an initial strong majority state, is typically much less than the consensus time. For voters on the two-clique graph, either reducing the density of interclique links or enhancing the influence of news sources again promotes polarization., Comment: 30 pages in IOP format, 14 figures. Version 2: reference added

We investigate simple strategies that embody the decisions that one faces when trying to park near a popular destination. Should one park far from the target (destination), where finding a spot is easy, but then be faced with a long walk, or should one attempt to look for a desirable spot close to the target, where spots may be hard to find? We study an idealized parking process on a one-dimensional geometry where the desired target is located at $x=0$, cars enter the system from the right at a rate $\lambda$ and each car leaves at a unit rate. We analyze three parking strategies---meek, prudent, and optimistic---and determine which is optimal., Comment: 16 pages, 10 figures, IOP format. Revised version has various small corrections and nearly coincides with the published version

We investigate the dynamics of the voter model in which the population itself changes endogenously via the birth-death process. There are two species of voters, labeled A and B, and the population of each species can grow or shrink by the birth-death process at equal rates $b$. Individuals of opposite species also undergo voter model dynamics in which an AB pair can equiprobably become AA or BB with rate $v$---neutral evolution. In the limit $b/v\to\infty$, the distribution of consensus times varies as $t^{-3}$ and the probability that the population size equals $n$ at the moment of consensus varies as $n^{-3}$. As the birth/death rate $b$ is increased, fixation occurs more more quickly; that is, population fluctuations promote consensus., 15 pages, 6 figures, IOP format. Version 2: minor changes in response to referee comments. For publication in JSTAT. Version 3: Various minor errors fixed

We study the dynamics of a \emph{myopic} forager that randomly wanders on a lattice in which each site contains one unit of food. Upon encountering a food-containing site, the forager eats all the food at this site with probability $p, 10 pages, 1o figures

International audience; We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verifications of our analytical results.

We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q=1-p. If the walk hops onto an empty site, there is no bias. For the 1-excited walk on the half-line (one cookie initially at each site), the probability of first returning to the starting point at time t scales as t^{-(2-p)}. Although the average return time to the origin is infinite for all p, the walk eats, on average, only a finite number of cookies until this first return when p3/4, where the walk is transient, including a mean displacement that grows as t^{nu}, with nu>1/2 dependent on p, and a breakdown of scaling for the probability distribution of the walk., 14 pages, 13 figures, 2-column revtex4 format, for submission to J. Phys. A

As a strategy to complete games quickly, we investigate one-dimensional random walks where the step length increases deterministically upon each return to the origin. When the step length after the kth return equals k, the displacement of the walk x grows linearly in time. Asymptotically, the probability distribution of displacements is a purely exponentially decaying function of |x|/t. The probability E(t,L) for the walk to escape a bounded domain of size L at time t decays algebraically in the long time limit, E(t,L) ~ L/t^2. Consequently, the mean escape time ~ L ln L, while ~ L^{2n-1} for n>1. Corresponding results are derived when the step length after the kth return scales as k^alpha$ for alpha>0., Comment: 7 pages, 6 figures, 2-column revtext4 format

We study opinion formation in a population that consists of leftists, centrists, and rightist. In an interaction between neighboring agents, a centrist and a leftist can become both centrists or leftists (and similarly for a centrist and a rightist). In contrast, leftists and rightists do not affect each other. The initial density of centrists rho_0 controls the evolution. With probability rho_0 the system reaches a centrist consensus, while with probability 1-rho_0 a frozen population of leftists and rightists results. In one dimension, we determine this frozen state and the opinion dynamics by mapping the system onto a spin-1 Ising model with zero-temperature Glauber kinetics. In the frozen state, the length distribution of single-opinion domains has an algebraic small-size tail x^{-2(1-psi)} and the average domain size grows as L^{2*psi}, where L is the system length. The approach to this frozen state is governed by a t^{-psi} long-time tail with psi-->2*rho_0/pi as rho_0-->0.

We introduce a network growth model based on complete redirection: a new node randomly selects an existing target node, but attaches to a random neighbor of this target. For undirected networks, this simple growth rule generates unusual, highly modular networks. Individual network realizations typically contain multiple macrohubs---nodes whose degree scales linearly with the number of nodes $N$. The size of the network "nucleus"---the set of nodes of degree greater than one---grows sublinearly with $N$ and thus constitutes a vanishingly small fraction of the network. The network therefore consists almost entirely of leaves (nodes of degree one) as $N\to\infty$., 21 pages, 10 figures, 2 appendices. Minor corrections in version2. For publication in JSTAT

The authors investigate the kinetics of a generalized monomer-monomer surface reaction model in which there is a variable excluded-volume interaction between same species particles (As and Bs). For no interaction, this system reduces to the monomer-monomer model for which there is a first-order transition between A- and B-saturated phases. As a function of the excluded volume interaction, the first-order transition line for weak interaction terminates at a tricritical point where two second-order transitions meet. These transitions, which appear to be in the Reggeon field theory universality class, separate the A-saturated, reactive, and B-saturated phases. Series expansions and various time-dependent Monte Carlo simulations are used to estimate the critical parameters and exponents associated with these transitions.

The authors introduce a lattice model of catalysis which involves a small particle (a dime) and a large particle (a dollar). The larger diameter of the dollar prevents its adsorption next to a site which is already occupied by a dollar. The catalytic process is the reaction and immediate desorption of nearest-neighbour dime-dollar adsorbates. As a function of the reaction rate and the relative deposition rates of the two species, this system exhibits a second-order kinetic phase transition which separates a dime-saturated phase from a reactive steady state. The features of the phase transition are examined by numerical simulations and analytic techniques, including a systematic mean-field cluster expansion.

The authors investigate the time evolution of a prototypical population biology reaction which involves reproduction, self-regulation and competitive annihilation of two distinct species. In one dimension, the authors use a quasistatic analysis to argue that for a system with equal initial densities of two strongly competing species, an alternating pattern of domains forms whose lengths grow logarithmically with time. A scaling analysis of the underlying master equation, as well as numerical integration of the reaction diffusion equations support this result. For unequal initial densities, the concentration of the minority species undergoes a power-law decay with a non-universal exponent. The authors generalize the model by allowing for a nonlinear self-regulation term in the rate equations. As a function of the exponent of this nonlinearity, the typical domain size may grow either as a power law with time or saturate at a finite value. This general approach also suggests that a coarsening domain mosaic occurs in arbitrary spatial dimensions. In two dimensions, numerical integration of the reaction diffusion equations indicates that the average domain area grows approximately as t0.84.

The authors investigate steady-state geometrical properties of a reaction interface in the two-species annihilation process, A+B to 0, when a flux j of A and B particles is injected at opposite extremities of a finite domain. By balancing the input flux with the number of reactions, they determine that the width w of the reaction zone scales as j-1/3 in the large flux limit, and that the concentration in this zone is proportional to j2/3. This same behaviour is deduced from the solution to the reaction-diffusion equation. In the low flux limit, the concentration is almost independent of position and is proportional to j. In the latter case, the local reaction rate reaches maximum at the edges of the system rather than at the midpoint. When the two species approach at a finite velocity, there exists a critical velocity, above which the reactants essentially pass through each other. Results similar to those in one dimension are found in two- and three-dimensional radial geometries. Finally, they apply the quasistatic approximation to their steady-state solution to recover the known time dependence for the reaction zone width for the case of initially separated components with no external input.

The authors have adapted the dimerization algorithm for enumerating self-avoiding walks to exploit the parallelism of the Connection Machine System. By this approach they have extended the series for the number of self-avoiding walks on the 2D square lattice by five terms. These data may permit more accurate estimates of critical properties.

The authors examine the dynamics of interface growth under the combined influence of a stabilizing and a destabilizing surface current in 1+1 dimensions. It has been shown that the surface is weakly stabilized if a newly deposited particle can move locally to maximize the total number of nearest-neighbour contacts with the surface (curvature-driven current). The introduction of a weak, destabilizing surface tension term results in an almost periodic, grooved surface. The width of the surface grows linearly in time and does not reach a steady state. It appears that such a formation cannot be described by a continuous Langevin equation.

The authors discuss the kinetics of an irreversible monomer-monomer model of heterogeneous catalysis. In this model, two reactive species, A and B, adsorb irreversibly onto single sites of a catalytic substrate; subsequently nearest-neighbour adsorbed AB pairs can bond to form a reaction product which desorbs from the substrate. The kinetics of this process is investigated in a mean-field approximation, where the catalytic substrate is considered to be an N-site complete graph. Two fundamental limits are identified: (a) the adsorption-controlled limit, where the reaction on the surface occurs quickly, so that the overall process is limited by the adsorption rate, and (b) the reaction-controlled limit, where adsorption occurs readily so that the overall reaction is limited by the conversion of unlike neighbouring monomers to an AB pair. By analysing the master equation for the probability density of coverage, they determine the rate at which the catalyst becomes 'saturated', i.e. completely covered by only one species. they show that the saturation time is proportional to N, and they also derive the probability distribution for the substrate coverage. Numerical simulations are performed on lattice substrates of finite spatial dimensionality d to test the range of validity of the mean-field approach. They find good agreement between simulations and mean-field theory for d=2 and 3, but not for d=1, suggesting that d=2 is a critical dimensionality for the monomer-monomer process.

The dynamics of a monomer-monomer and a monomer-dimer surface catalytic reaction is investigated. From the mean-field solution, finite systems eventually 'poison' at an exponential rate to a fully occupied, non-reactive state. For the monomer-monomer process, this poisoning is driven by concentration fluctuations of a diffusive nature, leading to poisoning times which vary as a power of the linear system size L. A comparison of the Monte Carlo simulations with the mean-field result suggests that the upper critical dimension for the monomer-monomer model is d, = 2. For the monomer-dimer process, there is an effective potential that needs to be surmounted by fluctuations, leading to poisoning times which grow at least as fast as eL. This gives rise to an apparent reactive steady state. Heterogeneous catalysis is a fundamental kinetic process in which the rate of a chemical reaction is enhanced by the presence of suitable catalyst material ( 13. A typical example is where reactants adsorb on a catalytic surface which then promotes the bonding of reactants. Once a reaction occurs, the reaction product desorbs, thereby allowing for continued operation of the system. This type of catalytic reaction underlies a host of technological processes and a substantial fraction of all chemical production ( 11. It has only been very recently, however, that investigations of microscopic models have begun to identify the general principles underlying the dynamical behaviour of heterogeneous catalysis. Ziff et a1 (2) introduced a simple lattice model which appears to describe various features of the surface catalytic reaction of carbon monoxide (monomers, which adsorb onto a single lattice site) and oxygen (which are deposited as dimers and disassociate upon adsorption). Depending on the relative deposition rate of the monomers and dimers, there may be 'poisoning', where the surface eventually becomes covered by only one species, or there may be an apparent reactive steady state. Novel kinetic phase transitions demarcate these two possibilities. This intriguing behaviour has stimulated further work on this and a simpler monomer-monomer process to be defined below (3-71. Our goal, in this letter, is to show that the mean-field solution, together with numerical simulations of very small systems, provide a rather complete description of the dynamics of idealised surface catalytic reactions. For the monomer-monomer process, we show that the final state of a finite system is always poisoned and that poisoning is approached at an exponential rate in mean-field theory. These facts are actually quite difficult to establish in numerical simulations of large systems (5).

A general discussion of the kinetics of continuous, irreversible fragmentation processes is presented. For a linear process, where particle breakup is driven by an external force, the authors develop a scaling theory to describe the evolution of the cluster size distribution. They treat the general case where the breakup rate of a cluster of mass x varies as xlambda . When lambda >0, corresponding to larger clusters more likely to break up, the scaled cluster size distribution, phi (x), decays with the scaled mass, x, as x-2 exp(-axlambda ), as x. For small mass, phi (x) has the log-normal form, exp(-a ln2 x), if the breakup kernel has a small-size cutoff, while phi (x) has a power-law tail in the absence of a cutoff. They also show that a conventional scaling picture applies only for the case lambda >0. For lambda >0, they develop an alternative formulation for the cluster size distribution, in which the typical mass scale is determined by the initial condition. In this regime, they also investigate the nature of a 'shattering' transition, where mass is lost to a 'dust' phase of zero-mass particles. They also study the kinetics of a nonlinear, collision-induced fragmentation process. They analyse the asymptotic behaviour of a simple-minded class of models in which a two-particle collision results in either: (1) both particles splitting into two equal pieces, (2) only the larger particle splitting in two, or (3) only the smaller particle splitting. They map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations. Scaling is found to hold for different ranges of homogeneity index for the three models.

We study the splitting probabilities for a one-dimensional Brownian motion in a cage whose two boundaries move at constant speeds c1 and c2. This configuration corresponds to the capture of a diffusing, but skittish lamb, with an approaching shepherd on the left and a precipice on the right. We derive compact expressions for these splitting probabilities when the cage is expanding. We also obtain the time-dependent first-passage probability to the left boundary, as well as the splitting probability to this boundary, when the cage is either expanding or contracting. The boundary motions have a non-trivial impact on the splitting probabilities, leading to multiple regimes of behavior that depend on the expansion or contraction speed of the cage. In particular, the probability to capture the lamb is maximized when the shepherd moves at a non-zero optimal speed if the initial lamb position and the ratio between the two boundary speeds satisfy certain conditions.

We introduce an agent-based acquaintance model in which social links are created by processes in which there is no explicit homophily. In spite of the homogeneous nature of the social interactions, highly-clustered social networks can arise. The crucial feature of our model is that of variable transitive interactions. Namely, when an agent introduces two unconnected friends, the rate at which a connection actually occurs between them depends on the number of their mutual acquaintances. As this transitive interaction rate is varied, the social network undergoes a dramatic clustering transition. Close to the transition, the network consists of a collection of well-defined communities. As a function of time, the network can also undergo an \emph{incomplete} gelation transition, in which the gel, or giant cluster, does not constitute the entire network, even at infinite time. Some of the clustering properties of our model also arise, but in a more gradual manner, in Facebook networks. Finally, we discuss a more realistic variant of our original model in which there is a soft cutoff in the rate of transitive interactions. With this variant, one can construct network realizations that quantitatively match Facebook networks., Comment: 16 pages, 14 figures, IOP format. Version 2: final form for publication in JSTAT

We analyze growing networks that are built by enhanced redirection. Nodes are sequentially added and each incoming node attaches to a randomly chosen 'target' node with probability 1-r, or to the parent of the target node with probability r. When the redirection probability r is an increasing function of the degree of the parent node, with r-->1 as the parent degree diverges, networks grown via this enhanced redirection mechanism exhibit unusual properties, including: (i) multiple macrohubs---nodes with degrees proportional to the number of network nodes N; (ii) non-extensivity of the degree distribution in which the number of nodes of degree k, N_k, scales as N^{nu-1}/k^{nu}, with 1=4(ln2)-2=0.77258..., Comment: 26 pages, 12 figures, IOP format. This manuscript is a comprehensive account of work reported briefly in arXiv:1307.3768. Version 2: two minor typos fixed and one other small change in response to referee report. Version 3: minor typos fixed. Final version of publication in JSTAT

We introduce the confident voter model, in which each voter can be in one of two opinions and can additionally have two levels of commitment to an opinion --- confident and unsure. Upon interacting with an agent of a different opinion, a confident voter becomes less committed, or unsure, but does not change opinion. However, an unsure agent changes opinion by interacting with an agent of a different opinion. In the mean-field limit, a population of size N is quickly driven to a mixed state and remains close to this state before consensus is eventually achieved in a time of the order of ln N. In two dimensions, the distribution of consensus times is characterized by two distinct times --- one that scales linearly with N and another that appears to scale as N^{3/2}. The longer time arises from configurations that fall into long-lived states that consist of two (or more) single-opinion stripes before consensus is reached. These stripe states arise from an effective surface tension between domains of different opinions., Comment: 13 pages, 8 figures, iop format. Version 2 has minor revisions in response to referee comments. For publication in JSTAT

We propose kinetic models for the spread of permanent innovations and transient fads by the mechanism of social reinforcement. Each individual can be in one of M+1 states of awareness 0,1,2,...,M, with state M corresponding to adopting an innovation. An individual with awareness k1. The fraction of the population that remains clueless about a transient fad after it has come and gone changes discontinuously as a function of the fad abandonment rate lambda for M>1. The fad dies out completely in a time that varies non-monotonically with lambda., Comment: 4 pages, 2 columns, 5 figures, revtex 4-1 format; revised version has been expanded and put into iop format, with one figure added

We investigate the effect of cooperative interactions on the asymmetric exclusion process, which causes the particle velocity to be an increasing function of the density. Within a hydrodynamic theory, initial density upsteps and downsteps can evolve into: (a) shock waves, (b) continuous compression or rarefaction waves, or (c) a mixture of shocks and continuous waves. These unusual phenomena arise because of an inflection point in the current versus density relation. This anomaly leads to a group velocity that can be either an increasing or a decreasing function of the density on either side of the inflection point, a property that underlies these localized wave singularities.

We study first-passage properties for a particle that diffuses either inside or outside of generalized paraboloids, defined by y=a(x_1^2+...+x_{d-1}^2)^{p/2} where p>1, with absorbing boundaries. When the particle is inside the paraboloid, the survival probability S(t) generically decays as a stretched exponential, ln(S) ~ -t^{(p-1)/(p+1)}, independent of the spatial dimensional. For a particle outside the paraboloid, the dimensionality governs the asymptotic decay, while the exponent p specifying the paraboloid is irrelevant. In two and three dimensions, S ~ t^{-1/4} and S ~(ln t)^{-1}, respectively, while in higher dimensions the particle survives with a finite probability. We also investigate the situation where the interior of a paraboloid is uniformly filled with non-interacting diffusing particles and estimate the distance between the closest surviving particle and the apex of the paraboloid., 7 pages, 2 figure, revtex4 format. Version 2 contains minor changes and one added figure in response to referee comments. For publication in JSTAT

In certain parliamentary democracies, there are two major parties that move in and out of power every few elections, and a third minority party that essentially never governs. We present a simple model to account for this phenomenon, in which minority party supporters sometimes vote ideologically (for their party) and sometimes strategically (against the party they like the least). The competition between these disparate tendencies reproduces the empirical observation of two parties that frequently exchange majority status and a third party that is almost always in the minority., Comment: 6 pages, 4 figures. Version 2: reference added. Version 3: final version

We determine the probability that a partially melted heteropolymer at the melting temperature will either melt completely or return to a helix state. This system is equivalent to the splitting probability for a diffusing particle on a finite interval that moves according to the Sinai model. When the initial fraction of melted polymer is f, the melting probability fluctuates between different realizations of monomer sequences on the polymer. For a fixed value of f, the melting probability distribution changes from unimodal to a bimodal as the strength of the disorder is increased., Comment: 4 pages, 5 figures

We introduce a toy model of the "rat race" in which individuals try to better themselves relative to the rest of the population. An individual is characterized by a real-valued fitness and each advances at a constant rate by an amount that depends on its standing in the population. The leader advances to remain ahead of its nearest neighbor, while all others advance by an amount that is set by the distance to the leader. A rich dynamics occurs as a function of the mean jump size of the trailing particles. For small jumps, the leader maintains its position, while for large jumps, there are long periods of stasis that are punctuated by episodes of explosive advancement and many lead changes. Intermediate to these two regimes, agents reach a common fitness and evolution grinds to a halt., 6 pages, 5 figures, 2-column revtex4 format. Slight revisions in response to referee comments

The authors investigate the repulsion of random walks (RWs) and self-avoiding walks (SAWs) induced by (a) excluding a single lattice point or (b) excluding all points on a half-line from - infinity to the origin. For SAWs, they use exact enumeration and Monte Carlo methods to study the asymptotic behaviour of the displacement away from the excluded set on three-, four- and five-dimensional hypercubic lattices. When the SAW begins one lattice site away from an excluded point along the x direction, the mean displacement after N steps, (xN) approaches a finite limit, at a power-law rate, as N to infinity . However, the distribution of projected displacements exhibits a residual asymmetry in the asymptotic limit, reflective of a long-range influence of the excluded point. This general pattern of behaviour also occurs for a purely random walk. For a SAW which starts one lattice spacing away from the axis of the excluded half-line, enumeration data suggest that the mean displacement along the axis diverges as N nu , with v=3/4 in two dimensions, but as Nz, with z approximately=0.35, in three dimensions. This unexpected behaviour appears to be corroborated by constant-fugacity Monte Carlo data. However, for the corresponding RW model, both the solution to the diffusion equation and a heuristic argument indicate that (xN) diverges as N1/2 in two dimensions, and as N1/2/ln N in three dimensions.

The authors study the kinetics of a diffusion-controlled reaction in which perfect traps (scavengers) diffuse and consume randomly distributed particles. They find that the number of particles remaining after time t decays as exp(- alpha ctd2/), for spatial dimensions d or=2, where alpha is a constant, and c is the trap concentration. These predictions are supported by Monte Carlo data in d=1 and d=2. They also discuss the differences between the scavenger reaction and the reaction of particles diffusing and being consumed by randomly distributed static traps. Finally, they treat the situation where both the particles and the traps diffuse.

Parisi and Sourlas (1981) have argued that the isotropic branched polymer problem in dimension d+1 is related to the Lee-Yang edge singularity problem in dimension d-1. To test if there is a relation with the directed branched polymer problem, the authors calculate the generating functions for both site and bond directed branched polymers for arbitrary dimension d to order smax=10 and bmax=8. Their analysis lends support to the proposal that theta (d+1)-1= theta D(d)= sigma LY(d-1)+1, where theta and theta D are the critical exponents for lattice animals and directed lattice animals, and sigma LY is the Lee-Yang edge singularity exponent. They also obtain expansions for the growth parameter in the variable sigma -1=(2d-1)-1 for bond and site directed lattice animals.

Studies percolation in a random resistor-diode network on the simple cubic lattice. Here the occupied bonds joining nearest-neighbour sites may be either resistor-like, transmitting information or connectivity in either direction along a bond, or diode-like, transmitting in only one direction. They consider a model in which there exist both positive diodes which transmit in the +x, +y or +z directions, and negative diodes which transmit oppositely. They apply positive-space renormalisation group methods to map out the phase diagram, and calculate the exponents associated with the phase transitions of this system. They find three novel types of transitions due to the presence of the diodes. With only one species of diodes and vacancies (unoccupied bonds) present, a directed threshold occurs at a critical concentration of diodes. Here an infinite cluster forms which transmits only in the direction of the average diode polarisation. A reverse threshold occurs when resistors and only one species of diodes exist. At this point the resistors mediate information flow in the infinite cluster opposite to the diode polarisation. Finally, with all bond elements present, a mixed threshold occurs.

We study the properties of a model system that exhibits a transition between ferromagnetic and helical order at a Lifshitz point, as interaction parameters R and S compete. Here R JJJ,, and S = J:/J,,, where J, and Jz denote interactions between nearest- neighbour and next-nearest-neighbour spin pairs respectively in the z direction, and J,, is a nearest-neighbour interaction between spin pairs in each xy plane. We calculate the high- temperature susceptibility series to order 8, 6, 5, and 35 respectively for the Ising, planar, Heisenberg, and spherical models (n = 1,2,3, and io). In order to verify our results, we derive rigorous results which provide strong checks on the series coefficients. Series analysis is focused on the ferromagnetic phase. In particular, we confirm scaling with respect to both parameters R and S. In addition, we find the critical region shrinks as the Lifshitz point is approached. This is indicated by analysing the spherical model series where asymptotic series behaviour is not evident, even at order 35. Finally, by exploiting simple geometric ideas about the dependence of the correlation length on R and S, we describe the full wavevector and temperature dependence of the structure factor.

The authors introduce and study the kinetics of 'cluster eating', in which a cluster of mass i and a cluster of mass j react to form a lighter cluster of mass mod i-j mod . They write the rate equations for this process, which describe the kinetics in the mean-field limit, where spatial fluctuations in cluster density and in cluster shape are neglected. An asymptotic solution to these equations is derived for the particular case in which the reaction rate is independent of the masses of the reacting clusters. At long times, they find that the density of clusters of mass k, ck(t) decays as Ak/( tau logk-1 tau ), where tau is proportional to the time, Ak=(N-1)!/(N-k)! and N is the largest cluster mass in the initial state. This very unusual behaviour is checked by numerical simulations. A more general situation where the reaction matrix depends on the parities of the masses of the two incident clusters is also discussed briefly and a wide variety of possible kinetic behaviours is delineated. Finally, the authors study cluster eating below the upper critical dimension, where fluctuations in cluster density give rise to a non-classical kinetic behaviour.

For a polymer chain in a good solvent, the author calculates the probability distribution functions between an endpoint and an interior point, and between two interior points, by using exact enumeration to study a lattice self-avoiding walk model. These distribution functions are different from the usual distribution function between endpoints. At small distance scales, the probability of nearest-neighbour contacts between two interior points is smaller than the probability of contact between two endpoints. FCC and triangular lattices are considered.

A universal combination of amplitudes for the end-to-end distance of N-step self-avoiding walks and for the number of N step self-avoiding polygons is estimated numerically for various lattices. In d=2, universality of this amplitude combination is confirmed to good accuracy, while in d=3, the data are consistent with universality but the error limits are rather large. For finite-width, square-lattice strips the authors calculate universal critical point correlation amplitudes by fixed-fugacity Monte Carlo simulations. In addition, these amplitudes are calculated analytically either exactly or approximately by Cardy's conformal mapping approach. Numerical values agree well with the theoretical predictions.

A generalisation of the pure site and pure bond percolation problems is studied, in which both the sites and bonds are independently occupied at random. This generalisation-the site-bond problem-is of current interest because of its application to the phenomenon of polymer gelation. Motivated by considerations of cluster connectivity, the authors have defined two distinct models for site-bond percolation.

The generating functions and mean displacements of various two-dimensional directed self-avoiding walk models are calculated exactly by a simple transfer-matrix approach. Asymptotically, the authors find (R/sub //N/) approximately RN2)1/2 approximately N, and (Rperpendicular to N2)1 approximately N1/2, where N is the number of steps in the walk, and // and perpendicular to refer to projections of the displacement parallel and perpendicular to the preferred axis of the walk respectively. Some general properties of directed self-avoiding walks for arbitrary dimensions are discussed as well.

The break-collapse method recently introduced for the q-state Potts model is adapted for resistor networks. This method greatly simplifies the calculation of the conductance of an arbitrary two-terminal d-dimensional array of conductances, obviating the use of either Kirchhoff's laws or the star-triangle transformation. In addition, a real-space renormalisation group based on a new type of averaging gives excellent results for the conductivity of the random-resistor network on the square lattice.

The authors investigate the behaviour of the maximum voltage drop across the bonds in a random resistor network above the percolation threshold. On the basis of numerical simulations on randomly diluted L*L square lattice networks, we find that the average value of the maximum voltage, (Vmax(p;L)), exhibits a peak as a function of the bond concentration, p, which is located above the percolation threshold. This peak value appears to grow logarithmically with L, while the location of the peak appears to approach the percolation threshold very slowly as L increases. To help understand these results, they introduce the 'bubble' model, a quasi-one-dimensional structure in which the system length varies exponentially in the system width. This model is exactly soluble and a rather good description of percolation in systems of greater than one dimension can be obtained. Moreover, within the bubble model the peak value of the maximum voltage increases as In L, while the peak location approaches the percolation threshold as In(In L) In L, in good agreement with our numerical results.

We treat a percolation model with anisotropic bond occupation probabilities in order to study the crossover behaviour of the effective spatial dimension of the system. Previous position-space renormalisation group (PSRG) studies of this problem on the square lattice show a crossover from two-dimensional to pseudo-one-dimensional critical behaviour. To investigate the possible reasons for this surprising result, which is the opposite of the behaviour observed in thermal critical phenomena, we first develop alternative PSRG schemes. We study the predictions of these groups for asymptotically large rescaling parameters, where the approximations in the PSRG are thought to become negligible. From a consideration of the approximations involved, we are led to a decimation transformation that uses an anisotropic cluster. The results of this method indicate that anisotropic percolation is in the same universality class as isotropic percolation, in complete analogy with thermal critical phenoma. Our results indicate several peculiar features both of this problem and of the PSRG methods.

The problem of understanding the asymptotic statistical behaviour of the winding angle, ON, of a self-avoiding walk of N steps on a planar lattice is posed. Exact series expansion data for the square lattice up to N = 21 are reported. These data and Monte Carlo estimates up to N G 170 steps are fitted well by a logarithmic growth law for (06). The ratio (@,,)/(Oh)* appears to approach a limit of 2.9 to 3.2, which is close to the Gaussian value, 3. Heuristic scaling arguments are consistent with simple logarithmic growth (and also illuminate the logarithmic behaviour known rigorously for free Brownian motion).

The author uses Monte Carlo methods to calculate the mean end-to-end distance of randomly branched polymer molecules. Molecular realisations are brown on the square and simple cubic lattices, and consist of N bifunctional monomers and Nf polyfunctional branching units of functionality f. Three cases are treated in which the monomer branches are either random walks, self-avoiding walks (SAW), or SAW's except that different branches may join at their end points to form closed loops.

Uses position-space renormalisation and exact enumeration methods to study configurational properties of directed lattice animals (clusters of directed bonds, in which the tail of a new bond must be added to the head of existing bonds). Exponents are found for the divergence of correlation lengths parallel and perpendicular to the preferred axis. In addition an exponent is found that characterises the singularity of the directed animal generating function and critical points are located.

The free energies of directed lattice animals in good and theta -solvents, and the free energy of directed percolation are found by the use of a simple Flory-type approximation, which accounts for the inherent anisotropy of these systems. From these free energies, the authors obtain the upper critical dimension below which mean-field theory breaks down. They also calculate closed-form, dimension-dependent expressions for the parallel and perpendicular correlation length exponents, which characterise the asymptotic cluster shapes. These exponents are in excellent agreement with existing numerical data.

The authors study the conductivity in directed percolation by an analogue experiment and by computer simulations. In the experiment, the dependence of the conductivity on concentration of conducting diodes, rho , shows very large fluctuations. This reflects the topological constraints of directed percolation where conduction is very sensitive to singly connected or cutting bonds. They have checked this hypothesis by studying the underlying directed backbone of the experiment using computer simulations. From the dependence of the conductivity on the backbone fraction, and the dependence of the backbone fraction on rho - rho c. They infer a directed-conductivity exponent considerably less than unity, consistent with earlier theoretical work.

The authors investigate the dispersion of a dynamically neutral tracer flowing in a self-similar, hierarchical model of a porous medium. They consider a purely convective limit in which the time for tracer particles to traverse a given bond is strictly proportional to the inverse of the flux in the bond; the effect of molecular diffusion is neglected. Their hierarchical model contains an adjustable 'asymmetry' parameter which controls the width of the distribution of transit times for a tracer to traverse the multiplicity of paths in the network. They derive a functional recursion relation for this distribution, from which exact expressions for the moments can be obtained. They find two regimes of behaviour which are governed by the value of the asymmetry parameter. In one regime, the transit time distribution is characterised by a single timescale, so that a localised pulse of tracer spreads out at the same rate at which the pulse is convected downstream. In the second regime, the small fraction of tracer passing through the slowest bond of the network dominates in the moments of the distribution, leading to highly enhanced dispersion. The consequences of these results for the dispersion coefficient are discussed, both for self-similar and homogeneous systems.

We develop a cell position-space renormalisation group (PSRG) with which we study the scaling properties of isolated polymer chains. We model a chain by a self-avoiding walk constrained to a lattice. For rescaling factors b S 6, we calculate recursion relations analytically on the square lattice with several different choices for the PSRG weight function. We also calculate implicit cell-to-cell transformations in which a cell of size b is rescaled to a cell of size b'. The results of these PSRGS improve both as b increases and as b/b'+ 1. In addition, we construct a true infinitesimal PSRG transformation, which appears to become exact as the dimensionality d approaches 1; we obtain the closed-form expression for the correlation length exponent, U = (d - l)/(d In d). The Flory formula deviates from this already at first order in (d - 1). We also develop a constant-fugacity Monte Carlo method which enables us to simu- late-in an unbiased way within the grand canonical ensemble-chains of up to lo3 bonds. With this method, we extend the PSRG to larger cells (b s 150) on the square lattice. Our numerical method provides high statistical accuracy for all cell sizes. However, in the range of b we study, the asymptotic behaviour of our results appears to depend on the choice of weight function. One weight function provides smooth behaviour as a function of b, and with it we extrapolate to find U = 0.756* 0.004. Further work is required to resolve the apparent anomalies in the results based on the other weight functions.