an l2 quotient algorithm for finitely presented groups abstract the paper develops algorithmic methods to enumerate all normal subgroups of a finitely presented group such that the factor groups are either isomorphic to psl 2 p n or to pgl 2 p n the case of two generators is treated in detail a range of examples starting with the free group on two generators and ending with groups having only finitely many normal subgroups of this type is discussed

the groups gm n p abstract we determine except for two outstanding cases for which values of m n and p the group g m n p defined by the presentation a b c a 2 b 2 a b 2 c 2 a c m b c n a b c p is finite these groups formed one of three families of groups first studied systematically by coxeter in 1939

covering of a finite lattice by a random walk abstract we study the covering process by a simple random walk of a d dimensional periodic hypercubic lattice of n sites in d 1 the probability l n x for site x to be the last site visited in this covering process does not depend on x as long as x is not the starting point of the walk we argue that in dimensions d 2 the probability l n x approaches a constant value according to a coulomb law l n x 1 2 1 const x d 2 valid for x small on the scale n 1 d whereas it behaves logarithmically in d 2 also there is a dimension dependent characteristic time scale on which the last site is visited the structure of the set of sites not yet visited on this characteristic time scale is fractal like in d 2 in d 3 on the other hand this set is essentially distributed randomly through the lattice

survival probability of a particle in a sea of mobile traps a tale of tails we study the long time tails of the survival probability p t of an a particle diffusing in d dimensional media in the presence of a concentration p of traps b that move subdiffusively such that the mean square displacement of each trap grows as t with 0 1 starting from a continuous time random walk description of the motion of the particle and of the traps we derive lower and upper bounds for p t and show that for 2 d 2 these bounds coincide asymptotically thus determining asymptotically exact results the asymptotic decay law in this regime is exactly that obtained for immobile traps this means that for sufficiently subdiffusive traps the moving a particle sees the traps as essentially immobile and lifshitz or trapping tails remain unchanged for 2 d 2 and d 2 the upper and lower bounds again coincide leading to a decay law equal to that of a stationary particle thus in this regime the moving traps see the particle as essentially immobile for d 2 however the upper and lower bounds in this regime no longer coincide and the decay law for the survival probability of the a particle remains ambiguous

effective permeability of sandstone shale reservoirs by a random walk method a random walk algorithm for calculating the effective permeability of random mixtures of two finite permeability components e g sandstone and shale is introduced it is an extension of the ant in a labyrinth algorithm which only applies when one of the components has zero permeability numerical results obtained from the proposed random walk algorithm are shown to be in close agreement with those obtained from the conventional finite difference method on three dimensional cubic lattices in contrast it is shown that the hybrid random walk algorithms which have been used in the past to study the critical behaviour of two component random conductor mixtures are inadequate for the problem under consideration the proposed random walk algorithm provides a computationally efficient method for calculating the effective permeability and does not require the explicit solution of the fluid flow through the random mixture

on the survival probability of a random walk in a finite lattice with a single trap we consider the survival of a random walker in a finite lattice with periodic boundary conditions the initial position of the random walker is uniformly distributed on the lattice with respect to the trap we show that the survival of a random walker un can be exactly related to the expected number of distinct sites visted on a trap free lattice by un 1 sn nd wherend is the number of lattice points ind dimensions we then analyze the behavior of sn in any number of dimensions by using tauberian methods we find that at sufficiently long times sn decays exponentially withn in all numbers of dimensions ind 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forn2 n 1 whend 1 andn lnn n 1 whend 2 no such crossover exists when z 3 the form of suggests that the single trap approximation is indeed a valid low concentration limit for survival on an infinite lattice with a finite concentration of traps

conductivity exponents from the analysis of series expansions for random resistor networks there has been considerable controversy in recent years over the value of the conductivity exponent t this exponent can be deduced from series expansions via the scaling relations t zeta d 2 nu where zeta is deduced from differences between the exponents of the resistive gamma r percolative gamma p and conductive gamma c susceptibility the author finds that allowance for non analytic confluent corrections to scaling and the use of recent pc estimates leads to estimates for gamma r gamma p and gamma c that are somewhat different to those of fisch and harris 1978 however the differences between these exponents do not change significantly moreover the change in accepted estimates of nu in the last five years cancels some of this remaining discrepancy and she concludes using the relation zeta gamma r gamma p that t 1 31 d 2 t 2 04 d 3 t 2 39 d 4 t 2 72 d 5 with an error of about or 0 10 in each case the d 2 estimate is in significantly better agreement with those of other methods than that of fisch and harris

evanescent continuous time random walks we study how an evanescence process affects the number of distinct sites visited by a continuous time random walker in one dimension we distinguish two very different cases namely when evanescence can only occur concurrently with a jump and when evanescence can occur at any time the first is characteristic of trapping processes on a lattice whereas the second is associated with spontaneous death processes such as radioactive decay in both of these situations we consider three different forms of the waiting time distribution between jumps namely exponential long tailed and ultraslow

characterization and quantification of the cluster hopping mechanism responsible for transport in two component random networks a cluster hopping mechanism for transport in two component random networks below pc is proposed and shown to account for observed conductivity exponents key structures forming paths of least resistance are identified a novel computationally efficient method for determining exponents results

narrow escape problem with a mixed trap and the effect of orientation we consider the mean first passage time mfpt of a two dimensional diffusing particle to a small trap with a distribution of absorbing and reflecting sections high order asymptotic formulas for the mfpt and the fundamental eigenvalue of the laplacian are derived which extend previously obtained results and show how the orientation of the trap affects the mean time to capture we obtain a simple geometric condition which gives the optimal trap alignment in terms of the gradient of the regular part of a regular part of a green s function and a certain alignment vector we find that subdividing the absorbing portions of the trap reduces the mean first passage time of the diffusing particle in the scenario where the trap undergoes prescribed motion in the domain the mfpt is seen to be particularly sensitive to the orientation of the trap

first encounters on combs we consider two random walkers embedded in a finite two dimension comb and we study the mean first encounter time mfet evidencing mainly numerically different scalings with the linear size of the underlying network according to the initial position of the walkers if one of the two players is not allowed to move then the first encounter problem can be recast into a first passage problem mfpt for which we also obtain exact results for different initial configurations by comparing mfet and mfpt we are able to figure out possible search strategies and in particular we show that letting one player be fixed can be convenient to speed up the search as long as we can finely control the initial setting while for a random setting on average letting one player rest would slow down the search

writing projective representations over subfields let g x be an absolutely irreducible subgroup of gl d k and let f be a proper subfield of the finite field k we present a practical algorithm to decide constructively whether or not g is conjugate to a subgroup of gl d f k where k denotes the centre of gl d k if the derived group of g also acts absolutely irreducibly then the algorithm is las vegas and costs o x d3 d2log f arithmetic operations in k this work forms part of a recognition project based on aschbacher s classification of maximal subgroups of gl d k

on coxeter s families of group presentations abstract in 1939 coxeter published three infinite families of group presentations he studied their properties in particular determining when groups defined by members of the families are infinite and the structure of finite ones eight presentations remained for which the finiteness question was unsettled we show that two of these eight presentations define finite groups for which we give comprehensive proofs and provide detailed structural information and that two of the presentations define infinite groups our results rely on substantial amounts of computer calculations in particular on coset enumeration to prove finiteness and on computation of automatic structures using knuth bendix rewriting to prove infiniteness

optimal intermittent search strategies smelling the prey we study the kinetics of the search of a single fixed target by a searcher walker that performs an intermittent random walk characterized by different states of motion in addition we assume that the walker has the ability to detect the scent left by the prey target in its surroundings our results in agreement with intuition indicate that the prey s survival probability could be strongly reduced increased if the predator is attracted or repelled by the trace left by the prey we have also found that for a positive trace the predator is guided towards the prey increasing the inhomogeneity s size reduces the prey s survival probability while the optimal value of the parameter that regulates intermittency ceases to exist the agreement between theory and numerical simulations is excellent

survival probability of a diffusing particle in the presence of poisson distributed mobile traps the problem of a diffusing particle moving among diffusing traps is analyzed in general space dimension d we consider the case where the traps are initially randomly distributed in space with uniform density ensuremath rho and derive upper and lower bounds for the probability q t averaged over all particle and trap trajectories that the particle survives up to time t we show that for 1l dl 2 the bounds converge asymptotically to give q t ensuremath sim mathrm exp ensuremath ensuremath lambda d t d 2 for 1l dl2 where ensuremath lambda d 2 ensuremath pi d mathrm sin ensuremath pi d 2 4 ensuremath pi d d 2 ensuremath rho and d is the diffusion constant of the traps and that q t ensuremath sim mathrm exp ensuremath 4 ensuremath pi ensuremath rho dt mathrm ln t for d 2 for dg2 bounds can still be derived but they no longer converge for large t for 1l dl 2 these asymptotic forms are independent of the diffusion constant of the particle the results are compared with simulation results obtained using a new algorithm v mehra and p grassberger phys rev e 65 050101 2002 which is described in detail deviations from the predicted asymptotic forms are found to be large even for very small values of q t indicating slowly decaying corrections whose form is consistent with the bounds we also present results in d 1 for the case where the trap densities on either side of the particle are different for this case we can still obtain exact bounds but they no longer converge

diffusion in superconducting percolation networks number of sites visited two models of diffusion in superconducting percolation networks are studied on a one dimensional system the authors use chains of 5 105 sites and study the scaling properties of the mean squared displacement of termites which execute random walks on the network mean number of distinct sites visited and mean number of visits to the origin they find that although these quantities can be described by well defined scaling laws the values of the critical exponents appear to depend on the time interval in which the walks are studied they observe at least four different scaling regimes the implications of these results for higher dimensional systems are discussed

on the joint residence time of n independent two dimensional brownian motions we study the behaviour of several joint residence times of n independent brownian particles in a disc of radius r in two dimensions we consider i the time tn t spent by all n particles simultaneously in the disc within the time interval 0 t ii the time t m n t which at least m out of n particles spend together in the disc within the time interval 0 t and iii the time m n t which exactly m out of n particles spend together in the disc within the time interval 0 t we obtain very simple exact expressions for the expectations of these three residence times in the limit t

some remarks on the critical behaviour of superconducting percolation networks recent relations that relate the exponent s of superconducting percolating networks to other percolation exponents are discussed the author analyses the available exact and numerical results for s to gain insight into the possible relation between s and the geometrical exponents of percolation and the structure of superconducting percolation networks the results are given geometrical interpretation the author also discusses the random walk statistics of the termite which executes a random walk on the superconducting percolation networks in particular the author proposes an expression for the mean number of distinct sites visited by the termite and interpret it in terms of the statistics of a random walk with a transition probability whose variance is infinite

algorithms determining finite simple images of finitely presented groups we address the question for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection we prove both positive and negative results for a collection of finite simple groups that contains infinitely many alternating groups or contains classical groups of unbounded dimensions we prove that there is no such algorithm on the other hand for families of simple groups of lie type of bounded rank we obtain positive results for example for any fixed untwisted lie type x there is an algorithm that determines whether or not any given finitely presented group has simple images of the form x q for infinitely many q and if there are finitely many the algorithm determines them

pascal principle for diffusion controlled trapping reactions in this paper we analyze the long time behavior of the survival probability p a t of an a particle that performs lattice random walk in the presence of randomly moving traps b we show that for both perfect and imperfect trapping reactions for arbitrary spatial dimension d and for a rather general class of random walks p a t is less than or equal to the survival probability of an immobile target a in the presence of randomly moving traps

group chase and escape with conversion from targets to chasers abstract we study the effect of converting caught targets into new chasers in the context of the recently proposed group chase and escape problem numerical simulations have shown that this conversion can substantially reduce the lifetimes of the targets when a large number of them are initially present at the same time it also leads to a non monotonic dependence on the initial number of targets resulting in the existence of a maximum lifetime as a counter effect for this conversion we further introduce self multiplying abilities to the targets we found that the longest lifetime exists when a suitable combination of these two effects is created

optimal search strategies of space time coupled random walkers with finite lifetimes we present a simple paradigm for detection of an immobile target by a space time coupled random walker with a finite lifetime the motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed we call these walkers mortal creepers a mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate m while still alive the creeper has a finite mean frequency of change of the direction of motion in particular we consider the efficiency of the target search process characterized by the probability that the creeper will eventually detect the target analytic results confirmed by numerical results show that there is an m dependent optimal frequency opt that maximizes the probability of eventual target detection we work primarily in one dimensional d 1 domains and examine the role of initial conditions and of finite domain sizes numerical results in d 2 domains confirm the existence of an optimal frequency of change of direction thereby suggesting that the observed effects are robust to changes in dimensionality in the d 1 case explicit expressions for the probability of target detection in the long time limit are given in the case of an infinite domain we compute the detection probability for arbitrary times and study its early and late time behavior we further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time we also consider a version of the standard target problem in which many creepers start at random locations at the same time

lattice theory of trapping reactions with mobile species we present a stochastic lattice theory describing the kinetic behavior of trapping reactions a stackrel ensuremath rightarrow b b in which both the a and b particles perform an independent stochastic motion on a regular hypercubic lattice upon an encounter of an a particle with any of the b particles a is annihilated with a finite probability finite reaction rate is taken into account by introducing a set of two state random variables char22 gates imposed on each b particle such that an open closed gate corresponds to a reactive passive state we evaluate here a formal expression describing the time evolution of the a particle survival probability which generalizes our previous results we prove that for quite a general class of random motion of the species involved in the reaction process for infinite or finite number of traps and for any time t the a particle survival probability is always larger in the case when a stays immobile than in situations when it moves

link indices of hurwitz maps abstract let m be a regular map of type m n with automorphism group a smooth quotient of the ordinary 2 m n triangle group then by recent work of melekoglu and singerman we can associate with m some positive integers called the link indices of m the number of link indices is always one two or three depending on the parity of m and n in the case where m and n are both odd which happens when m is a hurwitz map of type 3 7 there is a unique link index in this paper we prove that every positive integer divisible by 2 or 3 is the link index of some hurwitz map we also find the link indices of all hurwitz maps of genus up to 10000 and of all the hurwitz maps with automorphism group psl 2 q for some q 700 and we show that every integer between 2 and 360 is the link index of some hurwitz map

kinetics of stochastically gated diffusion limited reactions and geometry of random walk trajectories in this paper we study the kinetics of diffusion limited pseudo first order a stackrel ensuremath rightarrow b b reactions in situations in which the particles intrinsic reactivities are not constant but vary randomly in time that is we suppose that the particles are bearing gates which fluctuate in time randomly and independently of each other between two states char22 an active state when the reaction may take place between a and b particles appearing in close contact and a blocked state when the reaction is completely inhibited we focus here on two customary limiting cases of pseudo first order reactions char22 the so called target annihilation and the rosenstock trapping model char22 and consider four different particular models such that the a particle can be either mobile or immobile or gated or ungated and ungated or gated b particles can be fixed at random positions or move randomly all models are formulated on a d dimensional regular lattice and we suppose that the mobile species perform independent homogeneous discrete time lattice random walks the model involving a single immobile ungated target a and a concentration of mobile gated b particles is solved exactly for the remaining three models we determine exactly in the form of rigorous lower and upper bounds showing the same n dependence the large n asymptotical behavior of the probability that the a particle survives until the n mathrm th step we also realize that for all four models studied here the a particle survival probability can be interpreted as the moment generating function of some functionals of random walk trajectories such as e g the number of self intersections the number of sites visited exactly a given number of times the residence time on a random array of lattice sites etc our results thus apply to the asymptotic behavior of corresponding generating functions which are not known as yet

first passage time moments of asymmetric levy flights we investigate the first passage dynamics of symmetric and asymmetric l evy flights in a semi infinite and bounded intervals by solving the space fractional diffusion equation we analyse the fractional order moments of the first passage time probability density function for different values of the index of stability and the skewness parameter a comparison with results using the langevin approach to l evy flights is presented for the semi infinite domain in certain special cases analytic results are derived explicitly and in bounded intervals a general analytical expression for the mean first passage time of l evy flights with arbitrary skewness is presented these results are complemented with extensive numerical analyses

transport in one dimensional random resistor superconductor mixtures with random distribution of resistor strength the authors study the problem of transport in linear random resistor superconductor mixtures with a random distribution of resistor strength the superconductors with a concentration p are represented in the model as short circuits the resistor concentration is 1 p and their conductivity distribution is p sigma approximately sigma alpha alpha 0 the specific conductivity scales with the linear size l of the system as l alpha 1 alpha 1 p 1 1 alpha the mean square displacement x2 of a random walker in this system scales as x2 2 alpha 1 alpha 2 approximately 1 p 2 alpha 1 alpha t in the presence of a bias field they obtain x 1 1 alpha approximately 1 p 1 1 alpha t they present an exact enumeration method to study diffusion on those systems the numerical results confirm the above scaling relations

diffusion on percolation clusters at criticality the concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters the authors find from monte carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three dimensional space is d 3 3 or 0 1 where the size of the cluster is restricted to be larger than the span of the walk and is d 3 9 or 0 1 for a walk on clusters not subject to this restriction for two dimensional space they find d approximately d approximately 2 7 or p0 1 the exponent d and d is related to the scaling of the average length r of n steps via rd varies as n the fracton dimensionality which is related to the density of states was found to be 1 26 or 0 1 these results are in good agreement with the predictions of alexander and orbach 1982

first passage time statistics for two channel diffusion we present rigorous results for the mean first passage time and first passage time statistics for two channel markov additive diffusion in a 3 dimensional spherical domain inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes which is shown to effect a non trivial first passage behaviour we also address the scenario of intermittent immobilisation in both cases we prove that despite the perfectly non recurrent motion of two channel markov additive diffusion in 3 dimensions the first passage statistics at long times do not display poisson like behaviour if none of the phases has a vanishing diffusion coefficient this stands in stark contrast to the standard one channel markov diffusion counterpart we also discuss the relevance of our results in the context of cellular signalling

termite and ant diffusions in the d dimensional lattice trapping model the coherent medium approximation of odagaki and lax 1981 is generalised to the trapping model the frequency dependent diffusion constant in the d dimensional hypercubic lattice is studied when the jump rate obeys a bimodal distribution the coherent medium approximation gives the correct static diffusion constant the imaginary part of the ac part of the diffusion constant vanishes linearly in frequency omega when d 2 as omega ln omega when d 2 and as omega d 2 when d 4 as omega 2 ln omega when d 4 and as omega d 2 when d 4 the termite limit is studied by taking the limit that one of the two jump rates probability p becomes infinite in the termite diffusion the static diffusion constant is critical at p 1 and the critical exponents are the same as those for the termite diffusion in the hopping model the ant or ant lion limit is defined by the limit of one jump rate being zero the imaginary and real parts of the ac diffusion constant vanish linearly and quadratically in frequency respectively the critical exponent of the leading real part of the diffusion constant is one less than the corresponding exponent for the ant diffusion in the hopping model below the percolation threshold while the leading imaginary part has the same critical exponent as those in the hopping model

c groups of psl 2 q and pgl 2 q abstract we classify the c group representations of the groups psl 2 q and pgl 2 q we obtain that the c rank of psl 2 q and pgl 2 q is 3 except when q 7 9 11 19 31 for psl 2 q and when q 5 for pgl 2 q in which case it is 4 we provide all representations of rank four

first exit time of termites and random super normal conductor networks the resistance of a random super normal conductor network is calculated through the diffusion of a termite which performs a random walk on the normal bonds and has the same probability to exist from any point on a cluster of superconducting bonds the first time t that the termite exits at a distance r from the origin averaged over many configurations and runs is found to behave as t approximately r2k for r xi pc p s describes the divergence of the conductivity as p to pc p being the concentration of the superconductor scaling arguments are used to show that k 1 s 2 theta where 2 theta is the fractal dimension of random walks on single clusters at pc numerical simulations at two dimensions d 2 yields s 1 34 or 0 10 and k 1 34 or 0 03 in agreement with scaling the authors also show that the probability of return to the origin at time t behaves as t dk2 preliminary results at d 1 and other calculations methods are also discussed

trapping reactions with randomly moving traps exact asymptotic results for compact exploration trapping a1b b and recombination a1b 0 reactions tr and rr involving randomly moving a and b particles which react when they meet at a certain distance b are ubiquitous in nature a few stray examples include quenching of delocalized excitations coagulation recombination of radicals charge carriers or defects or biological processes related to population survival 1 in recent years there has been much interest in the longtime behavior of these processes following a remarkable discovery 2 10 of many particle effects which induce essential departures from the conventionally expected behavior 1 a pronounced deviation from the text book predictions was found for the diffusion controlled rr in case when initially the particles of the a and b species are all distributed at random with strictly equal mean densities n0 it has been first shown 2 and subsequently proven 3 4 that as t the mean density n t follows n t a d n 0 dt 2d 4 where d is the space dimensionality a d is a constant and d5d a 1db is the sum of particles diffusion coefficients this law contradicts the decay law obtained within the smoluchowski approach sa n t 1 f bd t 1 where as t

survival of an evasive prey abstract we study the survival of a prey that is hunted by n predators the predators perform independent random walks on a square lattice with v sites and start a direct chase whenever the prey appears within their sighting range the prey is caught when a predator jumps to the site occupied by the prey we analyze the efficacy of a lazy minimal effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range otherwise the prey stays still we show that if the sighting range of such a lazy prey is equal to 1 lattice spacing at least 3 predators are needed in order to catch the prey on a square lattice in this situation we establish a simple asymptotic relation lnp ev t n v 2lnp imm t between the survival probabilities of an evasive and an immobile prey hence when the density n v of the predators is low 1 the lazy evasion strategy leads to the spectacular increase of the survival probability we also argue that a short sighting prey its sighting range is smaller than the sighting range of the predators undergoes an effective superdiffusive motion as a result of its encounters with the predators whereas a far sighting prey performs a diffusive type motion

an l2 quotient algorithm for finitely presented groups on arbitrarily many generators abstract we generalize the plesken fabianska l 2 quotient algorithm for finitely presented groups on two or three generators to allow an arbitrary number of generators the main difficulty lies in a constructive description of the invariant ring of gl 2 k on m copies of sl 2 k by simultaneous conjugation by giving this description we generalize and simplify some of the known results in invariant theory an implementation of the algorithm is available in the computer algebra system magma

a single predator charging a herd of prey effects of self volume and predator prey decision making we study the degree of success of a single predator hunting a herd of prey on a two dimensional square lattice landscape we explicitly consider the self volume of the prey restraining their dynamics on the lattice the movement of both predator and prey is chosen to include an intelligent decision making step based on their respective sighting ranges the radius in which they can detect the other species prey cannot recognise each other besides the self volume interaction after spotting each other the motion of prey and predator turns from a nearest neighbour random walk into directed escape or chase respectively we consider a large range of prey densities and sighting ranges and compute the mean first passage time for a predator to catch a prey as well as characterise the effective dynamics of the hunted prey we find that the prey s sighting range dominates their life expectancy and the predator profits more from a bad eyesight of the prey than from his own good eye sight we characterise the dynamics in terms of the mean distance between the predator and the nearest prey it turns out that effectively the dynamics of this distance coordinate can be captured in terms of a simple ornstein uhlenbeck picture reducing the many body problem to a simple two body problem by imagining predator and nearest prey to be connected by an effective hookean bond all features of the model such as prey density and sighting ranges merge into the effective binding constant

computing matrix group decompositions with respect to a normal subgroup abstract we describe an algorithm which can be used to investigate whether a matrix group defined over a finite field decomposes with respect to a normal subgroup defined as the normal closure of a given set of matrices the possible decompositions correspond to classes in aschbacher s classification of subgroups of the general linear group

order statistics for first passage times in diffusion processes we consider the problem of the first passage times for absorption trapping of the firstj j 1 2 ofk j 1 our results are a special case of the theory of order statistics we show that in one dimension the mean time to absorption at a boundary for the first ofk diffusing particles 1 k goes as lnk 1 for the set of initial conditions in which none of thek particles is located at a boundary and goes ask 2 for the set of initial conditions in which some of thek particles may be located at the boundary we demonstrate that in one dimension our asymptotic results k21 are independent of the potential field in which the diffusion takes place for a wide class of potentials we conjecture that our results are independent of dimension and produce some evidence supporting this conjecture we conclude with a discussion of the possible import of these results on diffusion controlled rate processes

transport in a two component randomly composite material scaling theory and computer simulations of termite diffusion near the superconducting limit the question of transport in a random two component mixture is addressed to this end two models are precisely formulated that effectively extend to two component mixtures the de gennes ant model of one component systems we develop and test a scaling theory and we solve some of the problems associated with the limit where one of the components is superconducting our work provides the first practical realisation of the termite model of a random superconducting network which performs a normal walk in the normal regions of the material but performs a new and unique form of random walk in the superconducting regions we find that the divergence of the electrical conductivity at the percolation threshold can be described by this random walk and that the critical exponent s is given by s 1 3 f 0 1 in d 2 if we perturb away from the pure superconduct ing limit we find that the electrical properties can still be described by the random walk model with a crossover exponent i 0 4 d 2 moreover we find that the diffusion constant in this region is described by a scaling form so that data can be made to collapse upon a single curve whose form is governed by the exponents s and i

occupancy of a single site by many random walkers we consider an infinite number of noninteracting lattice random walkers with the goal of determining statistical properties of the time out of a total time t that a single site has been occupied by n random walkers initially the random walkers are assumed uniformly distributed on the lattice except for the target site at the origin which is unoccupied the random walk model is taken to be a continuous time random walk and the pausing time density at the target site is allowed to differ from the pausing time density at other sites we calculate the dependence of the mean time of occupancy by n random walkers as a function of n and the observation time t we also find the variance for the cumulative time during which the site is unoccupied the large t behavior of the variance differs according as the random walk is transient or recurrent it is shown that the variance is proportional to t at large t in three or more dimensions it is proportional to t 3 2 in one dimension and to t ln t in two dimensions

the shadow principle an optimal survival strategy for a prey chased by random predators abstract we consider a lattice model of the annihilation process a b b when a mobile prey a is chased by identical independent predators b performing random motions until one of them finds a and destroys it it is assumed that each predator follows some most probable trajectory around which it performs a random motion it is shown that if the random motion of the predators satisfies certain conditions the prey a can maximize its survival probability by following a specific trajectory which mimics the preferred trajectories of the predators we call this optimal trajectory as the shadow of the predator this is an extension of the so called pascal principle studied in the recent literature we discuss the conditions which allow for such extensions and give examples where they are realized

the random normal superconductor mixture in one dimension two models for random two conductor mixtures by diffusion processes are considered in the one dimensional case it is shown that the two models are not in the same universality class this is seen to be an artefact of systems with pc 1 however a scaling theory proposed earlier is tested for this simple one dimensional case and confirmed

hopping conduction in insulator conductor superconductor mixtures the coherent medium approximation is applied to study the frequency dependent diffusion constant in two and three dimensional ant termite like mixtures where three types of jump rates normal metallic superconducting and insulating are distributed randomly the dc diffusion constant vanishes when p r pc where p and r are the probabilities that a given bond is metallic and superconducting respectively and z is the coordination number of the lattice pc rc identical to 2 z in the low frequency limit the real and imaginary parts of the ac diffusion constant go as a z d p r f omega and b z d p r g omega respectively a behaviour similar to the pure ant limit the behaviour of f omega and g omega is examined below at and above the percolation thresholds in both two and three dimensions the coefficients a and b of these leading terms diverge as r to rc superconducting region percolates indicating enhancement in the dielectric constant