on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

time regularity for random walks on locally compact groups let g be a compactly generated locally compact group and let t be the operator of convolution with a probability measure on g our main results give sufficient conditions on for the operator t to be analytic in l p g 1 p where analyticity means that one has an estimate of form vert i t t n vert leq cn 1 for all n 1 2 in l p operator norm counterexamples show that analyticity may not hold if some of the conditions are not satisfied

a loop reversibility and subdiffusion of the rotor router walk the rotor router model on a graph describes a discrete time walk accompanied by the deterministic evolution of configurations of rotors randomly placed on vertices of the graph we prove the following property if at some moment of time the rotors form a closed clockwise contour on the planar graph then the clockwise rotations of rotors generate a walk which enters into the contour at some vertex v performs a number of steps inside the contour so that the contour formed by rotors becomes anti clockwise and then leaves the contour at the same vertex v this property generalizes the previously proved theorem for the case when the rotor configuration inside the contour is a cycle rooted spanning tree and all rotors inside the contour perform a full rotation we use the proven property for an analysis of the sub diffusive behavior of the rotor router walk

structure of metastable states in the hopfield model an upper bound for the number of metastable states in the hopfield model is calculated as a function of the hamming fraction from an input pattern for all finite values of alpha the ratio of number of patterns to nodes the hamming fraction from the input pattern to the nearest metastable state is infinite when alpha 0 113 the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the hamming fraction 0 5 from it

competition between hopfield and symmetry transform interactions in a neural net the authors consider a neural network proposed by coolen and kuijk 1989 that has interactions composed of two competing elements the first is the hopfield interaction for recall of a set of stored patterns the second is interactions between pairs of sites arranged so that the configuration undergoes a symmetry transformation at each parallel update first they consider the effect of the symmetry transform interactions alone they show that for sequential updating the symmetry transformation is no longer carried out faithfully but rather the spin configuration tends to a symmetry invariant in order to understand how the retrieval phase of the hopfield model is disrupted by the symmetry transform interactions they perform a replica symmetric analysis they demonstrate that the symmetry transform interactions generate a noise very similar to that of random external fields on the memory states the phase diagram suggests the possibility of symmetry invariant recognition for an extensive number of patterns and an optimal value for the symmetry transform interaction strength they present numerical simulations of the model under parallel dynamics to confirm these predictions

heat kernel estimates and riesz transforms on some riemannian covering manifolds consider a riemannian manifold m which is a galois covering of a compact manifold with nilpotent deck transformation group g for the laplace operator on m we prove a precise estimate for the gradient of the heat kernel and show that the riesz transforms are bounded in l p m 1 p we also obtain estimates for discrete oscillations of the heat kernel and boundedness of discrete riesz transform operators which are defined using the action of g on m

polynomial growth harmonic functions on groups of polynomial volume growth we consider harmonic functions of polynomial growth of some order d on cayley graphs of discrete groups of polynomial volume growth of order d w r t the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions more precisely the dimension of this space of harmonic functions is at most of order d d 1 as in the already known riemannian case this estimate is polynomial in the growth degree more generally our techniques also apply to graphs roughly isometric to cayley graphs of groups of polynomial volume growth

random walks on discrete groups of polynomial volume growth let mu be a probability measure with finite support on a discrete group gamma of polynomial volume growth the main purpose of this paper is to study the asymptotic behavior of the convolution powers mu n of mu if mu is centered then we prove upper and lower gaussian estimates we prove a central limit theorem and we give a generalization of the berry esseen theorem these results also extend to noncentered probability measures we study the associated riesz transform operators the main tool is a parabolic harnack inequality for centered probability measures which is proved by using ideas from homogenization theory and by adapting the method of krylov and safonov this inequality implies that the positive mu harmonic functions are constant finally we give a characterization of the mu harmonic functions which grow polynomially

onset of super retrieval phase and enhancement of the storage capacity in neural networks of nonmonotonic neurons analogue neural networks of associative memory with continuous time dynamics are studied for nonmonotonic transfer functions using the method of self consistent signal to noise analysis the hebb learning rule with unbiased random patterns is assumed for the synaptic couplings a novel phenomenon is found to occur as a result of a phase transition concerning the property of the local field distribution in retrieval states of the newly found phase which the authors refer to as the super retrieval phase noise in the local field vanishes and the memory retrieval without errors ensures even for an extensive number of memory patterns stored under the local learning rule the storage capacity is obtained as a function of the parameter representing the degree of nonmonotonicity of the transfer functions with the result that an enhancement of the storage capacity can also occur

storage capacity for hierarchically correlated patterns the storage capacity of a hopfield model with patterns forming a two level hierarchy is calculated without reference to a special learning rule if the number of different classes becomes very large strong correlations within the classes decrease the storage capacity in striking contrast to what is known about one level hierarchies i e patterns with magnetisation

comparative study of spurious state distribution in analogue neural networks and the boltzmann machine the authors conducted a comparative study of the density distribution of metastable states in analogue neural networks and the boltzmann machine by evaluating number densities of the attractors of the networks as functions of storage capacity analogue gain or temperature and pattern overlap the analysis is based on the fact that the boltzmann machine and the analogue neural network can be described by the thouless anderson palmer equations with and without the onsager reaction term respectively they found the remarkable result that the spurious state density around spin glass equilibrium states is much larger for the boltzmann machine than for the analogue neural network for a reasonably wide range of analogue gain or temperature which leads to an expectation that the analogue neural network should possess a much better potential for memory retrieval than the boltzmann machine

solutions to complex smoothing equations we consider smoothing equations of the form begin aligned x mathop limits text law sum j ge 1 t j x j c end aligned where c t 1 t 2 ldots is a given sequence of random variables and x 1 x 2 ldots are independent copies of x and independent of the sequence c t 1 t 2 ldots the focus is on complex smoothing equations i e the case where the random variables x c t 1 t 2 ldots are complex valued but also more general multivariate smoothing equations are considered in which the t j are similarity matrices under mild assumptions on c t 1 t 2 ldots we describe the laws of all random variables x solving the above smoothing equation these are the distributions of randomly shifted and stopped levy processes satisfying a certain invariance property called u alpha stability which is related to operator semi stability the results are applied to various examples from applied probability and statistical physics

content addressability and learning in neural networks the content addressability of patterns stored in ising spin neural network models with symmetric interactions is studied numerical results from simulations on the icl distributed array processor dap involving systems with up to 2048 neurons are presented behaviour consistent with finite size scaling characteristic of a first order phase transition is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it

metastable states of neural networks incorporating the physiological dale hypothesis physiologically it is likely that excitatory and inhibitory neurons are rather clearly distinguished or in other words each neuron has in most cases a unique excitatory or inhibitory property the dale hypothesis to study the consequence of the physiological constraint the author proposes a learning rule for neural networks which incorporates the constraint then the distribution of the metastable states is calculated and it is found that the retrieval states form a much larger group in the proposed model than in the hopfield model the author also studies the process of retrieval by considering the statistical dynamics of the overlaps the result suggests that excitatory neurons and inhibitory neurons are preferably balanced in number if the attraction basins of the stored patterns are to be sufficiently large

local limit theorem for symmetric random walks in gromov hyperbolic groups completing a strategy of gouezel and lalley we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non elementary gromov hyperbolic group denoting by r the inverse of the spectral radius of the random walk the probability to return to the identity at time n behaves like c r n n 3 2 an important step in the proof is to extend ancona s results on the martin boundary up to the spectral radius we show that the martin boundary for r harmonic functions coincides with the geometric boundary of the group in an appendix we explain how the symmetry assumption of the measure can be dispensed with for surface groups

structural matching by discrete relaxation this paper describes a bayesian framework for performing relational graph matching by discrete relaxation our basic aim is to draw on this framework to provide a comparative evaluation of a number of contrasting approaches to relational matching broadly speaking there are two main aspects to this study firstly we focus on the issue of how relational inexactness may be quantified we illustrate that several popular relational distance measures can be recovered as specific limiting cases of the bayesian consistency measure the second aspect of our comparison concerns the way in which structural inexactness is controlled we investigate three different realizations of the matching process which draw on contrasting control models the main conclusion of our study is that the active process of graph editing outperforms the alternatives in terms of its ability to effectively control a large population of contaminating clutter

convergence time and finite size effects in neural networks the time needed for convergence to a stable state starting from an initial state having macroscopic overlap with a stored state is studied numerically for networks up to n 105 neurons although such systems are much larger than those traditionally used in studying neural networks they are still too small to extract reliable information about the asymptotic behaviour of the convergence time

bayesian retrieval in associative memories with storage errors it is well known that for finite sized networks one step retrieval in the autoassociative willshaw net is a suboptimal way to extract the information stored in the synapses iterative retrieval strategies are much better but have hitherto only had heuristic justification we show how they emerge naturally from considerations of probabilistic inference under conditions of noisy and partial input and a corrupted weight matrix we start from the conditional probability distribution over possible patterns for retrieval we develop two approximate but tractable iterative retrieval methods one performs maximum likelihood inference to find the single most likely pattern using the conditional probability as a lyapunov function for retrieval the second method makes a mean field assumption to optimize a tractable estimate of the full conditional probability distribution in the absence of storage errors both models become very similar to the willshaw model

irrelevance of memory in the minority game by means of extensive numerical simulations we show that all the distinctive features of the minority game introduced by challet and zhang physica a 256 514 1998 are completely independent of the memory of the agents the only crucial requirement is that all the individuals must possess the same information irrespective of whether this information is true or false

emergence of quasiunits in the one dimensional zhang model we study the zhang model of sandpile on a one dimensional chain of length l where a random amount of energy is added at a randomly chosen site at each time step we show that in spite of this randomness in the input energy the probability distribution function of energy at a site in the steady state is sharply peaked and the width of the peak decreases as l 1 2 for large l we discuss how the energy added at one time is distributed among different sites by topplings with time we relate this distribution to the time dependent probability distribution of the position of a marked grain in the one dimensional abelian model with discrete heights we argue that in the large l limit the variance of energy at site x has a scaling form l 1 g x l where g xi varies as ln 1 xi for small xi which agrees very well with the results from numerical simulations

associative memory based on parametrically coupled chaotic elements abstract in this paper we propose an associative memory system based on parametrically coupled chaotic elements the proposed system is obtained by adding a new parameter control to our previously proposed system a chaotic activity in an early association stage makes an efficient association over the memories that are stored by means of autocorrelational learning when the system successfully recalls the target memory the system s motion is dominated by a spatially coherent oscillation while unstable motions remain when the system fails to make the association in addition the system has a large memory capacity a comparison between the proposed system and an approach with a nonmonotonic transfer function is also shown

asymptotic shape of the region visited by an eulerian walker we study an eulerian walker on a square lattice starting from an initial randomly oriented background using monte carlo simulations we present evidence that for a large number of steps n the asymptotic shape of the set of sites visited by the walker is a perfect circle the radius of the circle increases as n 1 3 for large n and the width of the boundary region grows as n ensuremath alpha 3 with ensuremath alpha 0 40 ifmmode pm else textpm fi 0 06 if we introduce stochasticity in the evolution rules the mean square displacement of the walker r n 2 ensuremath sim n 2 ensuremath nu shows a crossover from the eulerian ensuremath nu 1 3 to a simple random walk ensuremath nu 1 2 behavior

random mappings of scaled graphs given a connected finite graph with a fixed base point o and some graph g with a based point we study random 1 lipschitz maps of a scaled into g we are mostly interested in the case where g is a cayley graph of some finitely generated group where the construction does not depend on the choice of base points a particular case of being a graph on two vertices and one edge corresponds to the random walk on g and the case where is a graph on two vertices and two edges joining them corresponds to brownian bridge in g we show that unlike in the case g mathbb z d the asymptotic behavior of a random scaled mapping of into g may differ significantly from the asymptotic behavior of random walks or random loops in g in particular we show that this occurs when g is a free non abelian group also we consider the case when g is a wreath product of mathbb z with a finite group to treat this case we prove new estimates for transition probabilities in such wreath products for any group g generated by a finite set s we define a functor e from category of finite connected graphs to the category of equivalence relations on such graphs given a finite connected graph the value eg s can be viewed as an asymptotic invariant of g

the number of metastable states of a simple perceptron with gradient descent learning algorithm the number of metastable states of a simple perceptron with gradient descent learning algorithm has been calculated as a function of the storage capacity alpha and the gain parameter beta for alpha not 0 and beta 0 exponential large number of local minima similar to the spin glasses and analogue attractor networks has been found

carne varopoulos bounds for centered random walks we extend the carne varopoulos upper bound on the probability transitions of a markov chain to a certain class of nonreversible processes by introducing the definition of a centering measure in the case of random walks on a group we study the connections between different notions of centering

heavy tail phenomenon and convergence to stable laws for iterated lipschitz maps we consider the markov chain x n x n 0 infty on mathbb r d defined by the stochastic recursion x n x psi theta n x n 1 x starting at x in mathbb r d where 1 2 are i i d random variables taking their values in a metric space theta mathfrak r and psi theta n mathbb r d mapsto mathbb r d are lipschitz maps assume that the markov chain has a unique stationary measure under appropriate assumptions on psi theta n we will show that the measure has a heavy tail with the exponent 0 i e nu x in mathbb r d x t asymp t alpha using this result we show that properly normalized birkhoff sums s n x sum k 1 n x k x converge in law to an stable law for alpha in 0 2

information processing in three state neural networks we introduce networks of three state neurons where the additional state embodies the absence of information their dynamical behavior is studied from the standpoint of information processing these networks display strong pattern completion capabilities moreover inference naturally occurs between coherent patterns

maximal stability in unsupervised learning a perceptron can classify a set of random patterns into two groups the maximal gap between the two groups is calculated using replica theory the replica symmetric solution is shown to be unstable and gives results which are qualitatively different from the one step replica symmetry breaking solution rsb1 the results of a calculation without the replica method are given yielding an exact upper bound for the gap which almost coincides with the rsb1 solution finding the classification of a set of patterns with a maximal gap is a difficult combinatorial optimization problem an algorithm is developed which gives large gaps

retrieval and chaos in extremely diluted q ising neural networks using a probabilistic approach the deterministic and the stochastic parallel dynamics of aq ising neural network are studied at finiteq and in the limitq exact evolution equations are presented for the first time step these formulas constitute recursion relations for the parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis of the retrieval properties is carried out in terms of the gain parameter the loading capacity and the temperature the results for theq network are compared with those for theq 3 andq 4 models possible chaotic microscopic behavior is studied using the time evolution of the distance between two network configurations for arbitrary finiteq the retrieval regime is always chaotic in the limitq the network exhibits a dynamical transition toward chaos

euler walk on a cayley tree we describe two possible regimes dynamic phases of the euler walk on a cayley tree a condensed phase and a low density phase in the condensed phase the area of visited sites grows as a compact domain in the low density phase the proportion of visited sites decreases rapidly from one generation of the tree to the next we describe in detail returns of the walker to the root and growth of the domain of visited sites in the condensed phase we also investigate the critical behaviour of the model on the line separating the two regimes

self organizing height arrow model numerical and analytical results the recently introduced self organizing height arrow ha model is studied numerically on the square lattice and analytically on the bethe lattice the concentration of occupied sites and critical exponents of distributions of avalanches are evaluated for two slightly different versions of the model the exponents found for distributions of avalanches by mass area duration and appropriate fractal dimensions are close to those for the original sandpile model which suggests that the ha model belongs to the same universality class for comparison the concentration of occupied sites in the ha model is calculated exactly on the bethe lattice with coordination number q 4

statistical mechanics of the majority game the majority game modelling a system of heterogeneous agents trying to behave in a similar way is introduced and studied using methods of statistical mechanics the stationary states of the game are given by the local minima of a particular hopfield like hamiltonian on the basis of replica symmetric calculations we draw the phase diagram which contains the analogue of a retrieval phase the number of metastable states is estimated using the annealed approximation the results are confronted with extensive numerical simulations

pattern formation in fast growing sandpiles we study the patterns formed by adding n sand grains at a single site on an initial periodic background in the abelian sandpile models and relaxing the configuration when the heights at all sites in the initial background are low enough one gets patterns showing proportionate growth with the diameter of the pattern formed growing as n 1 d for large n in d dimensions on the other hand if sites with maximum stable height in the starting configuration form an infinite cluster we get avalanches that do not stop in this paper we describe our unexpected finding of an interesting class of backgrounds in two dimensions that show an intermediate behavior for any n the avalanches are finite but the diameter of the pattern increases as n for large n with 1 2 1 different values of can be realized on different backgrounds and the patterns still show proportionate growth the non compact nature of growth simplifies their analysis significantly we characterize the asymptotic pattern exactly for one illustrative example with 1

the number of metastable states in the projection rule neural network we calculate the direct average n of the number of metastable states in the projection rule neural network by means of the saddle point method n is obtained as a function of the hamming distance g to a given pattern stability gamma and energy e the critical storage capacity of the model is calculated the examination of the energy dependence of n leads to lower bounds for the energies of the spin glass states

convergence to stable laws for a class of multidimensional stochastic recursions we consider a markov chain x n n 0 infty on mathbb r d defined by the stochastic recursion xn mn xn 1 qn where qn mn are i i d random variables taking values in the affine group a mathbb r d mathbb r d rtimes rm gl mathbb r d assume that mn takes values in the group of similarities of mathbb r d and the markov chain has a unique stationary measure which has unbounded support we denote by mn the expansion coefficient of mn and we assume mathbb e m alpha 1 for some positive we show that the partial sums s n sum k 0 n x k properly normalized converge to a normal law 2 or to an infinitely divisible law which is stable in a natural sense 2 or of the tails of and of stationary measure for an associated markov operator 2 we extend the results to the situation where mn is a random generalized similarity

continuous phase transitions for dynamical systems we study the asymptotic expansion of the topological pressure of one parameter families of potentials at a point of non analyticity the singularity is related qualitatively and quantitatively to non gaussian limit laws and to slow decay of correlations with respect to the equilibrium measure

tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case we consider the stochastic recursion x n 1 m n 1 x n q n 1 n in mathbb n where q n x n in mathbb r d mn are similarities of the euclidean space mathbb r d and qn mn are i i d we study asymptotic properties at infinity of the invariant measure for the markov chain xn under assumption mathbb e log m 0 i e in the so called critical case

eulerian walkers as a model of self organized criticality we propose a new model of self organized criticality a particle is dropped at random on a lattice and moves along directions specified by arrows at each site as it moves it changes the direction of the arrows according to fixed rules on closed graphs these walks generate euler circuits on open graphs the particle eventually leaves the system and a new particle is then added the operators corresponding to particle addition generate an abelian group same as the group for the abelian sandpile model on the graph we determine the critical steady state and some critical exponents exactly using this equivalence

stable limits for sums of dependent infinite variance random variables the aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution conditions for this convergence to hold are known in the literature however most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process in this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence we will apply our results to some standard time series models including the garch 1 1 process and its squares the stochastic volatility models and solutions to stochastic recurrence equations

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

markov chain approximations to nonsymmetric diffusions with bounded coefficients we consider a certain class of nonsymmetric markov chains and obtain heat kernel bounds and parabolic harnack inequalities using the heat kernel estimates we establish a sufficient condition for the family of markov chains to converge to nonsymmetric diffusions as an application we approximate nonsymmetric diffusions in divergence form with bounded coefficients by nonsymmetric markov chains this extends the results by stroock and zheng to the nonsymmetric divergence forms 2012 wiley periodicals inc

replica field theory for deterministic models i binary sequences with low autocorrelation we study systems without quenched disorder with a complex landscape and we use replica symmetry theory to describe them we discuss the golay bernasconi derrida approximation of the low autocorrelation model and we reconstruct it by using replica calculations we then consider the full model its low t properties with the help of number theory and a hartree fock resummation of the high temperature series we show that replica theory allows us to solve the model in the high t phase our solution is based on one link integral techniques and is based on substituting a fourier transform with a generic unitary transformation we discuss this approach as a powerful tool to describe systems with a complex landscape in the absence of quenched disorder

multivalley structure of attractor neural networks we numerically investigate the structure of the basins of attraction and the nature of the spurious attractors of the pseudo inverse and the optimal weights attractor neural networks we show that the number of attractors in the optimal weights model increases as the margin parameter increases and that the basins of attraction of the stored patterns are not significantly enlarged by moreover the number of attractors is smaller than in the pseudo inverse model

precise capacity analysis in binary networks with multiple coding level inputs we compute retrieval probabilities as a function of pattern age for networks with binary neurons and synapses updated with the simple hebbian learning model studied in amit and fusi 1994 the analysis depends on choosing a neural threshold that enables patterns to stabilize in the neural dynamics in contrast to most earlier work where selective neurons for each pattern are drawn independently with fixed probability f here we analyze the situation where f is drawn from some distribution on a range of coding levels in order to set a workable threshold in this setting it is necessary to introduce a simple inhibition in the neural dynamics whose magnitude depends on the total activity of the network proper choice of the threshold depends on the value of the covariances between the synapses for which we provide an explicit formula retrieval probabilities depend on the distribution of the fields induced by a learned pattern we show that the field induced by the first learned pattern evolves as a markov chain during subsequent learning epochs leading to a recursive formula for the distribution alternatively the distribution can be computed using a normal approximation which involves the value of the synaptic covariances capacity is computed as the sum of the retrival probabilities over all ages we show through simulation that the chosen threshold enables retrieval with asynchronous dynamics even in the presence of significant noise in the initial state of the pattern the computed probabilities with both methods are shown to be very close to probabilities estimated from simulation the analysis is extended to randomly connected networks

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

exact occupation time distribution in a non markovian sequence and its relation to spin glass models we compute exactly the distribution of the occupation time in a discrete non markovian toy sequence that appears in various physical contexts such as the diffusion processes and ising spin glass chains the non markovian property makes the results nontrivial even for this toy sequence the distribution is shown to have non gaussian tails characterized by a nontrivial large deviation function that is computed explicitly an exact mapping of this sequence to an ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model at a given temperature what is the distribution over disorder of the thermally averaged number of spins that are aligned to their local fields we show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the sherrington kirkpatrick model with gaussian disorder

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

the local limit theorem on nilpotent lie groups a local limit theorem is proven on connected simply connected nilpotent lie groups for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization the result extends an earlier local limit theorem of alexopoulos which treated absolutely continuous measures with a continuous density of compact support and also extends local limit theorems of breuillard and diaconis hough which treated general measures on the heisenberg group

domains of attraction and the density of static metastable states in single pattern iterated neural networks the author calculates for a single pattern iterated network the density of static metastable states delta eta delta m as a function of the field distribution and the symmetry of the synaptic matrix the features of this function strongly suggest that the boundary upon which this density vanishes gives the critical overlap for the memory state i e gives a measure of the size of its domain of attraction this heuristic interpretation is shown to agree with the exact result for vanishing symmetry the only case where a direct calculation can be performed he explicitly calculates critical overlaps as a function of symmetry and mean field strength when the field distribution is a delta function and when it is a unit width gaussian

tail homogeneity of stationary measures for some multidimensional stochastic recursions we consider a stochastic recursion x n 1 m n 1 x n q n 1 n in mathbb n where q n m n are i i d random variables such that q n are translations m n are similarities of the euclidean space mathbb r d and x n in mathbb r d in the present paper we show that if the recursion has a unique stationary measure with unbounded support then the weak limit of properly dilated exists and defines a homogeneous tail measure the structure of is studied and the supports of and are compared in particular we obtain a product formula for

centered densities on lie groups of polynomial volume growth we study the asymptotic behavior of the convolution powers of a centered density on a connected lie group g of polynomial volume growth the main tool is a harnack inequality which is proved by using ideas from homogenization theory and by adapting the method of krylov and safonov applying this inequality we prove that the positive harmonic functions are constant we also characterise the harmonic functions which grow polynomially we give gaussian estimates for as well as for the differences and we give estimates similar to the ones given by the classical berry esseen theorem for and we use these estimates to study the associated riesz transforms

analysis on nilpotent groups soit g un groupe de lie nilpotent reel simplement connexe et soit l son algebre de lie soient l 1 l et l k 1 l k l k 1 2 soit d d g d l k 1 kdim r l k l k 1 d est la dimension de dirichlet de g soit h un groupe nilpotent discret finiment genere on demontre des theoremes d analyse pour g

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

the number of metastable states in the generalized random orthogonal model we calculate the number of metastable states in the generalized random orthogonal model the results obtained are verified by exact numerical enumeration for small system sizes taking into account finite size effects these results are compared with those for a hopfield model in order to examine the effect of strict orthonormality of neural network patterns on the number of metastable states

mean field equations for spin models with orthogonal interaction matrices we study the metastable states in ising spin models with orthogonal interaction matrices we focus on three realizations of this model the random case and two non random cases i e the fully frustrated model on an infinite dimensional hypercube and the so called sine model we use the mean field or tap equations which we derive by resumming the high temperature expansion of the gibbs free energy in some special non random cases we can find the absolute minimum of the free energy for the random case we compute the average number of solutions to the tap equations we find that the configurational entropy or complexity is extensive in the range trsb t tm we also present an apparently unrelated replica calculation which reproduces the analytical expression for the total number of tap solutions

the cover time of deterministic random walks for general transition probabilities abstract a deterministic random walk is a deterministic process analogous to a random walk while there are some results on the cover time of the rotor router model which is a deterministic random walk corresponding to a simple random walk nothing is known about the cover time of deterministic random walks emulating general transition probabilities this paper is concerned with the shortest remaining time srt router model with multiple tokens which is a deterministic process coping with general transition probabilities possibly containing irrational numbers for the model we give an upper bound of the cover time which is the first result on the cover time of deterministic random walks for general transition probabilities our upper bound also improves the existing bounds for the rotor router model in some cases

asymptotic entropy and green speed for random walks on countable groups we study asymptotic properties of the green metric associated with transient random walks on countable groups we prove that the rate of escape of the random walk computed in the green metric equals its asymptotic entropy the proof relies on integral representations of both quantities with the extended martin kernel in the case of finitely generated groups where this result is known benjamini and peres probab theory related fields 98 1994 91 112 we give an alternative proof relying on a version of the so called fundamental inequality relating the rate of escape the entropy and the logarithmic volume growth extended to random walks with unbounded support

distribution of the number of fitness maxima in fisher s geometric model fisher s geometric model describes biological fitness landscapes by combining a linear map from the discrete space of genotypes to an n dimensional euclidean phenotype space with a nonlinear single peaked phenotype fitness map genotypes are represented by binary sequences of length l and the phenotypic effects of mutations at different sites are represented by l random vectors drawn from an isotropic gaussian distribution recent work has shown that the interplay between the genotypic and phenotypic levels gives rise to a range of different landscape topographies that can be characterised by the number of local fitness maxima extending our previous study of the mean number of local maxima here we focus on the distribution of the number of maxima when the large l limit is taken at finite n we identify the typical scale of the number of maxima for general n and determine the full scaled probability density and two point correlation function of maxima for the one dimensional case we also elaborate on the close relation of the model to the anti ferromagnetic hopfield model with n random continuous pattern vectors and show that many of our results carry over to this setting more generally we expect that our analysis can help to elucidate the fluctuation structure of metastable states in various spin glass problems

dimension free lp estimates for single riesz transforms via an h joint functional calculus abstract by using an h joint functional calculus for strongly commuting operators we derive a scheme to deduce the l p boundedness of certain d dimensional riesz transforms from the l p boundedness of appropriate one dimensional riesz transforms moreover the l p bounds we obtain are independent of the dimension the scheme is applied to riesz transforms connected with orthogonal expansions and discrete riesz transforms on products of groups with polynomial growth

replica symmetry breaking in the minority game we extend and complete recent work concerning the analytic solution of the minority game nash equilibria ne of the game have been found to be related to the ground states of a disordered hamiltonian with replica symmetry breaking rsb signalling the presence of a large number of ne here we study the number of ne both analytically and numerically we then analyse the stability of the recently obtained replica symmetric solution and in the region where it becomes unstable derive the solution within one step rsb approximation we are finally able to draw a detailed phase diagram of the model

multiconnected neural network models a generalisation of the hopfield model which includes interactions between p 2 ising spins is considered the exact storage capacity behaves as np 1 2 p 1 ln n when the number of nodes n is large in the limit p to infinity the thermodynamics of the model can be solved exactly without using the replica method at zero temperature a solution which is completely correlated with the input pattern exists for alpha alpha c where alpha c to infinity as p to infinity and this solution has lower energy than the spin glass solution if alpha alpha 1 1 4 ln 2 where the number of patterns n 2 alpha p np 1 for finite values of p the correlation with the input pattern is not complete for p 3 and 4 approximate values of alpha c and alpha 1 are obtained and for p to infinity the replica symmetric approximation gives alpha c approximately p 4 ln p

self organized random walks and stochastic sandpile from linear to branched avalanches abstract in a model of self organized criticality unstable sites discharge to just one of their neighbours for constant discharge ratio and for a certain range of values of the input energy avalanches are simple branchless polya random walks and their scaling properties can be derived exactly if fluctuates widely enough avalanches become branched due to multiple discharges and behave like those of the stochastic sandpile at the threshold for branched behaviour peculiar scaling and anomalous diffusive transport are observed

the cluster index of regularly varying sequences with applications to limit theory for functions of multivariate markov chains we introduce the cluster index of a multivariate stationary sequence and characterize the index in terms of the spectral tail process this index plays a major role in limit theory for partial sums of sequences we illustrate the use of the cluster index by characterizing infinite variance stable limit distributions and precise large deviation results for sums of multivariate functions acting on a stationary markov chain under a drift condition

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

cycles in a synchronous neural network we extend the standard calculation of the number of metastable states fixed points so as to take into account cyclic attractors as well we calculate analytically the average number of cycles in the little model as a function of the fraction of flipping spins 1 q 2 we find that the cycles of length two are by far the most common type of attractors in particular for small values of the storage parameter the inverse cycles q 1 are the dominant attractors

central limit theorems for iterated random lipschitz mappings let m be a noncompact metric space in which every closed ball is compact and let g be a semigroup of lipschitz mappings of m denote by yn n 1 a sequence of independent g valued identically distributed random variables r v s and by z an m valued r v which is independent of the r v yn n 1 we consider the markov chain zn n 0 with state space m which is defined recursively by z0 z and zn 1 yn 1zn for n 0 let be a real valued function on g m the aim of this paper is to prove central limit theorems for the sequence of r v s yn zn 1 n 1 the main hypothesis is a condition of contraction in the mean for the action on m of the mappings yn we use a spectral method based on a quasi compactness property of the transition probability of the chain mentioned above and on a special perturbation theorem

three unfinished works on the optimal storage capacity of networks the optimal storage properties of three different neural network models are studied for two of these models the architecture of the network is a perceptron with or j interactions whereas for the third model the output can be an arbitrary function of the inputs analytic bounds and numerical estimates of the optimal capacities and of the minimal fraction of errors are obtained for the first two models the third model can be solved exactly and the exact solution is compared to the bounds and to the results of numerical simulations used for the two other models