the true self repelling motion we construct and study a continuous real valued random process which is of a new type it is self interacting self repelling but only in a local sense it only feels the self repellance due to its occu pation time measure density in the immediate neighbourhood of the point it is just visiting we focus on the most natural process with these properties that we call true self repelling motion this is the continuous counterpart to the integer valued true self avoiding walk which had been studied among others by the first author one of the striking properties of true self repelling motion is that although the coupleoxt occupation time measure of x at time tu is a continuous markov process x is not driven by a stochastic diaerential equation and is not a semi martingale it turns out for instance that it has a finite variation of order 3 2 which contrasts with the finite quadratic variation of semi martingales one of the key tools in the construction of x is a continuous system of coalescing brownian motions similar to those that have been constructed by arratia a1 a2 we derive various properties of x existence and properties of the occupation time densities ltoxu local variation etc and an identity that shows that the dynamics of x can be very loosely speaking described as follows dxt is equal to the gradient in space of ltoxu in a gener alized sense even though x7 ltoxu is not diaerentiable

no more than three favorite sites for simple random walk we prove that with probability 1 eventually there are no more than three favorite i e most visited sites of simple symmetric random walks this partially answers a relatively longstanding question of erdos and revesz

on the speed of a cookie random walk we consider the model of the one dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite we give a criterion to decide whether the limiting speed of the walk is non zero in particular we show that a positive speed may be obtained for just three cookies per site we also prove a result on the continuity of the speed with respect to the initial cookie distribution

monotonicity for excited random walk in high dimensions we prove that the drift d for excited random walk in dimension d is monotone in the excitement parameter beta in 0 1 when d is sufficiently large we give an explicit criterion for monotonicity involving random walk green s functions and use rigorous numerical upper bounds provided by hara private communication 2007 to verify the criterion for d 9

reinforced random walk leta i i 1 be a sequence of nonnegative numbers difine a nearest neighbor random motion overrightarrow x x 0 x 1 on the integers as follows initially the weight of each interval i i 1 i an integer equals 1 if at timen an interval i i 1 has been crossed exactlyk times by the motion its weight is 1 sum limits j 1 k a j given x 0 x 1 x n i0 i1 in the probability thatx n 1 isi n 1 ori n 1 is proportional to the weights at timen of the intervals i n 1 i n and i n iin 1 we prove that overrightarrow x either visits all integers infinitely often a s or visits a finite number of integers eventually oscillating between two adjacent integers a s and that mathop lim limits n to infty x n n 0 a s for much more general reinforcement schemes we provep overrightarrow x visits all integers infinitely often p overrightarrow x has finite range 1

the brownian web arratia arratia r 1979 ph d thesis university of wisconsin madison and unpublished work and later toth and werner toth b werner w 1998 probab theory relat fields111 375 452 constructed random processes that formally correspond to coalescing one dimensional brownian motions starting from every space time point we extend their work by constructing and characterizing what we call the brownian web as a random variable taking values in an appropriate metric space whose points are compact sets of paths this leads to general convergence criteria and in particular to convergence in distribution of coalescing random walks in the scaling limit to the brownian web

membership in constant time and almost minimum space this paper deals with the problem of storing a subset of elements from the bounded universe mathcal m 0 ldots m 1 so that membership queries can be performed efficiently in particular we introduce a data structure to represent a subset of n elements of mathcal m in a number of bits close to the information theoretic minimum b left lceil lg m choose n right rceil and use the structure to answer membership queries in constant time

multi excited random walks on integers we introduce a class of nearest neighbor integer random walks in random and non random media which includes excited random walks considered in the literature at each site the random walker has a drift to the right the strength of which depends on the environment at that site and on how often the walker has visited that site before we give exact criteria for recurrence and transience and consider the speed of the walk

continuously variable survival exponent for random walks with movable partial reflectors we study a one dimensional lattice random walk with an absorbing boundary at the origin and a movable partial reflector on encountering the reflector at site x the walker is reflected with probability r to x 1 and the reflector is simultaneously pushed to x 1 iteration of the transition matrix and asymptotic analysis of the probability generating function show that the critical exponent delta governing the survival probability varies continuously between 1 2 and 1 as r varies between 0 and 1 our study suggests a mechanism for nonuniversal kinetic critical behavior observed in models with an infinite number of absorbing configurations

vertex reinforced random walk this paper considers a class of non markovian discrete time random processes on a finite state space 1 d the transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix r which is real symmetric and nonnegative lets i n keep track of the number of visits to statei up to timen and form the fractional occupation vector v n where v i n s i n mathord left vphantom s i n left sum limits j 1 d s j n right right kern nulldelimiterspace left sum limits j 1 d s j n right it is shown thatv n converges to to a set of critical points for the quadratic formh with matrixr and that under nondegeneracy conditions onr there is a finite set of points such that with probability one v n p for somep in the set there may be more than onep in this set for whichp v n p 0 on the other handp v n p 0 wheneverp fails in a strong enough sense to be maximum forh

aspects on single symbol signaling on the frequency flat rayleigh fading channel the optimal single symbol detector on the rayleigh fading channel computes a functional quadratic form in the time continuous received signal a drawback is that closed form solutions of integral equations based on the channel statistics are required this makes simplified discrete receivers attractive a class of suboptimal receivers that transforms the received random process to a set of discrete observables is derived the set of observables constitutes a random vector in a finite dimensional receiver signal space given this vector the maximum likelihood detector computes a quadratic form in the received vector the discretization implies a loss of information therefore such a detector is not in general optimal given the received time continuous signal the purpose is however to achieve close to optimal performance when the number of observables becomes large this class of detectors is analyzed using exact error probability calculations which reveal several interesting properties the length of the observation interval and the number of discrete observables have significant influences on the error probability when the time variations of the fading process are rapid compared with the symbol duration by increasing the number of observables the error floor is lowered and the implicit diversity order is increased this implicit diversity arises as soon as more than one observable per symbol interval is used and is a consequence of the information bearing signal being a random process matched filter receivers use few discrete observables per symbol interval and thus suffer from high error floors and low implicit diversity orders on fast fading channels the error probability is highly dependent on the shapes and durations of the modulator waveforms for instance pulses of long duration give lower error probabilities than shorter pulses and for a certain type of orthogonal waveforms there is no error floor

the multivariate complex normal distribution a generalization the multivariate complex normal distribution usually employed in the literature is a special case since certain restrictions have been imposed on the covariances of the real and imaginary parts of its variables a more general distribution is proposed of which the usual distribution is shown to be a special case

succinct representations of graphs for a fixed graph property q the complexity of the problem given a graph g does g have property q is usually investigated as a function of v the number of vertices in g with the assumption that the input size is polynomial in v in this paper the complexity of these problems is investigated when the input graph is given by a succinct representation by a succinct representation it is meant that the input size is polylog in v it is shown that graph problems which are approached this way become intractable actually no nontrivial problem could be found which can be solved in polynomial time the main result is characterizing a large class of graph properties for which the respective succinct problem is np hard trying to locate these problems within the p time hierarchy shows that the succinct versions of polynomially equivalent problems may not be polynomially equivalent

vertex reinforced random walk on z has finite range a stochastic process called vertex reinforced random walk vrrw is defined in pemantle ann probab 16 1229 1241 we consider this process in the case where the underlying graph is an infinite chain i e process in the case where the underlying graph is an infinite chain i e the one dimensional integer lattice we show that the range is almost surely finite that at least five points are visited infinitely often almost surely and that with positive probability the range contains exactly five points there are always points visited infinitely often but at a set of times of zero density and we show that the number of visits to such a point to time n may be asymptotically n for a dense set of values 0 1 the power law analysis relies on analysis of a related urn model

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

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

weak limits of perturbed random walks and the equation y_t b_t alpha sup y_s colon s leq t beta inf y sb s colon s leq t let and be real numbers and f c o 0 we study the existence and uniqueness of solutions g of the equation g t f t sup o 8 t g s inf o 8 t g s carmona petit le gall and yor have shown existence or nonexistence and uniqueness for some we settle the remaining cases we study the nearest neighbor walk on the integers which behaves just like fair random walk unless one neighbor has been visited and the other has not when it jumps to the unvisited neighbor with probability p if p 2 3 we show these processes scaled converge to the solution of the equation above for brownian paths with 2p 1 p

joint density for the local times of continuous time markov chains we investigate the local times of a continuous time markov chain on an arbitrary discrete state space for fixed finite range of the markov chain we derive an explicit formula for the joint density of all local times on the range at any fixed time we use standard tools from the theory of stochastic processes and finite dimensional complex calculus we apply this formula in the following directions 1 we derive large deviation upper estimates for the normalized local times beyond the exponential scale 2 we derive the upper bound in varadhan s lemma for any measurable functional of the local times and 3 we derive large deviation upper bounds for continuous time simple random walk on large subboxes of z d tending to z d as time diverges we finally discuss the relation of our density formula to the ray knight theorem for continuous time simple random walk on z which is analogous to the well known ray knight description of brownian local times

successive approximations to solutions of stochastic differential equations abstract in the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions we note that lemma 3 in this paper is a generalization of gard s lemma and guarantees the existence of functions which satisfy the conditions of theorems 2 and 3 in this paper respectively theorem 3 includes as a special case a generalization of yamada s theorem which is proved by the method of the successive approximations

self avoiding random walk a brownian motion model with local time drift a natural model for a self avoiding brownian motion inr d when specialised and simplified tod 1 becomes the stochastic differential equation x t b t int limits 0 t g x s l s x s ds where l t x t 0 x r is the local time process ofx thoughx is not markovian an analogue of the ray knight theorem holds for l x x r which allows one to prove in many cases of interest that mathop lim limits t to infty x t t exists almost surely and to identify the limit

a fast general methodology for information theoretically optimal encodings of graphs we propose a fast methodology for encoding graphs with information theoretically minimum numbers of bits specifically a graph with property pi is called a em pi graph if pi satisfies certain properties then an n node m edge pi graph g can be encoded by a binary string x such that 1 g and x can be obtained from each other in o n log n time and 2 x has at most beta n o beta n bits for any continuous superadditive function beta n so that there are at most 2 beta n o beta n distinct n node pi graphs the methodology is applicable to general classes of graphs this paper focuses on planar graphs examples of such pi include all conjunctions over the following groups of properties 1 g is a planar graph or a plane graph 2 g is directed or undirected 3 g is triangulated triconnected biconnected merely connected or not required to be connected 4 the nodes of g are labeled with labels from 1 ldots ell 1 for ell 1 leq n 5 the edges of g are labeled with labels from 1 ldots ell 2 for ell 2 leq m and 6 each node respectively edge of g has at most ell 3 o 1 self loops respectively ell 4 o 1 multiple edges moreover ell 3 and ell 4 are not required to be o 1 for the cases of pi being a plane triangulation these examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs other than rooted trees with known polynomial time information theoretically optimal coding schemes

on a general many dimensional excited random walk in this paper we study a substantial generalization of the model of excited random walk introduced in electron commun probab 8 2003 86 92 by benjamini and wilson we consider a discrete time stochastic process xn n 0 1 2 taking values on zd d 2 described as follows when the particle visits a site for the first time it has a uniformly positive drift in a given direction l when the particle is at a site which was already visited before it has zero drift assuming uniform ellipticity and that the jumps of the process are uniformly bounded we prove that the process is ballistic in the direction l so that lim infn xn ln 0 a key ingredient in the proof of this result is an estimate on the probability that the process visits less than n1 2 distinct sites by time n where is some positive number depending on the parameters of the model this approach completely avoids the use of tan points and coupling methods specific to the excited random walk furthermore we apply this technique to prove that the excited random walk in an i i d random environment satisfies a ballistic law of large numbers and a central limit theorem

some extensions of the arc sine law as partial consequences of the scaling property of brownian motion the scaling property of brownian motion is exploited systematically in order to extend paul levy s arc sine law to brownian motion perturbed by its local time at 0 other important ingredients of the proofs are some ray knight theorems for local times

linear time compression of 2 manifold polygon meshes into information theoretically optimal number of bits the polygon mesh is an important primitive used in representing 3d models its specification consists of topological and geometrical quantities where the former describes the connection nodes edges and faces and the latter describes attributes such as node positions face colors etc our interest here lies with the encoding of the topology specifically this paper presents a linear time information theoretically optimal compression algorithm for encoding a planar 2 manifold polygon mesh to a bit string x

isoperimetric inequalities and markov chains soit a i j i j n la matrice stochastique d une chaine de markov reversible sur n 1 2 soit n 2 tel que f r c f d f c 0 n ou r 2n n 2 ou c 0 est independant de f et ou r indique la norme dans l r n d pour la mesure i i alors on a sup i j j 1 a i j 0 n 2

weak convergence of the localized disturbance flow to the coalescing brownian flow we define a new state space for the coalescing brownian flow also known as the brownian web on the circle the elements of this space are families of order preserving maps of the circle depending continuously on two time parameters and having a certain weak flow property the space is equipped with a complete separable metric a larger state space allowing jumps in time is also introduced and equipped with a skorokhod type metric also complete and separable we prove that the coalescing brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps each jump arising from a small localized disturbance of the circle a local version of this result is also obtained in which the weak limit law is that of the coalescing brownian flow on the line our set up is well adapted to time reversal and our weak limit result provides a new proof of time reversibility of the coalescing brownian flow we also identify a martingale associated with the coalescing brownian flow on the circle and use this to make a direct calculation of the laplace transform of the time to complete coalescence

self repelling walk on the sierpinski gasket we construct a one parameter family of self repelling processes on the sierpinski gasket by taking continuum limits of self repelling walks on the pre sierpinski gaskets we prove that our model interpolates between the brownian motion and the self avoiding process on the sierpinski gasket namely we prove that the process is continuous in the parameter in the sense of convergence in law and that the order of holder continuity of the sample paths is also continuous in the parameter we also establish a law of the iterated logarithm for the self repelling process finally we show that this approach yields a new class of one dimensional self repelling processes

succinct representation of balanced parentheses and static trees we consider the implementation of abstract data types for the static objects binary tree rooted ordered tree and a balanced sequence of parentheses our representations use an amount of space within a lower order term of the information theoretic minimum and support in constant time a richer set of navigational operations than has previously been considered in similar work in the case of binary trees for instance we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned the approach is applied to produce a succinct representation of planar graphs in which one can test adjacency in constant time

a law of large numbers for random walks in random environment we derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of kalikow the approach is based on the existence of a renewal structure under an assumption of transience in the direction l this extends to a multidimensional context previous work of kesten our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution

ballistic behavior in a 1d weakly self avoiding walk with decaying energy penalty we consider a weakly self avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp t s p wheret ands are the times at which the common site is visited andp is a parameter we prove that ifp 3 2 then the walk should have diffusive behavior the proof and the heuristic argument make use of a real space renormalization group transformation

regular edge labeling of 4 connected plane graphs and its applications in graph drawing problems abstract in this paper we extend the concept of the regular edge labeling for general plane graphs and for triconnected triangulated plane graphs to 4 connected triangulated plane graphs we present two different linear time algorithms for constructing such a labeling by using regular edge labeling we present a new linear time algorithm for constructing rectangular dual of planar graphs our algorithm is simpler than previously known algorithms the coordinates of the rectangular dual constructed by our algorithm are integers while the one constructed by known algorithms are real numbers our second regular edge labeling algorithm is based on canonical ordering of 4 connected triangulated plane graphs by using this technique we present a new algorithm for constructing visibility representation of 4 connected planar graphs our algorithm reduces the size of the representation by a factor of 2 for such graphs

enlacements du mouvement brownien autour des courbes de l espace limit theorems are proved for the winding numbers of a three dimensional brownian motion around certain curves in space in particular the joint asymptotic distribution of the winding numbers around two curves is obtained this joint distribution generalizes the asymptotic law of the winding numbers of a planar brownian motion around two points which has recently been given by pitman and yor the limiting distributions are closely related to the time spent by a linear brownian motion above and below a multiple of its maximum process proofs rely on stochastic calculus for continuous semi martingales

excited brownian motions as limits of excited random walks we obtain the convergence in law of a sequence of excited also called cookies random walks toward an excited brownian motion this last process is a continuous semi martingale whose drift is a function say of its local time it was introduced by norris rogers and williams as a simplified version of brownian polymers and then recently further studied by the authors to get our results we need to renormalize together the sequence of cookies the time and the space in a convenient way the proof follows a general approach already taken by toth and his coauthors in multiple occasions which goes through ray knight type results namely we first prove when is bounded and lipschitz that the convergence holds at the level of the local time processes this is done via a careful study of the transition kernel of an auxiliary markov chain which describes the local time at a given level then we prove a tightness result and deduce the convergence at the level of the full processes

brownian motion and random walk perturbed at extrema let bt be brownian motion we show there is a unique adapted process xt which satisfies dxt dbt except when xt is at a maximum or a minimum when it receives a push the magnitudes and directions of the pushes being the parameters of the process for some ranges of the parameters this is already known we show that if a random walk close to bt is perturbed properly its paths are close to those of xt

the brownian web characterization and convergence the brownian web bw is the random network formally consist ing of the paths of coalescing one dimensional brownian motions start ing from every space time point in r r we extend the earlier work of arratia and of toth and werner by providing characterization and convergence results for the bw distribution including convergence of the system of all coalescing random walks to the bw under diffusive space time scaling we also provide characterization and convergence results for the double brownian web which combines the bw with its dual process of coalescing brownian motions moving backwards in time with forward and backward paths reflecting off each other for the bw deterministic space time points are almost surely of type 0 1 zero paths into the point from the past and exactly one path out of the point to the future we determine the hausdorff dimension for all types that actually occur dimension 2 for type 0 1 3 2 for 1 1 and 0 2 1 for 1 2 and 0 for 2 1 and 0 3

the true self avoiding walk with bond repulsion on z limit theorems the true self avoiding walk with bond repulsion is a nearest neighbor random walk on z for which the probability of jumping along a bond of the lattice is proportional to exp g number of previous jumps along that bond first we prove a limit theorem for the distribution of the local time process of this walk using this result later we prove a local limit theorem as a for the distribution of a 2 3 x s a where s a is a random time distributed geometrically with mean e s a 1 e s a 1 a s 0 1 as a by product we also obtain an apparently new identity related to brownian excursions and bessel bridges

limiting behaviour of the occupation of wedges by complex brownian motion we prove a theorem which gives the lim inf behaviour ast tends to 0 for the amount of time a complex brownian motion spends in a wedge with apex at the origin the result is then shown to hold uniformly for all wedges a s

perturbation of singular equilibria of hyperbolic two component systems a universal hydrodynamic limit we consider one dimensional locally finite interacting particle systems with two conservation laws which under the eulerian hydrodynamic limit lead to two by two systems of conservation laws t x u 0 t u x u 0

asymptotic behavior of brownian polymers we consider a system that models the shape of a growing polymer our basic problem concerns the asymptotic behavior ofx t the location of the end of the polymer at timet we obtain bounds onx t in the physically uninteresting case thatd 1 and the interaction functionf x 0 if in addition f x behaves for largex likecx with 1 we obtain a strong law that gives the exact growth rate

coalescing brownian flows a new approach the coalescing brownian flow on r is a process which was introduced by arratia coalescing brownian motions on the line 1979 univ wisconsin madison and toth and werner probab theory related fields 111 1998 375 452 and which formally corresponds to starting coalescing brownian motions from every space time point we provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks this result holds under a finite variance assumption and is thus optimal in previous works by fontes et al ann probab 32 2004 2857 2883 newman et al electron j probab 10 2005 21 60 the topology and state space required a moment of order 3 e for this convergence to hold the proof relies crucially on recent work of schramm and smirnov on scaling limits of critical percolation in the plane our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work in particular similar results are obtained in the case of coalescing brownian motions on the sierpinski gasket this is the first such result where the limiting paths do not enjoy the noncrossing property

continuous time vertex reinforced jump processes we study the continuous time integer valued process which jumps to each of its two nearest neighbors at the rate of one plus the total time the process has previously spent at that neighbor we show that the proportion of the time before t which this process spends at integers j converges to positive random variables v j which sum to one and whose joint distribution is explicitly described we also show

exploration and trapping of mortal random walkers exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature and yet a finite lifetime is a frequent occurrence and its effects on a number of random walk properties may be profound for instance whereas the average number of distinct sites visited by an immortal walker grows with time without bound that of a mortal walker may depending on dimensionality and rate of evanescence remain finite or keep growing with the passage of time this number can in turn be used to calculate other classic quantities such as the survival probability of a target surrounded by diffusing traps if the traps are immortal the survival probability will vanish with increasing time however if the traps are evanescent the target may be spared a certain death we analytically calculate a number of basic and broadly used quantities for evanescent random walkers

diffusive limits for true or myopic self avoiding random walks and self repellent brownian polymers in d 3 the problems considered in the present paper have their roots in two different cultures the true or myopic self avoiding walk model tsaw was introduced in the physics literature by amit et al phys rev b 27 1635 1645 1983 this is a nearest neighbor non markovian random walk in mathbb z d which prefers to jump to those neighbors which were less visited in the past the self repelling brownian polymer model srbp initiated in the probabilistic literature by durrett and rogers probab theory relat fields 92 337 349 1992 independently of the physics community is the continuous space time counterpart a diffusion in mathbb r d pushed by the negative gradient of the mollified occupation time measure of the process in both cases similar long memory effects are caused by a path wise self repellency of the trajectories due to a push by the negative gradient of softened local time we investigate the asymptotic behaviour of tsaw and srbp in the non recurrent dimensions first we identify a natural stationary in time and ergodic distribution of the environment the local time profile as seen from the moving particle the main results are diffusive limits in the case of tsaw for a wide class of self interaction functions we establish diffusive lower and upper bounds for the displacement and for a particular more restricted class of interactions we prove full clt for the finite dimensional distributions of the displacement in the case of srbp we prove full clt without restrictions on the interaction functions these results settle part of the conjectures based on non rigorous renormalization group arguments equally valid for the tsaw and srbp cases in amit et al 1983 the proof of the clt follows the non reversible version of kipnis varadhan theory on the way to the proof we slightly weaken the so called graded sector condition

first passage properties of mortal random walks ballistic behavior effective reduction of dimensionality and scaling functions for hierarchical graphs we consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites the configuration comprising the graph the starting point of the walk and the locations of the trap sites is taken to be exactly self similar as one goes from one generation of the family to the next under these circumstances the total probability that the walker hits a trap is determined exactly as a function of the single step survival probability q of the mortal walker on the nth generation graph of the family this probability is shown to be given by the nth iterate of a certain scaling function or map q f q the properties of the map then determine in each case the behavior of the trapping probability the mean time to trapping the temporal scaling factor governing the random walk dimension on the graph and other related properties the formalism is illustrated for the cases of a linear hierarchical lattice and the sierpinski graphs in two and three euclidean dimensions we find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint the relevance of this finding for experiments involving travel times of particles in diffusion decay systems is discussed

generalized ray knight theory and limit theorems for self interacting random walks on mathbb z 1 we consider non markovian self interacting random walks sirw on the one dimensional integer lattice the walk starts from the origin and at each step jumps to a neighboring site the probability of jumping along a bond is proportional to number of previous jumps along that lattice bond where n r is a monotone weight function exponential and subexponential weight functions were considered in earlier papers in the present paper we consider weight functions with polynomial asymptotics these weight functions define variants of the reinforced random walk we prove functional limit theorems for the local time processes of these random walks and local limit theorems for the position of the random walker at late times a generalization of the ray knight theory of local time arises

doubly perturbed jump diffusion processes in this paper we study the existence and uniqueness of the solution to a class of doubly perturbed jump diffusion processes with non lipschitz coefficients some well known results are improved and generalized

true self avoiding walks with generalized bond repulsion on z we consider a nearest neighbor random walk on for which the probability of jumping along a bond of the lattice is proportional to exp g number of previous jumps along that bond k withg 0 k 0 1 after a review of earlier results obtained for the casek 1 we outline the generalizations fork 0 1 obtaining a whole range of anomalous diffusion limits

excited random walk against a wall we analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex a k a excited random walk we show that it is recurrent with an expected number of returns of sqrt log t

role of depletion on the dynamics of a diffusing forager we study the dynamics of a starving random walk in general spatial dimension d this model represents an idealized description for the fate of an unaware forager whose motion is not affected by the presence or absence of resources the forager depletes its environment by consuming resources and dies if it wanders too long without finding food in the exactly solvable case of one dimension we explicitly derive the average lifetime of the walk and the distribution for the number of distinct sites visited by the walk at the instant of starvation we also give a heuristic derivation for the averages of these two quantities we tackle the complex but ecologically relevant case of two dimensions by an approximation in which the depleted zone is assumed to always be circular and which grows incrementally each time the walk reaches the edge of this zone within this framework we derive a lower bound for the scaling of the average lifetime and number of distinct sites visited at starvation we also determine the asymptotic distribution of the number of distinct sites visited at starvation finally we solve the case of high spatial dimensions within a mean field approach

iterative diversity detection for correlated continuous time rayleigh fading channels an antenna array is proposed as a means of achieving a space diversity effect that partly overcomes the severity of continuous time rayleigh fading channels the investigated channel is assumed to be frequency nonselective with correlated diversity links where the correlation is related to the array geometry and the spatial and doppler dispersions further the error performance is improved by bit interleaving and channel coding where the encoders channel is viewed as a serially concatenated system a convolutional code constitutes the outer code whereas a differential encoder and the fading channel having truncated memory form a joint inner code in order to obtain a practical detector structure it is desirable to perform iterative decoding by applying some a posteriori probability app algorithms for this purpose we propose a novel generalization of the well known bahl cocke jelinek raviv 1974 algorithm that calculates the apps over channels having memory numerical results indicate that iterative decoding becomes more powerful when the exploited channel memory depth is extended also the error performance is significantly improved by introducing multiple antennas the interleaver gain is however seen to be quite moderate in contrast to additive white gaussian noise channels

on antenna array receiver principles for space time selective rayleigh fading channels optimum in the sense of minimum error probability single symbol diversity detection for fading noisy channels is too complex for practical implementation a simplified near optimum array receiver is proposed which is based on the statistics i e the covariance function matrix of the fading channel this detector is then analyzed by calculating the exact error probability when there is a spread of the direction of arrival of the incident radio waves the proposed detector significantly gains over an adaptive antenna array which forms a weighted sum of the received antenna signals also for this adaptive array a fundamental difference between maximum likelihood weights and least mean square weights is observed

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

orderly spanning trees with applications we introduce and study orderly spanning trees of plane graphs this algorithmic tool generalizes canonical orderings which exist only for triconnected plane graphs although not every plane graph admits an orderly spanning tree we provide an algorithm to compute an orderly pair for any connected planar graph g consisting of an embedded planar graph h isomorphic to g and an orderly spanning tree of h we also present several applications of orderly spanning trees 1 a new constructive proof for schnyder s realizer theorem 2 the first algorithm for computing an area optimal 2 visibility drawing of a planar graph and 3 the most compact known encoding of a planar graph with o 1 time query support all algorithms in this paper run in linear time

scaling limit of the recurrent biased random walk on a galton watson tree we show that the trace of the null recurrent biased random walk on a galton watson tree properly renormalized converges to the brownian forest our result extends to the setting of the random walk in random environment on a galton watson tree

the excited random walk in one dimension 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 p 3 4 where the walk is transient including a mean displacement that grows as t with 1 dependent on p and a breakdown of scaling for the probability distribution of the walk

superdiffusive bounds on self repellent brownian polymers and diffusion in the curl of the gaussian free field in d 2 we consider two models of random diffusion in random environment in two dimensions the first example is the self repelling brownian polymer this describes a diffusion pushed by the negative gradient of its own occupation time measure local time the second example is a diffusion in a fixed random environment given by the curl of massless gaussian free field

coalescing systems of non brownian particles a well known result of arratia shows that one can make rigorous the notion of starting an independent brownian motion at every point of an arbitrary closed subset of the real line and then building a set valued process by requiring particles to coalesce when they collide arratia noted that the value of this process will be almost surely a locally finite set at all positive times and a finite set almost surely if the initial value is compact the key to both of these facts is the observation that because of the topology of the real line and the continuity of brownian sample paths at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them we investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous we give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of brownian motions on the sierpinski gasket and stable processes on the real line with stable index greater than one

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

doubly perturbed neutral stochastic functional equations in this paper we prove the existence and uniqueness of the solution to a class of doubly perturbed neutral stochastic functional equations dpnsfes in short under some non lipschitz conditions the solution is constructed by successive approximation furthermore we give the continuous dependence of the solution on the initial value by means of the corollary of bihari inequality

on floor plan of plane graphs a floor plan is a rectangle partitioned into a set of disjoint rectilinear polygonal regions called modules a floor plan f represents a plane graph g as follows each vertex of g corresponds to a module of f and two vertices are adjacent in g iff their corresponding modules share a common boundary floor plans find applications in vlsi chip design if a module m is a union of k disjoint rectangles m is called a k rectangle module it was shown in k h yeap and m sarrafzadeh siam j comput 22 1993 pp 500 526 that every triangulated plane graph g has a floor plan using 1 2 and 3 rectangle modules in this paper we present a simple linear time algorithm that constructs a floor plan for g using only 1 and 2 rectangle modules

optimally frugal foraging we introduce the frugal foraging model in which a forager performs a discrete time random walk on a lattice in which each site initially contains mathcal s food units the forager metabolizes one unit of food at each step and starves to death when it last ate mathcal s steps in the past whenever the forager eats it consumes all food at its current site and this site remains empty forever no food replenishment the crucial property of the forager is that it is frugal and eats only when encountering food within at most k steps of starvation we compute the average lifetime analytically as a function of the frugality threshold and show that there exists an optimal strategy namely an optimal frugality threshold k that maximizes the forager lifetime

reinforced brownian motion a prototype we analyze the brownian motion limit of a prototypical unit step reinforced random walk on the half line a reinfoced random walk is one which changes the weight of any edge or vertex visited to increase the frequency of return visits the generating function for the discrete case is first derived for the joint probability distribution of s n the location of the walker at the n th step and a n the maximum location the walker achieved in n steps then the bulk of the analysis concerns the statistics of the limiting brownian walker and of its environment both parametrized by the amplitude delta of the reinforcement the walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement details depending very much on the value of 1 delta infty

the sde solved by local times of a brownian excursion or bridge derived from the height profile of a random tree or forest let b be a standard one dimensional brownian motion started at 0 let l t v b be the occupation density of b at level v up to time t the distribution of the process of local times l t v b v 0 conditionally given b 0 and l t 0 b l is shown to be that of the unique strong solution x of the ito sde dx v 4 x 2 a t 0 x v du 1 dv 2 x v dbv on the interval 0 v t x where v t x inf v f v 0 x u du t and x v 0 for all v v t x this conditioned form of the ray knight description of brownian local times arises from study of the asymptotic distribution as n and 2k n l of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points the sde is the continuous analog of a simple description of a galton watson branching process conditioned on its total progeny for l 0 corresponding to asymptotics of a uniform random tree the sde gives a description of the process of local times of a brownian excursion which is equivalent to jeulin s description of these local times as a time change of twice a brownian excursion another corollary is the biane yor description of the local times of a reflecting brownian ridge as a time changed reversal of twice a brownian meander of the same length

on self repellent one dimensional random walks we consider an ordinary one dimensional recurrent random walk on a self repellent random walk is defined by changing the ordinary law of the random walk in the following way a path gets a new relative weight by multiplying the old one with a factor 1 for every self intersection of the path 0 1 is a parameter

the brownian net the standard brownian web is a collection of coalescing one dimensional brownian mo tions starting from each point in space and time it arises as the diffusive scaling limit of a collection of coalescing random walks we show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability we call the limiting object the brownian net and study some of its elementary properties

repeated bond traversal probabilities for the simple random walk we consider the average number bm t of bonds traversed exactly m times by a t step simple random walk we determine bm t explicitly in the scaling limit t with m t fixed in dimension d 1 and m log t fixed in dimension d 2 the scaling function is an erfc in d 1 and an exponential in d 2

continuum sites stepping stone models coalescing exchangeable partitions and random trees analogues of stepping stone models are considered where the sitespace is continuous the migration process is a general markov process and the type space is infinite such processes were defined in previous work of the second author by specifying a feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing markov processes an alternative representation is obtained here in terms of a limit of interacting particle systems it is shown that under a mild condition on the migration process the continuum sites stepping stone process has continuous sample paths the case when the migration process is brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating brownian motion this duality relation is also used to show that a tree like random compact metric space that is naturally associated to an in nite family of coalescing brownian motions on the circle has hausdorff and packing dimension both almost surely equal to 1 2 and moreover this space is capacity equivalent to the middle 1 2 cantor set and hence also to the brownian zero set

perturbed brownian motions we study perturbed brownian motions that can be loosely speaking described as follows they behave exactly as linear brownian motion except when they hit their past maximum or and maximum where they get an extra push we define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain natural class of processes in the case where both perturbations at the maximum and at the minimum are self repelling we show that in fact more is true such a process can almost surely be constructed from brownian paths by a one to one measurable transformation this generalizes some results of carmona petit yor and davis we also derive some fine properties of perturbed brownian motions hausdorff dimension of points of monotonicity for example

a survey of one dimensional random polymers in the last decade there has been an enormous progress in the mathematical understanding of one dimensional polymer measures which are path measures that suppress self intersections we are currently in the situation that many interesting questions have either been answered or that essential new ideas are needed in this survey paper we discuss the most relevant results open questions and heuristics

once edge reinforced random walk on a tree we consider a nearest neighbor walk on a regular tree with transition probabilities proportional to weights or conductances of the edges initially all edges have weight 1 and the weight of an edge is increased to c 1 when the edge is traversed for the first time after such a change the weight of an edge stays at c forever we show that such a walk is transient for all values of c ge 1 and that the walk moves off to infinity at a linear rate we also prove an invariance principle for the height of the walk

scaling behavior for random walks with memory of the largest distance from the origin we study a one dimensional random walk with memory the behavior of the walker is modified with respect to the simple symmetric random walk only when he or she is at the maximum distance ever reached from his or her starting point home in this case having the choice to move farther or to move closer the walker decides with different probabilities if the probability of a forward step is higher then the probability of a backward step the walker is bold otherwise he or she is timorous we investigate the asymptotic properties of this bold timorous random walk showing that the scaling behavior varies continuously from subdiffusive timorous to superdiffusive bold the scaling exponents are fully determined with a new mathematical approach based on a decomposition of the dynamics in active journeys the walker is at the maximum distance and lazy journeys the walker is not at the maximum distance

pathwise uniqueness for perturbed versions of brownian motion and reflected brownian motion any solution of the functional equation wta bta asup st wsa 1 l w t 0 where b is a brownian motion behaves like a reflected brownian motion except when it attains a new maximum we call it an a per turbed reflected brownian motion similarly any solution of xta bta asup st xsa binft xs t 0 behaves like a brownian motion except when it attains a new maxi mum or minimum we call it an a b doubly perturbed brownian motion we complete some recent investigations by showing that for all permissible values of the parameters a a and b respectively these equations have pathwise unique solutions and these are adapted to the filtration of b