universality of high dimensional spanning forests and sandpiles we prove that the wired uniform spanning forest exhibits mean field behaviour on a very large class of graphs including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph several of our results are new even in the case of mathbb z d d ge 5 in particular we prove that every tree in the forest has spectral dimension 4 3 and walk dimension 3 almost surely and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1 2 respectively the past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest we obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth and is 1 on any bounded degree nonamenable graph we deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the abelian sandpile model are 2 and 1 2 respectively for any transitive graph with polynomial growth of dimension at least five and are 1 and 1 2 respectively for any bounded degree nonamenable graph in the case of mathbb z d d ge 5 some of our results regarding critical exponents recover earlier results of bhupatiraju et al electron j probab 22 85 51 2017 in this case we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant order multiplicative errors rather than the polylogarithmic order multiplicative errors present in that work

field theory conjecture for loop erased random walks we give evidence that the functional renormalization group frg developed to study disordered systems may provide a field theoretic description for the loop erased random walk lerw allowing to compute its fractal dimension in a systematic expansion in e 4 d up to two loop the frg agrees with rigorous bounds correctly reproduces the leading logarithmic corrections at the upper critical dimension d 4 and compares well with numerical studies we obtain the universal subleading logarithmic correction in d 4 which can be used as a further test of the conjecture

loop erased self avoiding random walk and the laplacian random walk comments that the laplacian random walk with eta 1 recently introduced by lyklema and evertsz 1986 is the same process as the loop erased self avoiding random walk analysed previously by the author 1980 some rigorous results about this process are reviewed

critical behaviour of self avoiding walk in five or more dimensions we use the lace expansion to prove that in five or more dimensions the standard self avoiding walk on the hypercubic integer lattice behaves in many respects like the simple random walk in particular it is shown that the leading asymptotic behaviour of the number of step self avoiding walks is purely exponential that the mean square displacement is asymptotically linear in the number of steps and that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion some related facts are also proved these results are optimal according to the widely believed conjecture that the self avoiding walk behaves unlike the simple random walk in dimensions two three and four

high dimensional incipient infinite clusters revisited the incipient infinite cluster iic measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite using the lace expansion we construct the iic measure for high dimensional percolation models in three different ways extending previous work by the second named author and jarai we show that each construction yields the same measure indicating that the iic is a robust object furthermore our constructions apply to spread out versions of both finite range and long range percolation models we also get estimates on structural properties of the iic such as the volume of the intersection between the iic and euclidean balls

critical exponents for the self avoiding random walk in three dimensions we compute by direct monte carlo simulation the main critical exponents 4 andv and the effective coordination number for the self avoiding random walk in three dimensions on a cubic lattice we find both hyperscaling relationsdv 2 anddv 2 4 0 satisfied ind 3

adaptive estimation for a class of nonlinearly parameterized dynamical systems the technical note presents a new technique for the adaptive parameter estimation in nonlinear parameterized dynamical systems the technique proposes an uncertainty set update approach that guarantees forward invariance of the true value of the parameters in addition it is shown that in the presence of sufficiently exciting state trajectories the parameter estimates converge to the true values and the uncertainty set vanishes around the true value of the parameters the set based approach is also shown to identify when the parameter estimates are biased two simulation examples are presented that demonstrate the effectiveness of the technique

estimation of states and parameters for linear systems with nonlinearly parameterized perturbations we consider systems that can be described by a linear part with a nonlinear perturbation where the perturbation is parameterized by a vector of unknown constant parameters under a set of technical assumptions about the perturbation and its relationship to the outputs we present a modular design technique for estimating the system states and the unknown parameters the design consists of a high gain observer that estimates the states of the system together with the full perturbation and a parameter estimator constructed by the designer to invert a nonlinear equation we illustrate the technique on a simulated dc motor with friction

the scaling limit of self avoiding random walk in high dimensions the brydges spencer lace expansion is used to prove that the scaling limit of the finite dimensional distributions of self avoiding random walk in the d dimensional cubic lattice mathbb z d is gaussian if d is sufficiently large it is also shown that the critical exponent gamma for the number of self avoiding walks is equal to 1 if d is sufficiently large

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

gaussian behavior of loop erased self avoiding random walk in four dimensions une marche auto evitante de longueur n sur le reseau entier z d est une suite ordonnee de points x 0 0 x 1 x n avec x i x i 1 1 et x i x j pour i j on etudie le comportement d une telle marche a d 4 quand on applique le processus de boucle effacee

k theorie bivariante pour les algebres de banach et conjecture de baum connes je suis parvenu dans ma these a construire une k theorie bivariante pour les algebres de banach cela m a permis de demontrer la conjecture de baum connes pour les groupes de lie semi simples et les groupes reductifs p adiques ainsi que pour les sous groupes discrets de type fini de sl 3 f avec f une extension finie de q p et pour les sous groupes discrets cocompacts de sp n 1 sl 3 r et sl 3 c j ai aussi demontre une conjecture analogue a la conjecture de baum connes pour les algebres l 1 et plus generalement toutes les bonnes completions des sous groupes fermes des groupes de lie semi simples et des groupes reductifs p adiques

a note on the critical dimension for weakly self avoiding walks we consider a weakly self avoiding t step random walk on zdwith possible nonnearest neighbor jumps we prove that for d 3 the scaling limit of the end point is cauchy distributed as t

self avoiding random walks on the hexagonal lattice we use the algorithm recently introduce by a berretti and a d sokal to compute numerically the critical exponents for the self avoiding random walk on the hexagonal lattice we find 1 3509 0 0057 0 0023v 0 7580 0 0049 0 0046 0 519 0 082 0 077 where the first error is the systematic one due to corrections to scaling and the second is the statistical error for the effective coordination number we find 1 84779 0 00006 0 0017 the results support the nienhuis conjecture 43 32 and provide a rough numerical check of the hyperscaling relationdv 2 an additional analysis taking the nienhuis value of 2 21 2 1 2 for granted gives 1 3459 0 0040 0 0008

loop erased self avoiding walks in two dimensions exact critical exponents and winding numbers abstract loop erased self avoiding walks lesaws are defined as walks resulting from sequential erasing of the loops of random walks the critical properties of lesaws are those of a c 2 conformal theory in two dimensions and geometrically those of subgraphs of spanning trees in this paper we derive the exact asymptotic behavior of the set of probabilities p k n that k k 2 lesaws of length n starting at neighbouring points do not intersect as well as the exact asymptotic winding angle distribution of a single two dimensional lesaw

intersections of random walks in four dimensions ii letf n be the probability that the paths of two simple random walks of lengthn starting at the origin in 4 have no intersection it has previously been shown thatf n c logn 1 2 here it is proved that for allr 1 2 mathop lim limits n to infty log n r f n infty

critical behavior and the limit distribution for long range oriented percolation i we consider oriented percolation on mathbb z d times mathbb z whose bond occupation probability is pd where p is the percolation parameter and d is a probability distribution on mathbb z d suppose that d x decays as x d for some 0 we prove that the two point function obeys an infrared bound which implies that various critical exponents take on their respective mean field values above the upper critical dimension d c 2 alpha wedge2 we also show that for every k the fourier transform of the normalized two point function at time n with a proper spatial scaling has a convergent subsequence to e c k alpha wedge2 for some c 0

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

intersections of random walks a direct renormalization approach various intersection probabilities of independent random walks ind dimensions are calculated analytically by a direct renormalization method adapted from polymer physics this heuristic approach based on edwards continuum model leads to a straightforward derivation and also to refinements of lawler s results for the simultaneous intersections of two walks in 4 or three walks in 3 these results are generalized top walks in d d frac 2p p 1 p 2 ford 4 an infinite set of universal critical exponents l l 1 are derived they govern the asymptotic probability z l sim s sigma l thatl star walks in d with a common origin do not intersect before times the l s are calculated up to ordero e2 whered 4 e this information is used to calculate the probability z g that a set of independent random walks in d or d d 4 respectivelyd 3 form a given topological networks g of multiple intersection points in the absence of any other double point respectively triple point this is generalized to a network in d leqq frac 2p p 1 dimension with exclusion ofp tuple points the method is quite general and can be used to calculate any critical intersection probability and provides the probabilist with a large variety of exact results yet to be proven rigorously

a new weighted metric the relative metric ii in the first part of this investigation we generalized a weighted distance function of r c li s and found necessary and sufficient conditions for it being a metric in this paper some properties of this so called m relative metric are established specifically isometries and quasiconvexity results are derived we also illustrate connections between our approach and generalizations of the hyperbolic metric

loop erased random walk on a torus in dimensions 4 and above we show that the statistics of loop erased random walks above the upper critical dimension 4 are different between the torus and the full space the typical length of the path connecting a pair of sites at distance l which scales as l2 in the full space changes under the periodic boundary conditions to ld 2 the results are precise for dimensions 5 for the dimension d 4 we prove an upper bound conjecturally sharp up to subpolyonmial factors

hyperbolicity cat 1 spaces and the ptolemy inequality using a four points inequality for the boundary of cat 1 spaces we study the relation between gromov hyperbolic spaces and cat 1 spaces

low density expansion for a two state random walk in a random environment a nearest neighbor random walk on z2 is considered where points of the lattice are labeled good or bad a particle takes a vertical step with probability ag or ab and a horizontal step with probability 1 ag or 1 ab depending on whether its present site is good or bad steps of size 1 are as probable as steps of size 1 if the good and bad sites are placed randomly with density of bad sites it is known that there exists an a such that for almost every placement of sites the random walk in the long run behaves like a homogeneous walk with vertical probability a and horizontal probability 1 a here the problem of estimating a as a function of is considered in particular when ag 1 2 ab fixed the first two terms of the expansion of a at 0 are rigorously derived

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

non coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster in this paper we study the abundance of self avoiding paths of a given length on a supercritical percolation cluster on mathbb z d more precisely we count z n the number of self avoiding paths of length n on the infinite cluster starting from the origin which we condition to be in the cluster we are interested in estimating the upper growth rate of z n limsup n rightarrow infty z n 1 n which we call the connective constant of the dilute lattice after proving that this connective constant is a s non random we focus on the two dimensional case and show that for every percolation parameter p in 1 2 1 almost surely z n grows exponentially slower than its expected value in other words we prove that limsup n rightarrow infty z n 1 n lim n rightarrow infty mathbb e z n 1 n where the expectation is taken with respect to the percolation process this result can be considered as a first mathematical attempt to understand the influence of disorder for self avoiding walks on a quenched dilute lattice our method which combines change of measure and coarse graining arguments does not rely on the specifics of percolation on mathbb z 2 so our result can be extended to a large family of two dimensional models including general self avoiding walks in a random environment

self attracting self avoiding walk this article is concerned with self avoiding walks saw on mathbb z d that are subject to a self attraction the attraction which rewards instances of adjacent parallel edges introduces difficulties that are not present in ordinary saw ueltschi has shown how to overcome these difficulties for sufficiently regular infinite range step distributions and weak self attractions ueltschi in probab theory relat fields 124 2 189 203 2002 this article considers the case of bounded step distributions for weak self attractions we show that the connective constant exists and in d ge 5 carry out a lace expansion analysis to prove the mean field behaviour of the critical two point function hereby addressing a problem posed by den hollander random polymers vol 1974 springer verlag berlin 2009

parameter estimation and compensation in systems with nonlinearly parameterized perturbations we consider a class of systems influenced by perturbations that are nonlinearly parameterized by unknown constant parameters and develop a method for estimating the unknown parameters the method applies to systems where the states are available for measurement and perturbations with the property that an exponentially stable estimate of the unknown parameters can be obtained if the whole perturbation is known the main contribution is to introduce a conceptually simple modular design that gives freedom to the designer in accomplishing the main task which is to construct an update law to asymptotically invert a nonlinear equation compensation for the perturbations in the system equations is considered for a class of systems with uniformly globally bounded solutions for which the origin is uniformly globally asymptotically stable when no perturbations are present we also consider the case when the parameters can only be estimated when the controlled state is bounded away from the origin and show that we may still be able to achieve convergence of the controlled state we illustrate the method through examples and apply it to the problem of downhole pressure estimation during oil well drilling

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

critical exponent for the loop erased self avoiding walk by monte carlo methods a monte carlo simulation was performed for loop erased self avoiding walks lesaw to ascertain the exponentv for the z2 and z3 lattices the estimated values were 2v 1 600 0 006 in two dimensions and 2v 1 232 0 008 in three dimensions leading to the conjecturev 4 5 for the two dimensional lesaw these results add to existing evidence that the loop erased self avoiding walks are not in the same universality class as self avoiding walks

proprietes d intersection des marches aleatoires i convergence vers le temps local d intersection we study intersection properties of multi dimensional random walks letx andy be two independent random walks with values in d d 3 satisfying suitable moment assumptions and letin denote the number of common points to the paths ofx andy up to timen the sequence in suitably normalized is shown to converge in distribution towards the intersection local time of two independent brownian motions results are applied to the proof of a central limit theorem for the range of a two dimensional recurrent random walk thus answering a question raised by n c jain and w e pruitt

a self avoiding walk with attractive interactions a self avoiding walk with small attractive interactions is described here the existence of the connective constant is established and the diffusive behavior is proved using the method of the lace expansion

the upper critical dimension of the abelian sandpile model the existing estimation of the upper critical dimension of the abelian sandpile model is based on a qualitative consideration of avalanches as self avoiding branching processes we find an exact representation of an avalanche as a sequence of spanning subtrees of two component spanning trees using equivalence between chemical paths on the spanning tree and loop erased random walks we reduce the problem to determination of the fractal dimension of spanning subtrees then the upper critical dimension du 4 follows from lawler s theorems for intersection probabilities of random walks and loop erased random walks

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

random subgraphs of finite graphs ii the lace expansion and the triangle condition in a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph in this paper we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph the latter is readily verified for several graphs with vertex set 0 1 r 1 n including the hamming cube on an alphabet of r letters the n cube for r 2 the n dimensional torus with nearest neighbor bonds and n sufficiently large and the n dimensional torus with n 6 and sufficiently spread out long range bonds the conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples

stable adaptive control of a class of first order nonlinearly parameterized plants stable adaptive control algorithms are designed for a class of first order nonlinearly parameterized plants containing nonlinearities similar to those arising in fermentation process models when the plant nonlinearity is conveniently parameterized and when the controller parameters are adjusted adaptively the output error is shown to converge to zero stability of the overall system is proved using the lyapunov function with a suitably chosen cubic term

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

strongly hyperbolic metrics on ptolemy spaces abstract recently strongly hyperbolic space as certain analytic enhancements of gromov hyperbolic space was introduced by b nica and j spakula in this paper we prove that the log metric log 1 d on the ptolemy space x d is a strongly hyperbolic metric using our result we construct two metrics on the ptolemy space and prove they are strongly hyperbolic

the infinite self avoiding walk in high dimensions a measure on infinite self avoiding walks is defined which is the natural limit of the uniform measure on finite self avoiding walks this limit is shown to exist in sufficiently large dimensions

the incipient infinite cluster for high dimensional unoriented percolation we consider bond percolation on mathbb z d at the critical occupation density p c for d 6 in two different models the first is the nearest neighbor model in dimension d 6 the second model is a spread out model having long range parameterized by l in dimension d 6 in the spread out case we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as x when d 6 and l is sufficiently large we also give a general criterion for this convergence to hold which is satisfied in the case d 6 in the nearest neighbor model by work of hara 12 we further give a second construction by taking p 6 the probability for invasion percolation to reach a site x is asymptotic to c x d 4 as x

the lace expansion approach to ballistic behaviour for one dimensional weakly self avoiding walks we prove ballistic behaviour in dimension one fora model of weakly self avoiding walks where loops of length m are penalized by a factor e mp with p 0 1 and sufficiently large furthermore we prove that the fluctuations around the linear drift satisfy a centrallimit theorem the proof uses a variant of the lace expansion together with an inductive analysis of the arising recursion relation

scaling for a random polymer letq n be the law of then step random walk on d obtained by weighting simple random walk with a factore for every self intersection domb joyce model of soft polymers it was proved by greven and den hollander 1993 that ind 1 and for every 0 there exist 0 1 and such that under the lawq n asn begin array l i theta beta is the lim it empirical speed of the random walk ii mu beta is the limit empirical distribution of the local times end array a representation was given for and in terms of a largest eigenvalue problem for a certain family of x matrices in the present paper we use this representation to prove the following scaling result as 0 begin array l i beta textstyle 1 over 3 theta beta to b ii beta textstyle 1 over 3 mu beta left left lceil cdot beta textstyle 1 over 3 right rceil right to l 1 eta cdot end array the limitsb 0 and are identified in terms of a sturm liouville problem which turns out to have several interesting properties the techniques that are used in the proof are functional analytic and revolve around the notion of epi convergence of functionals onl 2 our scaling result shows that the speed of soft polymers ind 1 is not right differentiable at 0 which precludes expansion techniques that have been used successfully ind 5 hara and slade 1992a b in simulations the scaling limit is seen for 10 2

convergence of self avoiding random walk to brownian motion in high dimensions it is proved that in sufficiently high dimensions the scaled self avoiding random walk on the hypercubic lattice converges in distribution to brownian motion convergence of the finite dimensional distributions was shown elsewhere in this paper tightness is shown

construction of the incipient infinite cluster for spread out oriented percolation above 4 1 dimensions we construct the incipient infinite cluster measure iic for sufficiently spread out oriented percolation on d for d 1 4 1 we consider two different constructions for the first construction we define n e by taking the probability of the intersection of an event e with the event that the origin is connected to x n d summing this probability over x d and normalising the sum to get a probability measure we let n and prove existence of a limiting measure the iic for the second construction we condition the connected cluster of the origin in critical oriented percolation to survive to time n and let n under the assumption that the critical survival probability is asymptotic to a multiple of n 1 we prove existence of a limiting measure with in addition we study the asymptotic behaviour of the size of the level set of the cluster of the origin and the dimension of the cluster of the origin under our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super brownian motion for d 1 4 1

a tree based scaling exponent for random cluster models for models of random trees in the hypercubic lattice or more general random clusters to which tree structure may be added the authors introduce a novel scaling exponent for the particular model of uniform random spanning trees in two and three dimensions its value is estimated by monte carlo simulation

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

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

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

a novel growing self avoiding walk in three dimensions the authors introduce a growing self avoiding walk in three dimensions 3d that can terminate only by returning to its point of origin this symmetric tricolour walk stw is a direct generalization of the smart kinetic walk to 3d and yet is readily simulated their monte carlo simulations strongly suggest that the fractal dimension of the stw is exactly 2 this conclusion is supported by a mapping of the stw onto a self attracting self avoiding walk at its collapse transition

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

simultaneous identification and stabilization of nonlinearly parameterized discrete time systems by nonlinear least squares algorithm this paper addresses the challenging problem of designing an adaptive feedback strategy for simultaneous identification and stabilization for a class of nonlinearly parameterized uncertain systems in discrete time the nonlinear least squares nls algorithm is applied to estimate the unknown parameters and it turns out to be the standard least squares ls algorithm whenever the model is linearly parameterized based on this algorithm both the strong consistency of the estimator and the global stability of the system are achieved with the output feedback design for the scalar parameter case

random surface correlation functions truncated pair functions for free random surface models and bernoulli ensembles are examined in both cases the pair function is shown to obey ornstein zernike scaling whenever various correlation lengths of the system satisfy a nonperturbative criterion under the same conditions the transverse displacement of surfaces contributing to the pair function is shown to be normally distributed a new type of transition which concerns the width of typical surfaces is introduced and studied whenever the system is below the melting transition temperature of a related lower dimensional model the width of typical surfaces is shown to be finite a thermodynamic formalism for free random surface models is developed the formalism is used to obtain sharp estimates of the entropy of surfaces contributing to the pair function

adaptive control of nonlinearly parameterized systems the smooth feedback case studies global adaptive control of nonlinearly parameterized systems with uncontrollable linearization using a parameter separation technique and the tool of adding a power integrator we develop a feedback domination design approach for the explicit construction of a smooth adaptive controller that solves the problem of global state regulation in contrast to the existing results in the literature a key feature of our adaptive regulator is its minimum order property namely no matter how big the number of unknown parameters is the order of the dynamic compensator is identical to one and is therefore minimal as a consequence global state regulation of feedback linearizable systems with nonlinear parameterization is achieved by one dimensional adaptive controllers without imposing any extra e g convex concave conditions on the unknown parameters

the scaling limit of loop erased random walk in three dimensions we show that the scaling limit exists and is invariant under dilations and rotations we give some tools that might be useful to show universality

polymers in a weak random potential in dimension four rigorous renormalization group analysis correlation functions of the edwards model of polymers at weak coupling are defined and studied at the critical point in dimension four by a rigorous renormalization group method which validates at any order perturbative renormalization group results on their behaviour at large distances remainders are controlled by a new argument which enlarges the use of methods of constructive field theory to models of statistical physics

adaptive control of a class of nonlinearly parameterized plants a stable adaptive control method is developed for a class of first order nonlinearly parameterized plants the method is based on modified adaptive algorithms that yield a stable system in which the output errors are guaranteed to converge to zero the stability analysis is carried out using suitably constructed lyapunov functions

decay of correlations in nearest neighbor self avoiding walk percolation lattice trees and animals we consider nearest neighbor self avoiding walk bond percolation lattice trees and bond lattice animals on z d the two point functions of these models are respectively the generating function for self avoiding walks from the origin to x z d the probability of a connection from the origin to x and the generating functions for lattice trees or lattice animals containing the origin and x using the lace expansion we prove that the two point function at the critical point is asymptotic to const x 2 d as x for d 5 for self avoiding walk for d 19 for percolation and for sufficiently large d for lattice trees and animals these results are complementary to those of ann probab 31 2003 349 408 where spread out models were considered in the course of the proof we also provide a sufficient and rather sharp if d 4 condition under which the two point function of a random walk on z d is asymptotic to const x 2 d as x

reflecting and absorbing boundary conditions on the tail of the laplacian random walk the authors introduce a new version of the laplacian random walk lrw for which the tail of the trajectory acts like a hard wall for incoming diffusing particles they show how to implement these reflecting boundary conditions through the use of a modified discrete laplace equation from an exact enumeration on the square lattice they find that reflecting boundary conditions on the tail of the trajectory give rise to a denser fractal compared with the recently studied laplacian random walk with absorbing boundary conditions

parameter convergence in nonlinearly parameterized systems a large class of problems in parameter estimation concerns nonlinearly parametrized systems over the past few years a stability framework for estimation and control of such systems has been established we address the issue of parameter convergence in such systems in this paper systems with both convex concave and general parameterizations are considered in the former case sufficient conditions are derived under which parameter estimates converge to their true values using a min max algorithm in the latter case to achieve parameter convergence a hierarchical min max algorithm is proposed where the lower level consists of a min max algorithm and the higher level component updates the bounds on the parameter region within which the unknown parameter is known to lie using this hierarchical algorithm a necessary and sufficient condition is established for global parameter convergence in systems with a general nonlinear parameterization in both cases the conditions needed are shown to be stronger than linear persistent excitation conditions that guarantee parameter convergence in linearly parametrized systems explanations and examples of these conditions and simulation results are included to illustrate the nature of these conditions a general definition of nonlinear persistent excitation that leads to parameter convergence is proposed at the end of this paper

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

the diffusion of self avoiding random walk in high dimensions we use the brydges spencer lace expansion to prove that the mean square displacement of at step strictly self avoiding random walk in thed dimensional square lattice is asymptotically of the formdt ast approaches infinity ifd is sufficiently large the diffusion constantd is greater than one

lace expansion for the ising model the lace expansion has been a powerful tool for investigating mean field behavior for various stochastic geometrical models such as self avoiding walk and percolation above their respective upper critical dimension in this paper we prove the lace expansion for the ising model that is valid for any spin spin coupling for the ferromagnetic case we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace expansion coefficients for self avoiding walk as a result we obtain gaussian asymptotics of the critical two point function for the nearest neighbor model with d gg 4 and for the spread out model with d 4 and l gg 1 without assuming reflection positivity

loop erased self avoiding random walk in two and three dimensions if n is the position of the self avoiding random walk in d obtained by erasing loops from simple random walk then it is proved that the mean square displacemente n 2 grows at least as fast as the flory predictions for the usual saw i e at least as fast asn3 2 ford 2 andn6 5 ford 3 in particular if the mean square displacement of the usual saw grows liken1 18 ind 3 as expected then the loop erased process is in a different universality class

a central limit theorem for a one dimensional polymer measure let s n n n 0 be a random walk on the integers having bounded steps the self repellent resp self avoiding walk is a sequence of transformed path measures which discourage resp forbid self intersections this is used as a model for polymers previously we proved a law of large numbers that is we showed the convergence of s n n toward a positive number under the polymer measure the present paper proves a classical central limit theorem for the self repellent and self avoiding walks that is we prove the asymptotic normality of s n n n for large n the proof refines and continues results and techniques developed previously

rigorous exponent inequalities for random walks the exponents for the resistance of a random walk path and the random walk on a random walk problem are related to a number of other exponents for random walks some rigorous inequalities for these exponents are then established

the triangle condition for percolation aizenman and newman introduced an unverified condition the triangle condition which has been shown to imply that a number of percolation critical exponents take their mean field values and which is expected to hold above six dimensions for nearest neighbour percolation we prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model and above six dimensions for a class of spread out models the proof uses an expansion which is related to the lace expansion for self avoiding walk percolation is a simple probabilistic model for which many interesting problems remain unsolved the basic objects in percolation are random graphs on an infinite lattice typically there is a critical density for the graph edges known as bonds below which there is zero probability that a fixed point in the lattice is part of an infinite connected graph but above which this probability is strictly positive this abrupt change in behaviour has been used in chemistry and statistical physics to model phase transitions in a variety of physical systems such as fluid flow through a porous medium hence the name percolation random resistor networks and the gelation of branched polymers for an introduction see 15 20 to define the models we are interested in we consider the of dimensional integer lattice z whose elements are referred to as sites and the set of pairs b x y of distinct sites the bonds to each bond b is associated an independent bernoulli random variable nb9 with nb 1 with probability p jb and nb 0 with probability p jh the jb are fixed and z invariant and p is a parameter the nearest neighbour model is defined by taking jb 1 for those b x y such that x y i 1 and jb 0 otherwise expectation with respect to the joint distribution of the nb is denoted by if nb 1 we say b is occupied and otherwise b is vacant given a bond configuration nb two sites x and y are said to be connected if there is a path from x to y which consists of occupied bonds the set of sites which are connected to x is called the connected cluster of x and is denoted c x the probability that x is connected to y is written tp x y 1 ip x y i x and y are connected i y e c x p where i denotes the indicator function denoting by c x the number of sites in c x the expected cluster size or susceptibility is given by

borel type bounds for the self avoiding walk connective constant let be the self avoiding walk connective constant on we show that the asymptotic expansion for c 1 in powers of 1 2d satisfies borel type bounds this supports the conjecture that the expansion is borel summable

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

the number and size of branched polymers in high dimensions we consider two models of branched polymers lattice trees on thed dimensional hypercubic lattice i the nearest neighbor model in sufficiently high dimensions and ii a spread out or long range model ford 8 in which trees are constructed from bonds of length less than or equal to a large parameterl we prove that for either model the critical exponent for the number of branched polymers exists and equals 5 2 and that the critical exponentv for the radius of gyration exists and equals 1 4 this improves our earlier results for the corresponding generating functions the proof uses the lace expansion together with an analysis involving fractional derivatives which has been applied previously to the self avoiding walk in a similar context

generalized approach to the non backtracking lace expansion the lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self avoiding walk percolation and lattice trees and animals the non backtracking lace expansion noble is a modification that allows us to improve its applicability in the nearest neighbor setting on the mathbb z d lattice for percolation lattice trees and lattice animals the noble gives rise to a recursive formula that we study in this paper at a general level we state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound in two related papers we show that these conditions are satisfied for percolation in d ge 11 for lattice trees in d ge 16 and for lattice animals in d ge 18

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper

critical two point functions and the lace expansion for spread out high dimensional percolation and related models we consider spread out models of self avoiding walk bond percolation lattice trees and bond lattice animals on mathbb z d having long finite range connections above their upper critical dimensions d 4 self avoiding walk d 6 percolation and d 8 trees and animals the two point functions for these models are respectively the generating function for self avoiding walks from the origin to x in mathbb z d the probability of a connection from 0 to x and the generating function for lattice trees or lattice animals containing 0 and x we use the lace expansion to prove that for sufficiently spread out models above the upper critical dimension the two point function of each model decays at the critical point as a multiple of x 2 d as x to infty we use a new unified method to prove convergence of the lace expansion the method is based on x space methods rather than the fourier transform our results also yield unified and simplified proofs of the bubble condition for self avoiding walk the triangle condition for percolation and the square condition for lattice trees and lattice animals for sufficiently spread out models above the upper critical dimension

logarithmic correction for the susceptibility of the 4 dimensional weakly self avoiding walk a renormalisation group analysis we prove that the susceptibility of the continuous time weakly self avoiding walk on mathbb z d in the critical dimension d 4 has a logarithmic correction to mean field scaling behaviour as the critical point is approached with exponent frac 1 4 for the logarithm the susceptibility has been well understood previously for dimensions d 5 using the lace expansion but the lace expansion does not apply when d 4 the proof begins by rewriting the walk two point function as the two point function of a supersymmetric field theory the field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers by providing a setting where the methods of the companion papers are applied together the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner