investigation of nonplanar perfectly matched absorbers for finite element mesh truncation we present a detailed theoretical and numerical investigation of the perfectly matched layer pml concept as applied to the problem of mesh truncation in the finite element method fem we show that it is possible to extend the cartesian pml concepts involving half spaces to cylindrical and spherical geometries appropriate for closed boundaries in two and three dimensions by defining lossy anisotropic layers in the relevant coordinate systems by using the method of separation of variables it is possible to solve the boundary value problems in these geometries the analytical solutions demonstrate that under certain conditions outgoing waves are absorbed with negligible reflection and the transmitted wave is attenuated within the pml to reduce the white space in radiation or scattering problems conformal pmls are constructed via parametric mappings it is also verified that the pml concept which was originally introduced for problems governed by maxwell s equations can be extended to cases governed by the scalar helmholtz equation finally numerical results are presented to demonstrate the use of the pml in fem mesh truncation

dilatation and order of contact for holomorphic self maps of strongly convex domains let d be a bounded strongly convex domain and let f be a holomorphic self map of d in this paper we introduce and study the dilatation alpha f of f defined if f has no fixed points in d as the usual boundary dilatation coefficient of f at its wolff point or if f has some fixed points in d as the ratio of shrinking of the kobayashi balls around a fixed point of f in particular we show that the map alpha defined as alpha f mapsto alpha f in 0 1 is lower semicontinuous among other things this allows us to study the limits of a family of holomorphic self maps of d in the case of an inner fixed point the dilatation is an intrinsic measure of the order of contact of f d to partial d finally using complex geodesics we define and study a directional dilatation which is a measure of the shrinking of the domain along a given direction again results of semicontinuity are given and applied to a family of holomorphic self maps

critical exponents of self avoiding walks on a family of truncated n simplex lattices the authors have studied self avoiding walks saws on a family of truncated n simplex lattices which provide a family of fractals in which the fractal dimension d can be varied to a wide range while the spectral dimension d is held almost fixed by means of exact renormalisation group transformations they have calculated the critical exponents nu alpha and gamma and the connectivity constant mu of the saws for n 5 and 6 they propose an approximate theory for calculating the critical exponents of the saws on fractals which is expected to be accurate at large values of n they show that the theory gives results which are in good agreement with exact values even for small values of n and leads to simple closed relations for the exponents

end to end distance from the green s function for a hierarchical self avoiding walk in four dimensions in bei92 we introduced a levy process on a hierarchical lattice which is four dimensional in the sense that the green s function for the process equals frac 1 x 2 if the process is modified so as to be weakly self repelling it was shown that at the critical killing rate mass squared c the green s function behaves like the free one now we analyze the end to end distance of the model and show that its expected value grows as a constant times sqrt t log frac 1 8 t left 1 o left frac log log t log t right right which is the same law as has been conjectured for self avoiding walks on the simple cubic lattice 4 the proof uses inverse laplace transforms to obtain the end to end distance from the green s function and requires detailed properties of the green s function throughout a sector of the complex plane these estimates are derived in a companion paper bi02

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

self avoiding walks on scale free networks several kinds of walks on complex networks are currently used to analyze search and navigation in different systems many analytical and computational results are known for random walks on such networks self avoiding walks saw s are expected to be more suitable than unrestricted random walks to explore various kinds of real life networks here we study long range properties of random saw s on scale free networks characterized by a degree distribution p k ensuremath sim k ensuremath ensuremath gamma in the limit of large networks system size n ensuremath rightarrow ensuremath infty the average number s n of saw s starting from a generic site increases as ensuremath mu n with ensuremath mu k 2 k ensuremath 1 for finite n s n is reduced due to the presence of loops in the network which causes the emergence of attrition of the paths for kinetic growth walks the average maximum length l increases as a power of the system size l ensuremath sim n ensuremath alpha with an exponent ensuremath alpha increasing as the parameter ensuremath gamma is raised we discuss the dependence of ensuremath alpha on the minimum allowed degree in the network a similar power law dependence is found for the mean self intersection length of nonreversal random walks simulation results support our approximate analytical calculations

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

high order accurate modeling of electromagnetic wave propagation across media grid conforming bodies the maxwell equations contain a dielectric permittivity e that describes the particular media for homogeneous materials at low temperatures this coefficient is constant within a material however it jumps at the interface between different media this discontinuity can significantly reduce the order of accuracy of the numerical scheme we solve the maxwell equations with an interface between two media using a fourth order accurate algorithm we regularize the discontinuous dielectric permittivity by a continuous function either locally near the interface or globally in the entire domain we study the effect of this regularization on the order of accuracy for a one dimensional time dependent problem we then implement this for the three dimensional maxwell equations in spherical coordinates with appropriate physical and artificial absorbing boundary conditions we use fourier filtering of the high frequency modes near the poles to increase the time step

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

plane wave analysis and comparison of split field biaxial and uniaxial pml methods as abcs for pseudospectral electromagnetic wave simulations in curvilinear coordinates in this paper we discuss and compare split field biaxial and uniaxial perfectly matched layer pml methods for absorbing outgoing vector waves in cylindrical and spherical coordinates we first extend berenger s split field formulation into spherical and cylindrical coordinates in such a way that it maintains all the desirable properties it exhibits in rectangular coordinates then we discuss the biaxial and the uniaxial medium pml methods in cartesian coordinates and extend them to spherical and cylindrical coordinates properties of plane wave solutions of the pml methods are analyzed in particular the decay and boundness properties of the solutions are considered in order to provide further insight into the different formulations presented herein moreover we propose a set of symmetric hyperbolic equations for both the biaxial and the uniaxial pml methods in the time domain which is fine tuned in numerical experiments and very suitable for time domain problems all three types of spherical and cylindrical pml methods are applied in simulations of plane wave scattering as well as radiating dipole problems we use a multidomain pseudospectral chebyshev numerical scheme and the effectiveness of the pml methods is demonstrated through the accurate numerical results obtained the order of outer boundary reflection is as low as 0 1 of the exact solution

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

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

conformal invariance predictions for the three dimensional self avoiding walk if the three dimensional self avoiding walk saw is conformally invariant then one can compute the hitting densities for the saw in a half space and in a sphere kennedy in phys rev lett 111 165703 2013 the ensembles of saw s used to define these hitting densities involve walks of arbitrary lengths and so these ensembles cannot be directly studied by the pivot monte carlo algorithm for the saw we show that these mixed length ensembles should have the same scaling limit as certain weighted ensembles that only involve walks with a single length thus providing a fast method for simulating these ensembles preliminary simulations which found good agreement between the predictions and monte carlo simulations for the saw were reported in kennedy phys rev lett 111 165703 2013 in this paper we present more accurate simulations testing the predictions and find even stronger support for the prediction that the saw is conformally invariant in three dimensions

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

critical behaviour of self avoiding walks that cross a square consider the set of all self avoiding walks in the square lattice which start at 0 0 end at l l and are entirely contained in the square 0 l 0 l associate a fugacity x with each step of the walk whittington and guttmann 1990 showed that the dominant walks have o l steps when x is small and o l2 steps when x is large and they conjectured that there is a single transition point at x mu 1 where mu is the inverse of the connective constant for unconstrained self avoiding walks we present a rigorous proof of this conjecture and its analogue in higher dimensions we also discuss what can be said rigorously about two scaling exponents associated with this phase transition and compare this with analogous results that have been obtained exactly and rigorously on the discrete sierpinski gasket by hattori hattori and kusuoka 1990

the pluricomplex poisson kernel for strongly convex domains let d be a bounded strongly convex domain in the complex space of dimension n for a fixed point p epsilon partial derivative d we consider the solution of a homogeneous complex monge ampere equation with a simple pole at p we prove that such a solution enjoys many properties of the classical poisson kernel in the unit disc and thus deserves to be called the pluricomplex poisson kernel of d with pole at p in particular we discuss extremality properties such as a generalization of the classical phragmen lindelof theorem relations with the pluricomplex green function of d uniqueness in terms of the associated foliation and boundary behaviors finally using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions

gaussian limit for critical oriented percolation in high dimensions in this paper we consider the spread out oriented bond percolation models inz d z withd 4 and the nearest neighbor oriented bond percolation model in sufficiently high dimensions let n n 1 2 be the random measures defined onr d by eta n a sum limits x in z d 1 a x sqrt n 1 0 0 to x n the mean of n denoted by bar eta n is the measure defined by bar eta n a e p eta n a we use the lace expansion method to show that the sequence of probability measures bar eta n r d 1 bar eta n converges weakly to a gaussian limit asn for everyp in the subcritical regime as well as the critical regime of these percolation models also we show that for these models the parallel correlation length xi p p c p 1 asp pc

on the critical behaviour of self avoiding walks for pt i see ibid vol 22 p 1839 1989 for the self avoiding walk problem the coefficients of the chain generating function and of the generating function for the sum of square end to end distances have been extended to 20 terms for the triangular lattice to 27 terms for the diamond lattice to 21 terms for the simple cubic lattice and to 16 terms for the bcc lattice precise estimates of the critical points are obtained and for the exponents we find that gamma 1 161 or 0 002 and nu 0 592 or 0 003 encompasses all the three dimensional lattice data

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

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

the fixed point of a generalization of the renormalization group maps for self avoiding paths on gaskets let w x y ax 3 bx 4 f 5 x 5 f 6 x 6 3 ax 2 2 y g 5 x 5 y h 3 x 3 y 2 h 4 x 4 y 2 n 3 x 3 y 3 a 24 x 2 y 4 a 05 y 5 a 15 xy 5 a 06 y 6 and x x frac partial w partial x y frac partial w partial y where the coefficients are non negative constants with a 0 such that x 2 x x 2 y x x 2 is a polynomial of x with non negative coefficients

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

backbone scaling for critical lattice trees in high dimensions we prove that for critical spread out lattice trees in dimensions the unique paths from the origin to vertices of large tree distance converge to brownian motion this provides an important ingredient for proving weak convergence of the corresponding historical processes

extremal analytic discs with prescribed boundary data this paper concerns the existence and uniqueness of extremal analytic discs with prescribed boundary data in a bounded strictly linearly convex domain d in cn we prove that for any two distinct points p q in dd respectively p e dd and a vector v such that f lv e tp dd and v p p 2 vji j p 0 where v p is the outward normal to dd at p there exists an extremal analytic disc passing through p q if dd e ck k 3 respectively l p l v if dd e ck k 14 consequently we can foliate d with these extremal analytic discs introduction in this paper we always assume that d is a bounded strictly linearly convex domain in cn see 1 for definitions let u denote the open unit disc in c an extremal analytic disc of d is a holomorphic map from u into d such that the kobayashi distance of any two points in f u is realized by lempert in 6 7 has proved the existence uniqueness and boundary regularity of such maps with data given at interior points following 7 we shall use the term e mapping for the extremal analytic disc it is also known 6 proposition 5 that if dd is of class ck k 3 then any e mapping of d is in ck 2 u

a note on mean field behavior for self avoiding walk on branching planes we consider the critical behavior of the susceptibility of the self avoiding walk on the grapht z wheret is a bethe lattice with degreek andz is the one dimensional lattice by directly estimating the two point function using a method of grimmett and newman we show that the bubble condition is satisfied whenk 2 and therefore the critical exponent associated with the susceptibility equals 1

self avoiding walks and trees in spread out lattices letgr be the graph obtained by joining all sites ofzd which are separated by a distance of at mostr let gr denote the connective constant for counting the self avoiding walks in this graph let gr denote the coprresponding constant for counting the trees embedded ingr then asr gr is asymptotic to the coordination numberkr ofgr while gr is asymptotic toekr however ifd is 1 or 2 then gr kr diverges to

scaling of self avoiding walks in high dimensions we examine self avoiding walks in dimensions 4 to 8 using high precision monte carlo simulations up to length n 16 384 providing the first such results in dimensions d 4 on which we concentrate our analysis we analyse the scaling behaviour of the partition function and the statistics of nearest neighbour contacts as well as the average geometric size of the walks and compare our results to 1 d expansions and to excellent rigorous bounds that exist in particular we obtain precise values for the connective constants 5 8 838 544 3 6 10 878 094 4 7 12 902 817 3 8 14 919 257 2 and give a revised estimate of 4 6 774 043 5 all of these are by at least one order of magnitude more accurate than those previously given from other approaches in d 4 and all approaches in d 4 our results are consistent with most theoretical predictions though in d 5 we find clear evidence of anomalous n 1 2 corrections for the scaling of the geometric size of the walks which we understand as a non analytic correction to scaling of the general form n 4 d 2 not present in pure gaussian random walks

the exponent for the mean square displacement of self avoiding random walk on the sierpinski gasket the authors rigorously prove that the exponent for the mean square displacement of self avoiding random walk on the sierpinski gasket is log 2 log left frac 7 sqrt 5 2 right 0 79862 0 5 log 2 log 5

self avoiding paths on the pre sierpinski gasket we study a statistical mechanics of self avoiding paths on the pre sierpinski gasket we first show the existence of the thermodynamic limit of the appropriately scaled free energy then we show that there are two domains in the weight parameters i e two phases between which the scaling differs i e there is a certain kind of phase transition in our model and we find the critical exponents of the free energy at the phase transition point we also show the convergence of the distribution of the scaled length of the paths at thermodynamic limit

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 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

four dimensional weakly self avoiding walk with contact self attraction we consider the critical behaviour of the continuous time weakly self avoiding walk with contact self attraction on mathbb z 4 for sufficiently small attraction we prove that the susceptibility and correlation length of order p for any p 0 have logarithmic corrections to mean field scaling and that the critical two point function is asymptotic to a multiple of x 2 this shows that small contact self attraction results in the same critical behaviour as no contact self attraction a collapse transition is predicted for larger self attraction the proof uses a supersymmetric representation of the two point function and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self avoiding walk without self attraction

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

hamiltonian walks on the sierpinski gasket we derive the exact number of hamiltonian walks h n on the two dimensional sierpinski gasket sg n at stage n whose asymptotic behavior is given by 3 23 3n 13 52 72 172212 35 13 16 n we also obtain the number of hamiltonian walks with one end at a specific outmost vertex of sg n with asymptotic behavior 3 23 3n 13 7 1724 33 4n the distribution of hamiltonian walks on sg n with one end at a specific outmost vertex and the other at an arbitrary vertex of sg n is investigated we rigorously prove that the exponent for the mean l displacement between the two end vertices of such hamiltonian walks on sg n is lln 2 ln 3 for l 0

end patterns of self avoiding walks consider a fixed end pattern a short self avoiding walk that can occur as the first few steps of an arbitrarily long self avoiding walk on d it is a difficult open problem to show that asn the fraction ofn step self avoiding walks beginning with this pattern converges it is shown that asn this fraction is bounded away from zero and that the ratio of the fractions forn andn 2 converges to one similar results are obtained when patterns are specified at both ends and also when the endpoints are fixed

monte carlo study of four dimensional self avoiding walks of up to one billion steps we study self avoiding walks on the four dimensional hypercubic lattice via monte carlo simulations of walks with up to one billion steps we study the expected logarithmic corrections to scaling and find convincing evidence in support the scaling form predicted by the renormalization group with an estimate for the power of the logarithmic factor of 0 2516 14 which is consistent with the predicted value of 1 4 we also characterize the behaviour of the pivot algorithm for sampling four dimensional self avoiding walks and conjecture that the probability of a pivot move being successful for an n step walk is o log n 1 4

a rigorous bound on the critical exponent for the number of lattice trees animals and polygons the number ofn site lattice trees up to translation is believed to behave asymptotically ascn 0 n where is a critical exponent dependent only on the dimensiond of the lattice we present a rigorous proof that d 1 d for anyd 2 the method also applies to lattice animals site animals and two dimensional self avoiding polygons we also prove that v whend 2 wherev is the exponent for the radius of gyration

semi flexible compact polymers on fractal lattices abstract semi flexible compact polymers modeled by hamiltonian walks with bending rigidity are studied on 3 and 4 simplex fractal lattices hamiltonian walks are weighted according to the number of bends in the walk and total weights are obtained by an exact recursive treatment asymptotic form of the partition function with temperature dependent scaling parameters as well as the corresponding critical exponents is determined various thermodynamic quantities are calculated numerically and presented graphically and the possibility of phase transition between a compact molten globule and an ordered crystal state is investigated no phase transition is found on either of these two lattices meaning that fractal geometry here prevents any kind of orientational order

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

constructive field theory and applications perspectives and open problems in this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory they could give work for future mathematical physicists trained with constructive methods well into the 21st century

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

compressed self avoiding walks bridges and polygons we study various self avoiding walks saws which are constrained to lie in the upper half plane and are subjected to a compressive force this force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half space in the case of bridges this is the unique end point in the case of saws or self avoiding polygons this corresponds to all vertices of maximal height we first use the conjectured relation with the schramm loewner evolution to predict the form of the partition function including the values of the exponents and then we use series analysis to test these predictions

the scaling limit of lattice trees in high dimensions we prove that above eight dimensions the scaling limit of sufficiently spread out lattice trees is the variant of super brownian motion known as integrated super brownian excursion ise as conjectured by aldous the same is true for nearest neighbour lattice trees in sufficiently high dimensions the proof uses the lace expansion

internal structure of polymer chains abstract the asymptotic behaviour of isolated long chain polymers in good solvents is modelled well by the self avoiding walk on a regular crystal lattice in this paper we investigate the typical scaling length and the probability distribution function for the position of the end point and an interior point for self avoiding walks on a simple cubic lattice we find that for n step walks the typical internal scaling length r n n associated with the position of the n th step has a similar functional form to the external scaling length r n the associated exponent is found to be the same for both scaling lengths within the confidence limits of our simulations the internal probability distribution function p n n r is shown to be skew gaussian and furthermore has a similar form to the external distribution function p n n r r r n n a exp r r n n where a 2 42 0 08 and 2 50 0 09

renormalization group analysis of the self avoiding paths on the d dimensional sierpinski gaskets notion of the renormalization group dynamical system the self avoiding fixed point and the critical trajectory are mathematically defined for the set of self avoiding walks on the d dimensional pre sierpinski gaskets n simplex lattices such that their existence imply the asymptotic behaviors of the self avoiding walks such as the existence of the limit distributions of the scaled path lengths of canonical ensemble the connectivity constant exponential growth of path numbers with respect to the length and the exponent for mean square displacement we apply the so defined framework to prove these asymptotic behaviors of the restricted self avoiding walks on the 4 dimensional pre sierpinski gasket

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

backward iteration in strongly convex domains we prove that a backward orbit with bounded kobayashi step for a hyperbolic parabolic or strongly elliptic holomorphic self map of a bounded strongly convex c 2 domain in c d necessarily converges to a repelling or parabolic boundary fixed point generalizing previous results obtained by poggi corradini in the unit disk and by ostapyuk in the unit ball of c d

eigenvalue spectrum of transition matrix of dual sierpinski gaskets and its applications the eigenvalue spectrum of the transition matrix of a network contains much information about its structural properties and is related to the behavior of various dynamical processes performed on it in this paper we study the eigenvalues of the transition matrix of the dual sierpinski gaskets embedded in d dimensional euclidean spaces we obtain all the eigenvalues as well as their corresponding degeneracies by making use of the spectral decimation technique we then apply the obtained eigenvalues to determine the exact number of spanning trees in the studied fractals and derive an explicit formula of the eigentime identity for random walks taking place on them

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

the use of the fft for the efficient solution of the problem of electromagnetic scattering by a body of revolution the enhancement of the computational efficiency of the body of revolution scattering problem is discussed with a view of making it practical for solving large body problems the problem of the electromagnetic scattering by a perfectly conducting body is considered although the methods provided can be extended to multilayered dielectric bodies as well typically the generation of the elements of the moment method matrix consumes a major portion of the computational time it is shown how this time can be significantly reduced by manipulating the expression for the matrix elements in a manner that allows one to compute them efficiently by using the fast fourier transform fft a technique for extracting the singularity of the green s function that appears within the integrands of the matrix diagonal is also presented further enhancing the usefulness of the fft it is shown that with the use of the method discussed here the computational time can be improved by at least an order of magnitude for large bodies in comparison to that for previous algorithms

on the probability that self avoiding walk ends at a given point we prove two results on the delocalization of the endpoint of a uniform self avoiding walk on z d for d 1 we show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity uniformly in x also for any fixed x in z d this probability decreases faster than n 1 4 epsilon for any epsilon 0 when x 1 we thus obtain a bound on the probability that self avoiding walk is a polygon

mean field behavior for long and finite range ising model percolation and self avoiding walk we consider self avoiding walk percolation and the ising model with long and flnite range by means of the lace expansion we prove mean fleld behavior for these models if d 2 fi 2 for self avoiding walk and the ising model and d 3 fi 2 for percolation where d denotes the dimension and fi the power law decay exponent of the coupling function we provide a simplifled analysis of the lace expansion based on the trigonometric approach in borgs et al 13

self testing algorithms for self avoiding walks we present a markov chain monte carlo algorithm for almost uniformly generating and approximately counting self avoiding walks in rectangular lattices zd these are classical problems that arise for example in the study of long polymer chains while there are a number of monte carlo algorithms used to solve these problems in practice these are heuristic and their correctness relies on unproven conjectures in contrast our algorithm is shown rigorously to produce answers with specified accuracy and confidence only the efficiency of the algorithm relies on a widely believed conjecture and a novel feature is that this conjecture can be tested as the algorithm proceeds with this self testing feature incorporated the algorithm has polynomially bounded running time and is completely reliable in the sense that it either outputs answers that are guaranteed to be within the specified accuracy and confidence bounds or finds a counter example to the conjecture

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

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

transition density of diffusion on the sierpinski gasket and extension of flory s formula some problems related to the transition density u t x of the diffusion on the sierpinski gasket are considerd based on recent rigorous results and detailed numerical calculations the main contents are an extension of flory s formula for the end to end distance exponent of self avoiding walks on the fractal spaces and an evidence of the oscillatory behavior of u t x on the sierpinski gasket

non intersection of transient branching random walks let g be a cayley graph of a nonamenable group with spectral radius rho 1 it is known that branching random walk on g with offspring distribution mu is transient i e visits the origin at most finitely often almost surely if and only if the expected number of offspring overline mu satisfies overline mu le rho 1 benjamini and muller groups geom dyn 6 231 247 2012 conjectured that throughout the transient supercritical phase 1 overline mu le rho 1 and in particular at the recurrence threshold overline mu rho 1 the trace of the branching random walk is tree like in the sense that it is infinitely ended almost surely on the event that the walk survives forever this is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely we prove this conjecture along with several other related conjectures made by the same authors a central contribution of this work is the introduction of the notion of local unimodularity which we expect to have several further applications in the future

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

new lower bounds on the self avoiding walk connective constant we give an elementary new method for obtaining rigorous lower bounds on the connective constant for self avoiding walks on the hypercubic lattice d the method is based on loop erasure and restoration and does not require exact enumeration data our bounds are best for highd and in fact agree with the first four terms of the 1 d expansion for the connective constant the bounds are the best to date for dimensionsd 3 but do not produce good results in two dimensions ford 3 4 5 and 6 respectively our lower bound is within 2 4 0 43 0 12 and 0 044 of the value estimated by series extrapolation

bounds on connective constants for self avoiding walks new lower bounds on the connective constant of the square and simple cubic lattice self avoiding walks are obtained by enumerating a particular subset of self avoiding walks and using a result of kesten the author finds mu sq 2 5680 and mu sc 4 352

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

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

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

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

monge ampere foliations with singularities at the boundary of strongly convex domains let d c n be a bounded strongly convex domain with smooth boundary we consider a monge ampere type equation in d with a simple pole at the boundary using the lempert foliation of d in extremal discs we construct a solution u whose level sets are bound aries of horospheres among other things we show that the biholomorphisms between strongly convex domains are exactly those maps which preserves our solution

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

self avoiding process on the sierpinski gasket we construct a self avoiding process taking values in the finite sierpinski gasket and study its properties we then study continuum limit processes that are suggested by the statistical mechanics of self avoiding paths on the pre sierpinski gasket we prove that there are three types of continuum limit processes according to the parameters defining the statistical mechanics of self avoiding paths i the self avoiding process we construct in this paper ii a deterministic motion along a peano curve on the finite sierpinski gasket iii a deterministic motion along a line segment

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