the scaling limit of the incipient infinite cluster in high dimensional percolation ii integrated super brownian excursion for independent nearest neighbor bond percolation on zd with d 6 we prove that the incipient infinite cluster s two point function and three point function converge to those of integrated super brownian excursion ise in the scaling limit the proof is based on an extension of the new expansion for percolation derived in a previous paper and involves treating the magnetic field as a complex variable a special case of our result for the two point function implies that the probability that the cluster of the origin consists of n sites at the critical point is given by a multiple of n 3 2 plus an error term of order n 3 2 e with e 0 this is a strong version of the statement that the critical exponent is given by 2

kinodynamic motion planning kinodynamic planning attempts to solve a robot motion problem subject to simultaneous kinematic and dynamics constraints in the general problem given a robot system we must find a minimal time trajectory that goes from a start position and velocity to a goal position and velocity while avoiding obstacles by a safety margin and respecting constraints on velocity and acceleration we consider the simplified case of a point mass under newtonian mechanics together with velocity and acceleration bounds the point must be flown from a start to a goal amidst polyhedral obstacles in 2d or 3d although exact solutions to this problem are not known we provide the first provably good approximation algorithm and show that it runs in polynomial time

nonuniform discretization for kinodynamic motion planning and its applications the first main result of this paper is a novel nonuniform discretization approximation method for the kinodynamic motion planning problem the kinodynamic motion planning problem is to compute a collision free time optimal trajectory for a robot whose accelerations and velocities are bounded previous approximation methods are all based on a uniform discretization in the time space on the contrary our method employs a nonuniform discretization in the configuration space thus also a nonuniform one in the time space compared to the previously best algorithm of donald and xavier the running time of our algorithm reduces in terms of 1 varepsilon roughly from o 1 varepsilon 6d 1 to o 1 varepsilon 4d 2 in computing a trajectory in a d dimensional configuration space such that the time length of the trajectory is within a factor of 1 varepsilon of the optimal more importantly our algorithm is able to take advantage of the obstacle distribution and is expected to perform much better than the analytical result this is because our nonuniform discretization has the property that it is coarser in regions that are farther from all obstacles so for situations where the obstacles are sparse or the obstacles are unevenly distributed the size of the discretization is significantly smaller our second main result is the first known polynomial time approximation algorithm for the curvature constrained shortest path problem in three and higher dimensions we achieved this by showing that the approximation techniques for the kinodynamic motion planning problem are applicable to this problem

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

approximate kinodynamic planning using l2 norm dynamic bounds in this paper we address the issue of kinodynamic motion planning given a point that moves with bounded acceleration and velocity we wish to find the time optimal trajectory from a start configuration to a goal configuration a configuration consists of both a position and a velocity as finding exact optimal solutions to this problem seems very hard we present a provably good approximation algorithm we obtain a provably good approximation algorithm using the l 2 norm to bound acceleration and velocity our results are an extension of the earlier work of canny donald reif and xavier 1 who present similar results where the dynamic bounds can be examined in each dimension independently they use the l infty norm to bound acceleration and velocity

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

collision detection for moving polyhedra we consider the collision detection problem for a three dimensional solid object moving among polyhedral obstacles the configuration space for this problem is six dimensional and the traditional representation of the space uses three translational parameters and three angles typically euler angles the constraints between the object and obstacles then involve trigonometric functions we show that a quaternion representation of rotation yields constraints which are purely algebraic in a seven dimensional space by simple manipulation the constraints may be projected down into a six dimensional space with no increase in complexity the algebraic form of the constraints greatly simplifies computation of collision points and allows us to derive an efficient exact intersection test for an object which is translating and rotating among obstacles

random walks with short range interaction and mean field behavior we introduce a model of self repelling random walks where the short range interaction between two elements of the chain decreases as a power of the difference in proper time the model interpolates between the lattice edwards model and the ordinary random walk we show by means of monte carlo simulations in two dimensions that the exponentvmf obtained through a mean field approximation correctly describes the numerical data and is probably exact as long as it is smaller than the corresponding exponent for self avoiding walks we also compute the exponent and present a numerical study of the scaling functions

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

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

tree based trajectory optimization based on local linearity of continuous non linear dynamics this technical note addresses a tradeoff in the tree based trajectory optimization algorithms for open loop optimal control problem of rigid body system in this technical note linear prediction based uniform state sampling method lpuss which relaxes the tradeoff between solutions quality and computational efficiency is proposed on the basis of the local linearity of motion equations lpuss has a lower calculation order than randomized kinodynamic planning rkp and rapid semi optimal motion planning rasmo in the validation using multi degree of freedom dof under actuated manipulator models the solutions quality and the computational efficiency of lpuss were better than those of rkp and rasmo lpuss finished the optimization for the 6 dof model within 40 minutes this was the world s first success of the optimization of the swing up motion of a 6 dof under actuated manipulator model

curvature constrained shortest paths in a convex polygon let b be a point robot moving in the plane whose path is constrained to have curvature at most 1 and let poly be a convex polygon with n vertices we study the collision free optimal path planning problem for b moving between two configurations inside poly a configuration specifies both a location and a direction of travel we present an o n2 log n time algorithm for determining whether a collision free path exists for b between two given configurations if such a path exists the algorithm returns a shortest one we provide a detailed classification of curvature constrained shortest paths inside a convex polygon and prove several properties of them which are interesting in their own right for example we prove that any such shortest path is comprised of at most eight segments each of which is a circular arc of unit radius or a straight line segment some of the properties are quite general and shed some light on curvature constrained shortest paths amid obstacles

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

minimum time speed optimisation over a fixed path in this paper we investigate the problem of optimising the speed of a vehicle over a fixed path for minimum time traversal we utilise a change of variables that has been known since the 1980s although the resulting convexity of the problem was not noted until recently the contributions of this paper are three fold first we extend the convexification of the problem to a more general framework second we identify a wide range of vehicle models and constraints which can be included in this expanded framework third we develop and implement an algorithm that allows these problems to be solved in real time on embedded systems with a high degree of accuracy

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

spatial planning a configuration space approach this paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects this problem arises in applications that require choosing how to arrange or how to move objects without collisions the approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space in which each coordinate represents a degree of freedom in the position or orientation of the object the configurations forbidden to this object due to the presence of other objects can then be characterized as regions in the configuration space called configuration space obstacles the paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra

the effect of convolving families of l functions on the underlying group symmetries let fn and gm be families of primitive automorphic l functions for gln aq and glm aq respectively such that as n m the statistical behavior 1 level density of the low lying zeros of l functions in fn resp gm agrees with that of the eigenvalues near 1 of matrices in g1 resp g2 as the size of the matrices tend to infinity where each gi is one of the classical compact groups unitary u symplectic sp or orthogonal o so even so odd assuming that the convolved families of l functions fn gm are automorphic we study their 1 level density we also study convolved families of the form f gm for a fixed f under natural assumptions on the families which hold in many cases we can associate to each family l of l functions a symmetry constant cl equal to 0 resp 1 or 1 if the corresponding low lying zero statistics agree with those of the unitary resp symplectic or orthogonal group our main result is that cf g cf cg the symmetry type of the convolved family is the product of the symmetry types of the two families a similar statement holds for the convolved families f gm we provide examples built from dirichlet l functions and holomorphic modular forms and their symmetric powers an interesting special case is to convolve two families of elliptic curves with positive rank in this case the symmetry group of the convolution is independent of the ranks in accordance with the general principle of multiplicativity of the symmetry constants but the ranks persist before taking the limit n m as lower order terms

a new inductive approach to the lace expansion for self avoiding walks we introduce a new inductive approach to the lace expansion and apply it to prove gaussian behaviour for the weakly self avoiding walk on d where loops of length m are penalised by a factor e m p 0 4 p 0 2 d 4 in particular we derive results first obtained by brydges and spencer and revisited by other authors for the case d 4 p 0 in addition we prove a local central limit theorem with the exception of the case d 4 p 0

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