thick points for planar brownian motion and the erdős taylor conjecture on random walk and conjectured that the limit exists and equals 1 tr a s the importance of determining the value of this limit is clarified in 1 3 below where this value appears in the power laws governing the local time of the walk the erd6s taylor conjecture was quoted in the book by r v sz 19 w but to the best of our knowledge the bounds in 1 1 were not improved prior to the present paper as it turns out an important step towards our solution of the erd6s taylor conjecture was the formulation by perkins and taylor 17 of an analogous problem on the maximal occupation measure that planar brownian motion run for unit time can

representations of the lie superalgebra and parastatistics fock spaces the algebraic structure generated by the creation and annihilation operators of a system of m parafermions and n parabosons satisfying the mutual parafermion relations is known to be the lie superalgebra osp 2m 1 vertical bar 2n the fock spaces of such systems are then certain lowest weight representations of osp 2m 1 vertical bar 2n in the current paper we investigate what happens when the number of parafermions and parabosons becomes infinite in order to analyze the algebraic structure and the fock spaces we first need to develop a new matrix form for the lie superalgebra b n n osp 2n 1 vertical bar 2n and construct a new gelfand zetlin basis of the fock spaces in the finite rank case the new structures are appropriate for the situation n infinity the algebra generated by the infinite number of creation and annihilation operators is b infinity infinity a well defined infinite rank version of the orthosymplectic lie superalgebra the fock spaces are lowest weight representations of b infinity infinity with a basis consisting of particular row stable gelfand zetlin patterns

the multifractal spectrum of brownian intersection local times let l be the projected intersection local time of two independent brownian paths in d for d 2 3 we determine the lower tail of the random variable ell mathbb u where mathbb u is the unit ball the answer is given in terms of intersection exponents which are explicitly known in the case of planar brownian motion we use this result to obtain the multifractal spectrum or spectrum of thin points for the intersection local times

wigner quantum systems two particles interacting via a harmonic potential i two dimensional space a noncanonical quantum system consisting of two nonrelativistic particles interacting via a harmonic potential is considered the center of mass position and momentum operators obey the canonical commutation relations whereas the internal variables are assumed to be the odd generators of the lie superalgebra sl 1 2 this assumption implies a set of constraints in the phase space which are explicitly written in the paper all finite dimensional irreducible representations of sl 1 2 are considered particular attention is paid to the physical representations i e the representations corresponding to hermitian position and momentum operators the properties of the physical observables are investigated in particular the operators of the internal hamiltonian the relative distance the internal momentum and the orbital momentum commute with each other the spectrum of these operators is finite the distance between the constituents is preserved in time it can take no more than three different values for any non negative integer or half integer l there exists a representation where the orbital momentum is l in units of 2 the position of any one of the particles cannot be localized since the operators of the coordinates do not commute with each other the constituents are smeared with a certain probability within a finite surface which moves with a constant velocity together with the center of mass

cover times for brownian motion and random walks in two dimensions lett x denote the rst hitting time of the disc of radius centered at x for brownian motion on the two dimensional torus t 2 we prove that sup x2t2t x j log j 2 2 as 0 the same applies to brownian motion on any smooth compact connected two dimensional riemannian manifold with unit area and no boundary as a consequence we prove a conjecture due to aldous 1989 that the number of steps it takes a simple random walk to cover all points of the lattice torus z 2 is asymptotic to 4n 2 logn 2 determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete so far this structure was only studied non rigorously in the physics literature we also establish a conjecture due to kesten and r ev esz that describes the asymptotics for the number of steps needed by simple random walk in z 2 to cover the disc of radius n

extension of the mathfrak o mathfrak s mathfrak p m n sim mathfrak s mathfrak o m n correspondence to the infinite dimensional chiral spinors and self dual tensors the spinor representations of the orthosymplectic lie superalgebras are considered and constructed these are infinite dimensional irreducible representations of which the superdimension coincides with the dimension of the spinor representation of next we consider the self dual tensor representations of and their generalizations these are also infinite dimensional and correspond to the highest irreducible component of the pth power of the spinor representation we determine the character of these representations and deduce a superdimension formula from this it follows that also for these representations the correspondence holds

gel fand zetlin basis and clebsch gordan coefficients for covariant representations of the lie superalgebra gl m n a gel fand zetlin basis is introduced for the irreducible covariant tensor representations of the lie superalgebra gl m n explicit expressions for the generators of the lie superalgebra acting on this basis are determined furthermore clebsch gordan coefficients corresponding to the tensor product of any covariant tensor representation of gl m n with the natural representation v 1 0 0 of gl m n with highest weight 1 0 0 are computed both results are steps for the explicit construction of the parastatistics fock space

holomorphic discrete series and harmonic series unitary irreducible representations of non compact lie groups sp 2n r u p q and so 2n generalised characters of the infinite dimensional holomorphic discrete series unitary irreducible representations of the non compact groups u p q sp 2n r and so 2n are explicitly expressed in terms of characters of finite dimensional unitary group representations these formulae are remarkably succinct despite involving certain infinite series of schur functions similar formulae are derived for harmonic series unitary representation of both u p q and sp 2n r consideration of the branching rules from u p q to u q p and from sp 2n r to u n enables holomorphic representations to be identified as a subset of the harmonic representations the branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations in the case of the former a known result is recast in terms of closed formulae involving schur functions and for the latter various generalisations of these formulae are given a conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of sp 2n r and u p q illustrative examples are presented

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

parastatistics and conformal field theories in two dimensions the relation between parafermion field theories of order q and the corresponding fermion field theories with so q symmetry is studied it is shown that these theories are related but not identical the explicit relation between the states and the observables of the two classes of theories are given without using the klein transformations the formalism is applied to the free conformally invariant parafermion theories in two dimensions their virasoro algebra and so n kac moody algebra are given the equivalence of their canonical form of the energy momentum tensor with the sugawara sommerfield form is also elucidated

exact packing dimension in random recursive constructions we explore the exact packing dimension of certain random recursive constructions in case of polynomial decay at 0 of the distribution function of random variable x associated with the construction we prove that it does not exist and in case of exponential decay it is t log logt where is the fractal dimension of the limit set and 1 is the rate of exponential decay

a strong invariance principle for nonconventional sums in kifer and varadhan nonconventional limit theorems in discrete and continuous time via martingales 2010 we obtained a functional central limit theorem known also as a weak invariance principle for sums of the form sum n 1 nt f big x n x 2n x kn x q k 1 n x q k 2 n x q ell n big normalized by 1 sqrt n where x n n 0 is a sufficiently fast mixing vector process with some moment conditions and stationarity properties f is a continuous function with polynomial growth and certain regularity properties and qi i k are positive functions taking on integer values on integers with some growth conditions which are satisfied for instance when qi s are polynomials of growing degrees this paper deals with strong invariance principles known also as strong approximation theorems for such sums which provide their uniform in time almost sure approximation by processes built out of brownian motions with error terms growing slower than sqrt n this yields in particular an invariance principle in the law of iterated algorithm for the above sums among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems as well as to some questions in metric number theory and they can be considered as a natural follow up of a series of papers dealing with nonconventional ergodic averages

multiple intersection exponents for planar brownian motion let p 2 n1 np be positive integers and b 1 1 ldots b n 1 1 ldots b 1 p ldots b n p p be independent planar brownian motions started uniformly on the boundary of the unit circle we define a p fold intersection exponent p n1 np as the exponential rate of decay of the probability that the packets bigcup j 1 n i b j i 0 t 2 i 1 p have no joint intersection the case p 2 is well known and following two decades of numerical and mathematical activity lawler et al acta math 187 275 308 2001 rigorously identified precise values for these exponents the exponents have not been investigated so far for p 2 we present an extensive mathematical and numerical study leading to an exact formula in the case n1 1 n2 2 and several interesting conjectures for other cases

a new approach to strong embeddings we revisit strong approximation theory from a new perspective culminating in a proof of the komlos major tusnady embedding theorem for the simple random walk the proof is almost entirely based on a series of soft arguments and easy inequalities the new technique inspired by stein s method of normal approximation is applicable to any setting where stein s method works in particular one can hope to take it beyond sums of independent random variables

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

harmonic oscillators coupled by springs discrete solutions as a wigner quantum system we consider a quantum system consisting of a one dimensional chain of m identical harmonic oscillators with natural frequency coupled by means of springs such systems have been studied before and appear in various models in this paper we approach the system as a wigner quantum system not imposing the canonical commutation relations but using instead weaker relations following from the compatibility of hamilton s equations and the heisenberg equations in such a setting the quantum system allows solutions in a finite dimensional hilbert space with a discrete spectrum for all physical operators we show that a class of solutions can be obtained using generators of the lie superalgebra gl 1 m then we study from a mathematical point of view the properties and spectra of the physical operators in a class of unitary representations of gl 1 m these properties are both interesting and intriguing in particular we can give a complete analysis of the eigenvalues of the hamiltonian and of the pos

the parafermion fock space and explicit so 2n 1 representations the defining relations triple relations of n pairs of parafermion operators f j j 1 n are known to coincide with a set of defining relations for the lie algebra so 2n 1 in terms of 2n generators with the common hermiticity conditions this means that the parafermions of order p correspond to a finite dimensional unitary irreducible representation w p of so 2n 1 with highest weight p 2 p 2 p 2 although the dimension and the character of w p is known by classical formulae there is no explicit basis of w p available in which the parafermion operators have a natural action in this paper we construct an orthogonal basis for w p and present the explicit actions of the parafermion generators on these basis vectors we use group theoretical techniques in which the u n subalgebra of so 2n 1 plays a crucial role a set of gelfand zetlin patterns of u n will be used to label the basis vectors of w p and also in the explicit action matrix elements certain u n clebsch gordan coefficients are essential

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

the exact packing measure of brownian double points let d subset mathbb r 3 be the set of double points of a three dimensional brownian motion we show that if 3 2 2 is the intersection exponent of two packets of two independent brownian motions then almost surely the packing measure of d is zero if int 0 r 1 xi phi r xi dr infty and infinity otherwise as an important step in the proof we show up to constants estimates for the tail at zero of brownian intersection local times in dimensions two and three

the paraboson fock space and unitary irreducible representations of the lie superalgebra mathfrak osp 1 2n it is known that the defining relations of the orthosymplectic lie superalgebra mathfrak osp 1 2n are equivalent to the defining triple relations of n pairs of paraboson operators b pm i in particular with the usual star conditions this implies that the parabosons of order p correspond to a unitary irreducible infinite dimensional lowest weight representation v p of mathfrak osp 1 2n apart from the simple cases p 1 or n 1 these representations had never been constructed due to computational difficulties despite their importance in the present paper we give an explicit and elegant construction of these representations v p and we present explicit actions or matrix elements of the mathfrak osp 1 2n generators the orthogonal basis vectors of v p are written in terms of gelfand zetlin patterns where the subalgebra mathfrak u n of mathfrak osp 1 2n plays a crucial role our results also lead to character formulas for these infinite dimensional mathfrak osp 1 2n representations furthermore by considering the branching mathfrak osp 1 2n supset mathfrak sp 2n supset mathfrak u n we find explicit infinite dimensional unitary irreducible lowest weight representations of mathfrak sp 2n and their characters

a braided look at green ansatz for parabosons bosons and parabosons are described as associative superalgebras with an infinite number of odd generators bosons are shown to be a quotient superalgebra of parabosons establishing thus an even algebra epimorphism which is an immediate link between their simple modules parabosons are a super hopf algebra the super hopf algebraic structure of parabosons combined with the projection epimorphism previously stated provides a braided interpretation of the green ansatz device and of the parabosonic fock like representations this braided interpretation combined with an old problem leads to the construction of straightforward generalizations of the green ansatz

brownian intersection local times upper tail asymptotics and thick points we equip the intersection of p independent brownian paths in r d d 2 with the natural measure l defined by projecting the intersection local time measure via one of the brownian motions onto the set of intersection points given a bounded domain u c r d we show that as a the probability of the event l u a decays with an exponential rate of a 1 p where is described in terms of a variational problem in the important special case when u is the unit ball in r 3 and p 2 we characterize in terms of an ordinary differential equation we apply our results to the problem of finding the hausdorff dimension spectrum for the thick points of the intersection of two independent brownian paths in r 3