eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

martingale hardy spaces for 0 p 1 the relation between the usually considered martingale hardy spaces is examined it will be shown that if the stochastic basis is regular then all these hardy spaces are equivalent an atomic description of the spacesh andp p will be given for 0 p 1 and with the help of this we shall consider the duals of these spaces moreover the dual space ofvmo p generated by the dual space ofh p will be determined in the case when all algebras are generated by countably many atoms furthermore we shall show in this case that the dual ofvmo 1 ish 1

breaking records in the evolutionary race abstract we explore some aspects of the relationship between biological evolution processes and the mathematical theory of records for eigen s quasispecies model with an uncorrelated fitness landscape we show that the evolutionary trajectories traced out by a population initially localized at a randomly chosen point in sequence space can be described in close analogy to record dynamics with two complications first the increasing number of genotypes that become available with increasing distance from the starting point implies that fitness records are more frequent than for the standard case of independent identically distributed random variables second fitness records can be bypassed which strongly reduces the number of genotypes that take part in an evolutionary trajectory for exponential and gaussian fitness distributions this number scales with sequence length n as n and it is of order unity for distributions with a power law tail this is in strong contrast to the number of records which is of order n for any fitness distribution

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

levy flights on the half line we study the probability distribution function pdf of the position of a levy flight of index 0 2 in the presence of an absorbing wall at the origin the solution of the associated fractional fokker planck equation can be constructed using a perturbation scheme around the brownian solution corresponding to 2 as an expansion in e 2 we obtain an explicit analytical solution exact at the first order in e which allows us to conjecture the precise asymptotic behavior of this pdf including the first subleading corrections for any careful numerical simulations as well as an exact computation for 1 confirm our conjecture

convex hull of n planar brownian motions exact results and an application to ecology we compute exactly the mean perimeter and area of the convex hull of n independent planar brownian paths each of duration t both for open and closed paths we show that the mean perimeter alpha n sqrt t and the mean area beta n t for all t the prefactors alpha n and beta n computed exactly for all n increase very slowly logarithmically with increasing n this slow growth is a consequence of extreme value statistics and has interesting implications in an ecological context in estimating the home range of a herd of animals with a population size n

martingale weak orlicz karamata hardy spaces associated with concave functions abstract in this paper we introduce the weak orlicz karamata hardy spaces associated with concave functions and establish the atomic decompositions as applications we prove several martingale inequalities and obtain a duality theorem

the convex hull for a random acceleration process in two dimensions we compute exactly the mean perimeter l t and the mean area a t of the convex hull of a random acceleration process of duration t in two dimensions we use an exact mapping that relates via cauchy s formulae the computation of the perimeter and the area of the convex hull of an arbitrary two dimensional stochastic process x t y t to the computation of the extreme value statistics of the associated one dimensional component process x t the latter can be computed exactly for the one dimensional random acceleration process even though the process is non markovian physically our results are relevant in describing the average shape of a semi flexible ideal polymer chain in two dimensions

le spectre de certaines mosaiques poissoniennes du plan et l enveloppe convexe du pont brownien the spectral empirical function of a homogeneous isotropic poisson mosaic process is a functional of the perimeter of the convex hull of planar brownian bridge some geometrical identities follow

persistence in nonequilibrium surface growth persistence probabilities of the interface height in 1 1 and 2 1 dimensional atomistic solid on solid stochastic models of surface growth are studied using kinetic monte carlo simulations with emphasis on models that belong to the molecular beam epitaxy mbe universality class both the initial transient and the long time steady state regimes are investigated we show that for growth models in the mbe universality class the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady state regimes for the mbe universality class the positive and negative persistence exponents in the steady state are found to be 0 66 0 02 and 0 78 0 02 respectively in 1 1 dimensions and 0 76 0 02 and 0 85 0 02 respectively in 2 1 dimensions the noise reduction technique is applied on some of the 1 1 dimensional models in order to obtain accurate values of the persistence exponents we show analytically that a relation between the steady state persistence exponent and the dynamic growth exponent found earlier to be valid for linear models should be satisfied by the smaller of the two steady state persistence exponents in the nonlinear models our numerical results for the persistence exponents are consistent with this prediction we also find that the steady state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state the dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form

semiflexible polymer in the half plane and statistics of the integral of a brownian curve a continuum model of a polymer with non zero bending energy fluctuating without overhangs in the half plane is considered the exact partition function is obtained from the marshall watson solution of the klein kramers equation for brownian motion in the half space the partition function contains information on probabilities associated with the integral of a brownian curve and reproduces sinai s t 5 4 result for the asymptotic first passage time density the t 5 2 dependence of a different passage probability implies a first order polymer adsorption transition for short range pinning potentials

mean perimeter of the convex hull of a random walk in a semi infinite medium we study various properties of the convex hull of a planar brownian motion defined as the minimum convex polygon enclosing the trajectory in the presence of an infinite reflecting wall recently in a rapid communication phys rev e textbf 91 050104 r 2015 we announced that the mean perimeter of the convex hull at time t rescaled by sqrt dt is a non monotonous function of the initial distance to the wall in the present article we first give all the details of the derivation of this mean rescaled perimeter in particular its value when starting from the wall and near the wall we then determine the physical mechanism underlying this surprising non monotonicity of the mean rescaled perimeter by analyzing the impact of the wall on two complementary parts of the convex hull finally we provide a further quantification of the convex hull by determining the mean length of the portion of the reflecting wall visited by the brownian motion as a function of the initial distance to the wall

universal first passage properties of discrete time random walks and levy flights on a line statistics of the global maximum and records abstract in these lecture notes i will discuss the universal first passage properties of a simple correlated discrete time sequence x 0 0 x 1 x 2 x n up to n steps where x i represents the position at step i of a random walker hopping on a continuous line by drawing independently at each time step a random jump length from an arbitrary symmetric and continuous distribution it includes e g the levy flights i will focus on the statistics of two extreme observables associated with the sequence i its global maximum and the time step at which the maximum occurs and ii the number of records in the sequence and their ages i will demonstrate how the universal statistics of these observables emerge as a consequence of pollaczek spitzer formula and the associated sparre andersen theorem

finite size scaling of the bayesian perceptron we study numerically the properties of the bayesian perceptron through a gradient descent on the optimal cost function the theoretical distribution of stabilities is deduced it predicts that the optimal generalizer lies close to the boundary of the space of error free solutions the numerical simulations are in good agreement with the theoretical distribution the extrapolation of the generalization error to infinite input space size agrees with the theoretical results finite size corrections are negative and exhibit two different scaling regimes depending on the training set size the variance of the generalization error vanishes for n confirming the property of self averaging s1063 651x 97 15406 x

punctuated evolution for the quasispecies model abstract biological evolution in a sequence space with random fitnesses is studied within eigen s quasispecies model a strong selection limit is employed in which the population resides at a single sequence at all times evolutionary trajectories start at a randomly chosen sequence and proceed to the global fitness maximum through a small number of intermittent jumps the distribution of the total evolution time displays a universal power law tail with exponent 2 simulations show that the evolutionary dynamics is very well represented by a simplified shell model in which the sub populations at local fitness maxima grow independently the shell model allows for highly efficient simulations and provides a simple geometric picture of the evolutionary trajectories

intrinsic volumes of sobolev balls with applications to brownian convex hulls a formula due to sudakov relates the first intrinsic volume of a convex set in a hilbert space to the maximum of the isonormal gaussian process over this set using this formula we compute the first intrinsic volumes of infinite dimensional convex compact sets including unit balls with respect to sobolev type seminorms and ellipsoids in the hilbert space we relate the distribution of the random one dimensional projections of these sets to the distributions s1 s2 c1 c2 studied by biane pitman yor bull ams 38 2001 we show that the k th intrinsic volume of the set of all functions on 0 1 which have lipschitz constant bounded by 1 and which vanish at 0 respectively which have vanishing integral is given by vk k 2 3 2 k 1 respectively vk k 1 2 2 3 2 k 3 2 this is related to the results of gao and vitale discrete comput geom 26 2001 elect comm probab 8 2003 who considered a similar question for functions with a restriction on the total variation instead of the lipschitz constant using the results of gao and vitale we give a new proof of the formula for the expected volume of the convex hull of the d dimensional brownian motion which is due to eldan elect j probab to appear additionally we prove an analogue of eldan s result for the brownian bridge similarly we show that the results on the intrinsic volumes of the lipschitz balls can be translated into formulae for the expected volumes of zonoids aumann integrals generated by the brownian motion and the brownian bridge also these results have discrete versions for gaussian random walks and bridges our proofs exploit sudakov s and tsirelson s theorems which establish a connection between the intrinsic volumes and the isonormal gaussian process

the predual and john nirenberg inequalities on generalized bmo martingale spaces in this paper we introduce the generalized bmo martingale spaces by stopping time sequences which enable us to characterize the dual spaces of martingale hardy lorentz spaces h s p q for 0 p 1 1 q moreover by duality we obtain a john nirenberg theorem for the generalized bmo martingale spaces when the stochastic basis is regular we also extend the boundedness of fractional integrals to martingale hardy lorentz spaces

universality and time scale invariance for the shape of planar levy processes for a broad class of planar markov processes viz l evy processes satisfying certain conditions valid e g in the case of brownian motion and l evy flights we establish an exact universal formula describing the shape of the convex hull of sample paths we show indeed that the average number of edges joining paths points separated by a time lapse ensuremath delta ensuremath tau ensuremath in left ensuremath delta ensuremath tau 1 ensuremath delta ensuremath tau 2 right is equal to 2 phantom rule 0 16em 0ex text ln left ensuremath delta ensuremath tau 2 ensuremath delta ensuremath tau 1 right regardless of the specific distribution of the process s increments and regardless of its total duration t the formula also exhibits invariance when the time scale is multiplied by any constant apart from its theoretical importance our result provides insights regarding the shape of two dimensional objects e g polymer chains modeled by the sample paths of stochastic processes generally more complex than brownian motion in particular for a total time or parameter duration t the average number of edges on the convex hull cut off to discard edges joining points separated by a time lapse shorter than some ensuremath delta ensuremath tau lt will be given by 2 phantom rule 0 16em 0ex text ln frac t ensuremath delta ensuremath tau thus it will only grow logarithmically rather than at some higher pace

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

persistence of a continuous stochastic process with discrete time sampling we introduce the concept of discrete time persistence which deals with zero crossings of a continuous stochastic process x t measured at discrete times t n delta t for a gaussian markov process with relaxation rate mu we show that the persistence no crossing probability decays as rho a n for large n where a exp mu delta t and we compute rho a to high precision we also define the concept of alternating persistence which corresponds to a 1 corresponding to motion in an unstable potential mu 0 there is a nonzero probability of having no zero crossings in infinite time and we show how to calculate it

time at which the maximum of a random acceleration process is reached we study the random acceleration model which is perhaps one of the simplest yet nontrivial non markov stochastic processes and is key to many applications for this non markov process we present exact analytical results for the probability density p tm t of the time tm at which the process reaches its maximum within a fixed time interval 0 t we study two different boundary conditions which correspond to the process representing respectively i the integral of a brownian bridge and ii the integral of a free brownian motion our analytical results are also verified by numerical simulations

atomic decompositions of vector valued weak musielak orlicz martingale spaces and applications abstract in this paper some vector valued weak martingale hardy spaces of musielak orlicz type are introduced and several atomic decompositions are established for them as applications of the atomic decompositions a sufficient condition for sublinear operators defined on vector valued weak musielak orlicz martingale spaces to be bounded is given using the sufficient condition some martingale inequalities are deduced these results closely depend on the geometrical properties of the banach space in which the martingales take values the results obtained here generalize the corresponding known results of scalar valued and vector valued weak martingale hardy spaces

embeddings between weak orlicz martingale spaces abstract let be an increasing and convex function on 0 with 0 0 satisfying that for any 0 there exists a positive constant c such that t c t t 0 let w l denote the corresponding weak orlicz space we obtain some embeddings between vector valued weak orlicz martingale spaces by establishing the w l inequalities for martingale transform operators with operator valued multiplying sequences these embeddings are closely related to the geometric properties of the underlying banach space in particular for any scalar valued martingale f f n n 1 we claim that sup n f n w l n 1 d f n 2 1 2 w l where d f n f n f n 1 and only depends on

extreme value statistics and traveling fronts various applications abstract an intriguing connection between extreme value statistics and traveling fronts has been found recently in a number of diverse problems in this short review we outline a few such problems and consider their various applications

maximum of n independent brownian walkers till the first exit from the half space we consider the one dimensional target search process that involves an immobile target located at the origin and n searchers performing independent brownian motions starting at the initial positions all on the positive half space the process stops when the target is first found by one of the searchers we compute the probability distribution of the maximum distance m visited by the searchers till the stopping time and show that it has a power law tail for large m thus all moments of m up to the order n 1 are finite while the higher moments diverge the prefactor bn increases with n faster than exponentially our solution gives the exit probability of a set of n particles from a box 0 l through the left boundary incidentally it also provides an exact solution of laplace s equation in an n dimensional hypercube with some prescribed boundary conditions the analytical results are in excellent agreement with monte carlo simulations

distribution of the time at which n vicious walkers reach their maximal height we study the extreme statistics of n nonintersecting brownian motions vicious walkers over a unit time interval in one dimension using path integral techniques we compute exactly the joint distribution of the maximum m and of the time m at which this maximum is reached we focus in particular on nonintersecting brownian bridges watermelons without wall and nonintersecting brownian excursions watermelons with a wall we discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium dprm with one free end point on the other hand we also check our results using numerical simulations of dyson s brownian motion and confront them with numerical simulations of the polynuclear growth model png and of a model of dprm on a discrete lattice some of the results presented here were announced in a recent letter j rambeau and g schehr europhys lett 91 60006 2010

a convex hull based estimator of home range size this paper considers the problem of estimating the home range size of an animal from locational data which are usually collected by use of radio telemetry with convex hull based estimators it is shown that the widely used minimum convex polygon has some undesirable characteristics as an estimator and thus should be used with caution a method based on convex hull peeling of the data to estimate convex density contours is studied as an alternative simple home range size estimator monte carlo simulations show that this estimator has very similar statistical properties if the point fixes are generated from a bivariate normal distribution a uniform on a disc distribution and two other distributions with intermediate tail behaviour these standard distributions are used as a reference to define a sample size bias corrected estimator of home range size and to provide an approximate method for calculating the precision of this estimator the technique is applied to the estimation of the home range sizes of deer

universal properties of a run and tumble particle in arbitrary dimension we consider an active run and tumble particle rtp in d dimensions starting from the origin and evolving over a time interval 0 t we examine three different models for the dynamics of the rtp the standard rtp model with instantaneous tumblings a variant with instantaneous runs and a general model in which both the tumblings and the runs are noninstantaneous for each of these models we use the sparre andersen theorem for discrete time random walks to compute exactly the probability that the x component does not change sign up to time t showing that it does not depend on d as a consequence of this result we compute exactly other x component properties namely the distribution of the time of the maximum and the record statistics showing that they are universal i e they do not depend on d moreover we show that these universal results hold also if the speed v of the particle after each tumbling is random drawn from a generic probability distribution our findings are confirmed by numerical simulations some of these results have been announced in a recent letter phys rev lett 124 090603 2020 10 1103 physrevlett 124 090603

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

large deviations of convex hulls of self avoiding random walks a global picture of a random particle movement is given by the convex hull of the visited points we obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self avoiding random walks namely the classical self avoiding walk the smart kinetic self avoiding walk and the loop erased random walk to obtain a comprehensive description of the measured random quantities we applied sophisticated large deviation techniques which allowed us to obtain the distributions over a large range of support down to probabilities far smaller than p 10 ensuremath 100 we give an approximate closed form of the so called large deviation rate function mathrm ensuremath phi which generalizes above the upper critical dimension to the previously studied case of the standard random walk further we show correlations between the two observables also in the limits of atypical large or small values

a gaussian scenario for unsupervised learning we consider random patterns on the n sphere which are uniformly distributed with the exception of a single symmetry breaking orientation along which they are gaussian distributed the unsupervised recognition of this orientation by different learning rules is studied in the large n limit using the replica method the model is simple enough to be analytically tractable and rich enough to exhibit most of the phenomena observed with other pattern distributions a learning algorithm based on the minimization of a cost function is identified which reaches the upper theoretical limit imposed by the optimal bayes learning scenario an implementation of this algorithm is proposed and tested numerically

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

convex hull of n planar brownian paths an exact formula for the average number of edges we establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent brownian paths in the plane this requires the introduction of a counting criterion which amounts to cutting off edges that are in a specific sense small the main argument consists in a mapping between planar brownian convex hulls and configurations of constrained independent linear brownian motions this new formula is confirmed by retrieving an existing exact result on the average perimeter of the boundary of brownian convex hulls in the plane

joint distributions of partial and global maxima of a brownian bridge we analyze the joint distributions and temporal correlations between the partial maximum m and the global maximum m achieved by a brownian bridge on the subinterval and on the entire interval respectively we determine three probability distribution functions the joint distribution of both maxima the distribution p m of the partial maximum and the distribution of the gap between the maxima we present exact results for the moments of these distributions and quantify the temporal correlations between m and m by calculating the pearson correlation coefficient

universal survival probability for a correlated random walk and applications to records we consider a model of space continuous one dimensional random walk with simple correlation between the steps the probability that two consecutive steps have same sign is q with 0 leq q leq 1 the parameter q allows thus to control the persistence of the random walk we compute analytically the survival probability of a walk of n steps showing that it is independent of the jump distribution for any finite n this universality is a consequence of the sparre andersen theorem for random walks with uncorrelated and symmetric steps we then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk which show the same universality in particular we show that the distribution of the number of records for a walk of n gg 1 steps is the same as for a random walk with n rm eff q n 2 1 q uncorrelated and symmetrically distributed steps we also show that in the regime where n to infty and q to 1 with y n 1 q this model converges to the run and tumble particle a persistent random walk often used to model the motion of bacteria our theoretical results are confirmed by numerical simulations

hardy lorentz spaces for b valued martingales abstract in this paper we study the hardy lorentz spaces for banach space valued martingales the dual spaces are characterized and several martingale inequalities are established the proofs mainly depend on the classical tool of atomic decompositions as usual these conclusions are closely related to the geometrical properties of the underlying banach spaces

inelastic collapse of a ball bouncing on a randomly vibrating platform a theoretical study is undertaken of the dynamics of a ball which is bouncing inelastically on a randomly vibrating platform of interest are the distributions of the number of flights n f and the total time ensuremath tau c until the ball has effectively collapsed i e coalesced with the platform in the strictly elastic case both distributions have power law tails characterized by exponents that are universal i e independent of the detail of the platform noise distribution however in the inelastic case both distributions have exponential tails p n f ensuremath sim mathrm exp ensuremath ensuremath theta 1 n f and p ensuremath tau c ensuremath sim mathrm exp ensuremath ensuremath theta 2 ensuremath tau c the decay exponents ensuremath theta 1 and ensuremath theta 2 depend continuously on the coefficient of restitution and are nonuniversal however as one approaches the elastic limit they vanish in a manner which turns out to be universal an explicit expression for ensuremath theta 1 is provided for a particular case of the platform noise distribution

unsupervised learning of binary vectors a gaussian scenario we study a model of unsupervised learning where the real valued data vectors are isotropically distributed except for a single symmetry breaking binary direction bin 1 1 n onto which the projections have a gaussian distribution we show that a candidate vector j undergoing gibbs learning in this discrete space approaches the perfect match j b exponentially in addition to the second order retarded learning phase transition for unbiased distributions we show that first order transitions can also occur extending the known result that the center of mass of the gibbs ensemble has bayes optimal performance we show that taking the sign of the components of this vector clipping leads to the vector with optimal performance in the binary space these upper bounds are shown generally not to be saturated with the technique of transforming the components of a special continuous vector except in asymptotic limits and in a special linear case simulations are presented which are in excellent agreement with the theoretical results

statistical mechanics of unsupervised structure recognition a model of unsupervised learning is studied where the environment provides n dimensional input examples that are drawn from two overlapping gaussian clouds we consider the optimization of two different objective functions the search for the direction of the largest variance in the data and the largest separating gap stability between clusters of examples respectively by means of a statistical mechanics analysis we investigate how well the underlying structure is inferred from a set of examples the performances of the learning algorithms depend crucially on the actual shape of the input distribution a generic result is the existence of a critical number of examples needed for successful learning the learning strategies are compared with methods different in spirit such as the estimation of parameters in a model distribution and an information theoretical approach

last passage time for linear diffusions and application to the emptying time of a box we study the statistics of last passage time for linear diffusions first we present an elementary derivation of the laplace transform of the probability density of the last passage time thus recovering known results from the mathematical literature we then illustrate them on several explicit examples in a second step we study the spectral properties of the schrodinger operator associated to such diffusions in an even potential u x u x unveiling the role played by the so called weyl coefficient indeed in this case our approach allows us to relate the last passage times for dual diffusions i e diffusions driven by opposite force fields and to obtain new explicit formulae for the mean last passage time we further show that for such even potentials the small time t expansion of the mean last passage time on the interval 0 t involves the korteveg de vries invariants which are well known in the theory of schrodinger operators finally we apply these results to study the emptying time of a one dimensional box of size l containing n independent brownian particles subjected to a constant drift in the scaling limit where both n rightarrow infty and l rightarrow infty keeping the density rho n l fixed we show that the limiting density of the emptying time is given by a gumbel distribution our analysis provides a new example of the applications of extreme value statistics to out of equilibrium systems

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

flux to a trap the flux of particles to a single trap is investigated for two systems 1 particles in 3d space which jump a fixed step lengthl the rayleigh flight and are adsorbed by a spherical surface and 2 particles on a lattice jumping to nearest neighbor sites with a single adsorbing site initially the particles are uniformly distributed outside the traps when the jump length goes to zero both processes go over to regular diffusion and the first case yields the diffusive flux to a sphere as solved by smoluchowski for nonzero step length the flux for large times is given by a modified form of smoluchowski s result with the effective radius replaced byr cl wherec 0 29795219 andcl is the milne extrapolation length for this problem for the second problem a similar expression for the flux is found with the effective trap radius a function of the lattice sc bcc fcc being considered

dual spaces of weak orlicz hardy spaces for vector valued martingales abstract the dual spaces of weak orlicz hardy spaces for b valued martingales which are defined by quasi concave functions are identified to obtain the results the so called weak type generalized campanato spaces are introduced and the classical technique of atomic decompositions is employed the results obtained here extend the corresponding known results in the literature from real valued setting to vector valued setting and from strong type spaces to weak type spaces

persistence exponents and the statistics of crossings and occupation times for gaussian stationary processes we consider the persistence probability the occupation time distribution and the distribution of the number of zero crossings for discrete or equivalently discretely sampled gaussian stationary processes gsps of zero mean we first consider the ornstein uhlenbeck process finding expressions for the mean and variance of the number of crossings and the partial survival probability we then elaborate on the correlator expansion developed in an earlier paper g c m a ehrhardt and a j bray phys rev lett 88 070602 2002 to calculate discretely sampled persistence exponents of gsps of known correlator by means of a series expansion in the correlator we apply this method to the processes d n x dt n ensuremath eta t with ng 3 incorporating an extrapolation of the series to the limit of continuous sampling we then extend the correlator method to calculate the occupation time and crossing number distributions as well as their partial survival distributions and the means and variances of the occupation time and number of crossings we apply these general methods to the d n x dt n ensuremath eta t processes for n 1 random walk n 2 random acceleration and larger n and to simple diffusion from random initial conditions in one to three dimensions the results for discrete sampling are extrapolated to the continuum limit where possible

bayesian learning versus optimal learning abstract we consider the optimal performance that may be reached in the problem of learning the symmetry breaking direction of a cloud of p n points in a n dimensional space the performance is measured through the overlap ropt between the true symmetry breaking direction and the learnt one depending on the problem the learning curves ropt may present discontinuities we show that close to these bayesian learning is not optimal

on a random area variable arising in discrete time queues and compact directed percolation a well known discrete time single server queueing system with mean arrival rate and mean departure rate is considered from the perspective of the area a swept out by the queue occupation process during a busy period we determine the exact form of the tail of the distribution pr a x in particular we show that pr a x cx 1 4 exp dx1 2 for all 1 where and expressions for c and d are given for the critical case 1 we show that pr a x c x 1 3 with c also given a simple mapping used in the derivation establishes a connection with compact directed percolation on a square lattice as a corollary therefore we are also able to specify the large area asymptotic behaviour of this model at all points in the phase diagram this extends previous scaling results which are only valid close to the percolation threshold

optimal unsupervised learning we introduce an inferential approach to unsupervised learning which allows us to define an optimal learning strategy applying these ideas to a simple previously studied model we show that it is impossible to detect structure in data until a critical number of examples have been presented an effect which will be observed in all problems with certain underlying symmetries thereafter the advantage of optimal learning over previously studied learning algorithms depends critically upon the distribution of patterns optimal learning may be exponentially faster models with more subtle correlations are harder to analyse but in a simple limit of one such problem we calculate exactly the efficacy of an algorithm similar to some used in practice and compare it to that of the optimal prescription

estimation of a convex density contour in two dimensions abstract if a density in two dimensions has a convex contour containing probability the contour may be estimated from a sample by finding the convex polygon of smallest area containing a proportion of the sample points an algorithm for finding a particular contour is given that takes o n 2 space and o n 3 time for n sample points

leapover lengths and first passage time statistics for levy flights exact results for the first passage time and leapover statistics of symmetric and one sided levy flights lfs are derived lfs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one sided lfs and surprisingly with an index alpha 2 for symmetric lfs the first passage time distribution scales like a power law with an index 1 2 as required by the sparre andersen theorem for symmetric lfs whereas one sided lfs have a narrow distribution of first passage times the exact analytic results are confirmed by extensive simulations

record statistics for multiple random walks we study the statistics of the number of records r n n for n identical and independent symmetric discrete time random walks of n steps in one dimension all starting at the origin at step 0 at each time step each walker jumps by a random length drawn independently from a symmetric and continuous distribution we consider two cases i when the variance ensuremath sigma 2 of the jump distribution is finite and ii when ensuremath sigma 2 is divergent as in the case of l evy flights with index 0l ensuremath mu l2 in both cases we find that the mean record number ensuremath langle r n n ensuremath rangle grows universally as ensuremath sim ensuremath alpha n sqrt n for large n but with a very different behavior of the amplitude ensuremath alpha n for ng1 in the two cases we find that for large n ensuremath alpha n ensuremath approx 2 sqrt mathrm ln n independently of ensuremath sigma 2 in case i in contrast in case ii the amplitude approaches to an n independent constant for large n ensuremath alpha n ensuremath approx 4 sqrt ensuremath pi independently of 0l ensuremath mu l2 for finite ensuremath sigma 2 we argue and this is confirmed by our numerical simulations that the full distribution of r n n sqrt n ensuremath 2 sqrt mathrm ln n sqrt mathrm ln n converges to a gumbel law as n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty in case ii our numerical simulations indicate that the distribution of r n n sqrt n converges for n ensuremath rightarrow ensuremath infty and n ensuremath rightarrow ensuremath infty to a universal nontrivial distribution independently of ensuremath mu we discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the standard poor s 500 index