the true self repelling motion we construct and study a continuous real valued random process which is of a new type it is self interacting self repelling but only in a local sense it only feels the self repellance due to its occu pation time measure density in the immediate neighbourhood of the point it is just visiting we focus on the most natural process with these properties that we call true self repelling motion this is the continuous counterpart to the integer valued true self avoiding walk which had been studied among others by the first author one of the striking properties of true self repelling motion is that although the coupleoxt occupation time measure of x at time tu is a continuous markov process x is not driven by a stochastic diaerential equation and is not a semi martingale it turns out for instance that it has a finite variation of order 3 2 which contrasts with the finite quadratic variation of semi martingales one of the key tools in the construction of x is a continuous system of coalescing brownian motions similar to those that have been constructed by arratia a1 a2 we derive various properties of x existence and properties of the occupation time densities ltoxu local variation etc and an identity that shows that the dynamics of x can be very loosely speaking described as follows dxt is equal to the gradient in space of ltoxu in a gener alized sense even though x7 ltoxu is not diaerentiable

a fast fourier transform on multipoles fftm algorithm for solving helmholtz equation in acoustics analysis this article presents a fast algorithm for the efficient solution of the helmholtz equation the method is based on the translation theory of the multipole expansions here the speedup comes from the convolution nature of the translation operators which can be evaluated rapidly using fast fourier transform algorithms also the computations of the translation operators are accelerated by using the recursive formulas developed recently by gumerov and duraiswami siam j sci comput 25 1344 1381 2003 it is demonstrated that the algorithm can produce good accuracy with a relatively low order of expansion efficiency analyses of the algorithm reveal that it has computational complexities of o na where a ranges from 1 05 to 1 24 however this method requires substantially more memory to store the translation operators as compared to the fast multipole method hence despite its simplicity in implementation this memory requirement issue may limit the application of this algorithm to solving very lar

efficient fmm accelerated vortex methods in three dimensions via the lamb helmholtz decomposition vortex methods are used to efficiently simulate incompressible flows using lagrangian techniques use of the fmm fast multipole method allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods both equations require field evaluation of constrained divergence free vector valued quantities velocity vorticity and cross terms from these these are usually evaluated by performing several fmm accelerated sums of scalar harmonic functions we present a formulation of vortex methods based on the lamb helmholtz decomposition of the velocity in terms of two scalar potentials in its original form this decomposition is not invariant with respect to translation violating a key requirement for the fmm one of the key contributions of this paper is a theory for translation for this representation the translation theory is developed by introducing conversion operators which enable the representation to be restored in an arbitrary reference frame using this form efficient vortex element computations can be made which need evaluation of just two scalar harmonic fmm sums for evaluating the velocity and vorticity evolution terms details of the decomposition translation and conversion formulae and sample numerical results are presented

fluctuation theory of connectivities for subcritical random cluster models we develop a fluctuation theory of connectivities for subcritical random cluster models the theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one dimensional chains of irreducible objects statistics of local observables for example displacement over such chains obey classical limit laws and our construction leads to an effective random walk representation of percolation clusters the results include a derivation of a sharp ornstein zernike type asymptotic formula for two point functions a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling in two dimensions duality considerations enable a reformulation of these results for supercritical nearest neighbor random cluster measures in particular for nearest neighbor potts models in the phase transition regime accordingly we prove that in two dimensions potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive thus no roughening transition is possible in the whole regime where our results apply our results hold under an assumption of exponential decay of finite volume wired connectivities assumption 1 2 below in rectangular domains that is conjectured to hold in the whole subcritical regime the latter is known to be true in any dimensions when q 1 q 2 and when q is sufficiently large in two dimensions assumption 1 2 holds whenever there is an exponential decay of connectivities in the infinite volume measure by duality this includes all supercritical nearest neighbor potts models with positive surface tension between ordered phases

quantum correlations and distinguishability of quantum states a survey of various concepts in quantum information is given with a main emphasis on the distinguishability of quantum states and quantum correlations covered topics include generalized and least square measurements state discrimination quantum relative entropies the bures distance on the set of quantum states the quantum fisher information the quantum chernoff bound bipartite entanglement the quantum discord and geometrical measures of quantum correlations the article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them and for mathematicians looking for an up to date introductory course on these subjects which are mainly developed in the physics literature

high performance blas formulation of the multipole to local operator in the fast multipole method the multipole to local m2l operator is the most time consuming part of the far field computation in the fast multipole method for laplace equation its natural expression though commonly used does not respect a sharp error bound we here first prove the correctness of a second expression we then propose a matrix formulation implemented with basic linear algebra subprograms blas routines in order to speed up its computation for these two expressions we also introduce special data storages in memory to gain greater computational efficiency this blas scheme is finally compared for uniform distributions to other m2l improvements such as block fft fft with polynomial scaling rotations and plane wave expansions when considering runtime extra memory storage numerical stability and common precisions for laplace equation the blas version appears as the best one

optimal secondary source position in exterior spherical acoustical holophony abstract exterior spherical acoustical holophony is a branch of spatial audio reproduction that deals with the rendering of a given free field radiation pattern the primary field by using a compact spherical loudspeaker array the secondary source more precisely the primary field is known on a spherical surface surrounding the primary and secondary sources and since the acoustic fields are described in spherical coordinates they are naturally subjected to spherical harmonic analysis besides the inverse problem of deriving optimal driving signals from a known primary field is ill posed because the secondary source cannot radiate high order spherical harmonics efficiently especially in the low frequency range as a consequence a standard least squares solution will overload the transducers if the primary field contains such harmonics here this is avoided by discarding the strongly decaying spherical waves which are identified through inspection of the radiation efficiency curves of the secondary source however such an unavoidable regularization procedure increases the least squares error which also depends on the position of the secondary source this paper deals with the above mentioned questions in the context of far field directivity reproduction at low and medium frequencies in particular an optimal secondary source position is sought which leads to the lowest reproduction error in the least squares sense without overloading the transducers in order to address this issue a regularization quality factor is introduced to evaluate the amount of regularization required it is shown that the optimal position improves significantly the holophonic reconstruction and maximizes the regularization quality factor minimizes the amount of regularization which is the main general contribution of this paper therefore this factor can also be used as a cost function to obtain the optimal secondary source position

acoustic scattering by arbitrary distributions of disjoint homogeneous cylinders or spheres a t matrix formulation is presented to compute acoustic scattering from arbitrary disjoint distributions of cylinders or spheres each with arbitrary uniform acoustic properties the generalized approach exploits the similarities in these scattering problems to present a single system of equations that is easily specialized to cylindrical or spherical scatterers by employing field expansions based on orthogonal harmonic functions continuity of pressure and normal particle velocity are directly enforced at each scatterer using diagonal analytic expressions to eliminate the need for integral equations the effect of a cylinder or sphere that encloses all other scatterers is simulated with an outer iterative procedure that decouples the inner object solution from the effect of the enclosing object to improve computational efficiency when interactions among the interior objects are significant numerical results establish the validity and efficiency of the outer iteration procedure for nested objects two and three dimensional methods that employ this outer iteration are used to measure and characterize the accuracy of two dimensional approximations to three dimensional scattering of elevation focused beams

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

bayes cost reduction algorithm in quantum hypothesis testing corresp an iterative procedure is described for reducing the bayes cost in decisions among m 2 quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses the resulting decision strategy is a projection valued measure and yields an upper bound to the minimum attainable bayes cost from it is derived an algorithm for finding the optimum measurement states for choosing among m linearly independent pure states with minimum probability of error the method is also applied to decisions among m unimodal coherent quantum signals in thermal noise

on weak mixing in lattice models for lattice models on d weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region for a wide class of models on 2 including all finite range models we show that weak mixing is a consequence of gibbs uniqueness exponential decay of an appropriate form of connectivity and a natural coupling property in particular on 2 the fortuin kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially and the q state potts model above the critical temperature is weak mixing whenever correlations decay exponentially a hypothesis satisfied if q is sufficiently large ratio weak mixing is the property that uniformly over events a and b occurring on subsets and respectively of the lattice p a b p a p b 1 decreases exponentially in the distance between and we show that under mild hypotheses for example finite range weak mixing implies ratio weak mixing

discord of response the presence of quantum correlations in a quantum state is related to the state s response to local unitary perturbations such a response is quantified by the distance between the unperturbed and perturbed states minimized with respect to suitably identified sets of local unitary operations in order to be a bona fide measure of quantum correlations the distance function must be chosen among those that are contractive under completely positive and trace preserving cptp maps the most relevant instances of such physically well behaved metrics include the trace the bures and the hellinger distance to each of these metrics one can associate the corresponding discord of response namely the trace or hellinger or bures minimum distance from the set of unitarily perturbed states all these three discords of response satisfy the basic axioms for a proper measure of quantum correlations in the present work we focus in particular on the bures distance which enjoys the unique property of being both riemannian and contractive under cptp maps and admits important operational interpretations in terms of state distinguishability we compute analytically the bures discord of response for two qubit states with maximally mixed marginals and we compare it with the corresponding bures geometric discord namely the geometric measure of quantum correlations defined as the bures distance from the set of classical quantum states finally we investigate and identify the maximally quantum correlated two qubit states according to the bures discord of response these states exhibit a remarkable nonlinear dependence on the global state purity

universality classes for extreme value statistics the equilibrium low temperature physics of disordered systems is governed by the statistics of extremely low energy states it is thus relevant to discuss the possible universality classes for extreme value statistics we compare the usual probabilistic classification to the results of the replica approach we show in detail for several problems including the random energy model and the decaying burgers turbulence that one class of independent variables corresponds exactly to the so called one step replica symmetry breaking solution in the replica language we argue that this universality class holds if the correlations are sufficiently weak and propose a conjecture on the level of correlations which leads to different universality classes

calculation of acoustical scattering from a cluster of scatterers the problem of determining the field scattered by a cluster of scatterers when they are insonified by a known acoustical field is addressed the problem is formulated by using the t matrix method and the resulting system of linear equations is solved by using the multilevel fast multipole algorithm mlfma and the fast multipole method fast fourier transform fmmfft method and the efficiency of the two methods is compared it was observed that in general the mlfma performs better than the fmmfft algorithm however when the scatterers are distributed uniformly on a rectangular grid the fmmfft algorithm performs as good as the mlfma the accuracy of the methods is evaluated by modeling a spherical scatterer as composed of many small spheres

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

dobrushin kotecký shlosman theorem up to the critical temperature we develop a non perturbative version of the dobrushin koteck shlosman theory of phase separation in the canonical 2d ising ensemble the results are valid for all temperatures below critical

geometric quantum discord with bures distance the qubit case the minimal bures distance of a quantum state of a bipartite system ab to the set of classical states for subsystem a defines a geometric measure of quantum discord when a is a qubit we show that this geometric quantum discord is given in terms of the eigenvalues of a 2nb 2nb hermitian matrix nb being the hilbert space dimension of the other subsystem b as a first application we calculate the geometric discord for the output state of the dqc1 algorithm we find that it takes its highest value when the unitary matrix from which the algorithm computes the trace has its eigenvalues uniformly distributed on the unit circle modulo a symmetry with respect to the origin as a second application we derive an explicit formula for the geometric discord of a two qubit state with maximally mixed marginals and compare it with other measures of quantum correlations we also determine the closest classical states to

airy distribution function from the area under a brownian excursion to the maximal height of fluctuating interfaces the airy distribution function describes the probability distribution of the area under a brownian excursion over a unit interval surprisingly this function has appeared in a number of seemingly unrelated problems mostly in computer science and graph theory in this paper we show that this distribution function also appears in a rather well studied physical system namely the fluctuating interfaces we present an exact solution for the distribution p h m l of the maximal height h m measured with respect to the average spatial height in the steady state of a fluctuating interface in a one dimensional system of size l with both periodic and free boundary conditions for the periodic case we show that p h m l l 1 2f h m l 1 2 for all l 0 where the function f x is the airy distribution function this result is valid for both the edwards wilkinson ew and the kardar parisi zhang interfaces for the free boundary case the same scaling holds p h m l l 1 2f h m l 1 2 but the scaling function f x is different from that of the periodic case we compute this scaling function explicitly for the ew interface and call it the f airy distribution function numerical simulations are in excellent agreement with our analytical results our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables some of these results were announced in a recent letter s n majumdar and a comtet phys rev lett 92 225501 2004

quantum discord a measure of the quantumness of correlations two classically identical expressions for the mutual information generally differ when the systems involved are quantum this difference defines the quantum discord it can be used as a measure of the quantumness of correlations separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord thus showing that absence of entanglement does not imply classicality we relate this to the quantum superposition principle and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system i e the pointer states

unifying approach to the quantification of bipartite correlations by bures distance the notion of distance defined on the set of states of a composite quantum system can be used to quantify total quantum and classical correlations in a unifying way we provide new closed formulae for classical and total correlations of two qubit bell diagonal states by considering the bures distance complementing the known corresponding expressions for entanglement and more general quantum correlations we thus complete the quantitative hierarchy of bures correlations for bell diagonal states we then explicitly calculate bures correlations for two relevant families of states werner states and rank 2 bell diagonal states highlighting the subadditivity which holds for total correlations with respect to the sum of classical and quantum ones when using bures distance finally we analyse a dynamical model of two independent qubits locally exposed to non dissipative decoherence channels where both quantum and classical correlations measured by bures distance exhibit freezing phenomena in analogy with other known quantifiers of correlations

recursions for the computation of multipole translation and rotation coefficients for the 3 d helmholtz equation we develop exact expressions for the coefficients of series representations of translations and rotations of local and multipole fundamental solutions of the helmholtz equation in spherical coordinates these expressions are based on the derivation of recurrence relations some of which to our knowledge are presented here for the first time the symmetry and other properties of the coefficients are also examined and based on these efficient procedures for calculating them are presented our expressions are direct and do not use the clebsch gordan coefficients or the wigner 3 j symbols although we compare our results with methods that use these to prove their accuracy for evaluating an nt term truncation of the translated series involving o n t 2 multipoles our expressions require o n t 3 evaluations compared to previous exact expressions that require o n t 5 operations

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

fluctuation of a planar brownian loop capturing a large area we consider a planar brownian loop b that is run for a time t and conditioned on the event that its range encloses the unusually high area of t 2 with t 0 being large the conditioned process denoted by x was proposed by senya shlosman as a model for the fluctuation of a phase boundary we study the deviation of the range of x from a circle of radius t this deviation is measured by the inradius r in x and outradius r out x which are the maximal radius of a disk enclosed by the range of x and the minimal radius of a disk that contains this range we prove that in a typical realization of the conditioned measure each of these quantities differs from t by at most t 2 3 e

taylor expansion based fast multipole method for 3 d helmholtz equations in layered media abstract in this paper we develop a taylor expansion te based fast multipole method fmm for low frequency 3d helmholtz green s function in layered media two forms of taylor expansions with either non symmetric or symmetric derivatives of layered media green s functions are used for the implementations of the proposed te fmm in the implementation with non symmetric derivatives an algorithm based on discrete complex image approximations and recurrence formulas is shown to be very efficient and accurate in computing the high order derivatives meanwhile the implementation based on symmetric derivatives is more robust and pre computed tables for the high order derivatives in translation operators are used numerical tests in layered media have validated the accuracy and o n complexity of the proposed algorithms

regularization methods for near field acoustical holography the reconstruction of the pressure and normal surface velocity provided by near field acoustical holography nah from pressure measurements made near a vibrating structure is a linear ill posed inverse problem due to the existence of strongly decaying evanescentlike waves regularization provides a technique of overcoming the ill posedness and generates a solution to the linear problem in an automated way we present four robust methods for regularization the standard tikhonov procedure along with a novel improved version landweber iteration and the conjugate gradient approach each of these approaches can be applied to all forms of interior or exterior nah problems planar cylindrical spherical and conformal we also study two parameter selection procedures the morozov discrepancy principle and the generalized cross validation which are crucial to any regularization theory in particular we concentrate here on planar and cylindrical holography these forms of nah which rely on the discrete fourier transform are important due to their popularity and to their tremendous computational speed in order to use regularization theory for the separable geometry problems we reformulate the equations of planar cylindrical and spherical nah into an eigenvalue problem the resulting eigenvalues and eigenvectors couple easily to regularization theory which can be incorporated into the nah software with little sacrifice in computational speed the resulting complete automation of the nah algorithm for both separable and nonseparable geometries overcomes the last significant hurdle for nah

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

unified view of quantum and classical correlations we discuss the problem of the separation of total correlations in a given quantum state into entanglement dissonance and classical correlations using the concept of relative entropy as a distance measure of correlations this allows us to put all correlations on an equal footing entanglement and dissonance whose definition is introduced here jointly belong to what is known as quantum discord our methods are completely applicable for multipartite systems of arbitrary dimensions we investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states

multiple scattering by multiple spheres a new proof of the lloyd berry formula for the effective wavenumber we provide the first classical derivation of the lloyd berry formula for the effective wavenumber of an acoustic medium filled with a sparse random array of identical small scatterers our approach clarifies the assumptions under which the lloyd berry formula is valid more precisely we derive an expression for the effective wavenumber which assumes the validity of lax s quasi crystalline approximation but makes no further assumptions about scatterer size and then we show that the lloyd berry formula is obtained in the limit as the scatterer size tends to zero

efficient recursive generation of the scalar spherical multipole translation matrix new efficient recursive procedures for generating the translation matrix of the scalar spherical multipole field are described they are based on a new set of recurrence relations that result when the angular momentum operator is applied to the spherical multipole field their efficiency and accuracy are compared analytically and through a computer experiment with those of the brute force method and an existing recursive procedure

phase separation in random cluster models ii the droplet at equilibrium and local deviation lower bounds we study the droplet that results from conditioning the planar subcritical fortuin kasteleyn random cluster model on the presence of an open circuit 0 encircling the origin and enclosing an area of at least or exactly n2 we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary conv 0 of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length namely the length of the longest line segment of which the polygon conv 0 is formed we prove that there exists a constant c 0 such that the conditional probability that the normalised quantity n 1 3 log n 2 3 mlr 0 exceeds c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mfl 0 to obtain these bounds we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a markov chain that resamples sections of the circuit boundary we analyse the chain at equilibrium to prove the local roughness lower bounds alongside complementary upper bounds provided in 14 the fluctuations mlr 0 and mfl 0 are determined up to a constant factor

a scalar potential formulation and translation theory for the time harmonic maxwell equations we develop a computational method based on the debye scalar potential representation which efficiently reduces the solution of maxwell s equations to the solution of two scalar helmholtz equations one of the key contributions of this paper is a theory for the translation of maxwell solutions using such a representation since the scalar potential form is not invariant with respect to translations the translation theory is developed by introducing conversion operators which enable the representation of the electric and magnetic vector fields via scalar potentials in an arbitrary reference frame advantages of this representation include the fact that only two helmholtz equations need be solved and moreover the divergence free constraints are satisfied automatically by construction truncation error bounds are also presented the availability of a translation theory and error bounds for this representation can find application in methods such as the fast multipole method for illustration of the use of the representation and translation theory we implemented an algorithm for the simulation of mie scattering off a system of spherical objects of different sizes and dielectric properties using a variant of the t matrix method the resulting system was solved using an iterative method based on gmres the results of the computations agree well with previous computational and experimental results

short note a fast and stable method for rotating spherical harmonic expansions in this paper we present a simple and efficient method for rotating a spherical harmonic expansion this is a well studied problem arising in classical scattering theory quantum mechanics and numerical analysis usually addressed through the explicit construction of the wigner rotation matrices we show that rotation can be carried out easily and stably through pseudospectral projection without ever constructing the matrix entries themselves existing fast algorithms based on recurrence relations are subject to a variety of instabilities limiting the effectiveness of the approach for expansions of high degree

phase separation in random cluster models iii circuit regularity we study the droplet that results from conditioning the subcritical fortuin kasteleyn random cluster model on the presence of an open circuit encircling the origin and enclosing an area of at least or exactly n 2 in this paper we prove that the resulting circuit is highly regular we define a notion of a regeneration site in such a way that for any such site v the circuit cuts through the radial line segment through v only at v we show that provided that the conditioned circuit is centred at the origin in a natural sense the set of regeneration sites reaches into all parts of the circuit with maximal distance from one such site to the next being at most logarithmic in n with high probability the result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations and as such it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in hammond arxiv 1001 1527 2009 arxiv 1001 1528 2009

lower bounds for boundary roughness for droplets in bernoulli percolation we consider boundary roughness for the droplet created when supercritical two dimensional bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l 2 for large l the maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull we show that for large l this maximum is at least of order l 1 3 logl 2 3 this complements the upper bound of order l 1 3 logl 2 3 proved in al3 for the average local roughness the exponent 1 3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions

isoperimetry in two dimensional percolation we study isoperimetric sets i e sets with minimal boundary for a prescribed volume on the unique infinite connected component of supercritical bond percolation on the square lattice in the limit of the volume tending to infinity properly scaled isoperimetric sets are shown to converge in the hausdorff metric to the solution of an isoperimetric problem in 2 with respect to a particular norm as part of the proof we also show that the anchored isoperimetric profile as well as the cheeger constant of the giant component in finite boxes scale to deterministic quantities this settles a conjecture of itai benjamini for the square lattice 2015 wiley periodicals inc

a matrix version of the fast multipole method we present a matrix interpretation of the three dimensional fast multipole method fmm the fmm is for efficient computation of gravitational electrostatic potentials and fields it has found various applications and inspired the design of many efficient algorithms the one dimensional fmm is well interpreted in terms of matrix computations the three dimensional matrix version reveals the underlying matrix structures and computational techniques used in fmm it also provides a unified view of algorithm variants as well as existing and emerging implementations of the fmm

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

constrained brownian motion fluctuations away from circular and parabolic barriers motivated by the polynuclear growth model we consider a brownian bridge b t with b t 0 conditioned to stay above the semicircle ct t p t 2 t2 in the limit of large t the fluctuation scale of b t ct t is t 1 3 and its time correlation scale is t 2 3 we prove that in the sense of weak convergence of path measures the conditioned brownian bridge when properly rescaled converges to a stationary diffusion process with a drift explicitly given in terms of airy functions the dependence on the reference point t t 2 1 1 is only through the second derivative of ct t at t t we also prove a corresponding result where instead of the semicircle the barrier is a parabola of height t 1 2 the fluctuation scale is then t 2 3 more general conditioning shapes are briefly discussed 1 introduction and main results we consider the brownian bridge b t over the time interval t t t 0 b t b t 0 conditioned to lie above the semicircle ct t t 2 t 2 let b t be the conditioned brownian bridge and let xt t b t ct t be the deviation of b t away from ct t see figure 1 clearly xt t 0 xt t xt t 0 and the path measure of the process is defined on c t t r c t t the space of continuous functions over the interval t t equipped with the supremum norm the issue is to understand the statistical properties of xt t for large t a well studied special case is when ct t is replaced by the function zero the brownian bridge is then constrained to stay positive a stochastic process known as brownian excursion in the limit of large t it converges to the three dimensional bessel process time dependent barriers like the circle seem to be hardly studied an exception is the parabola gt 2 t t 2 t 2 for

mixing properties and exponential decay for lattice systems in finite volumes an infinite volume mixing or exponential decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions but in some circumstances where such properties hold propagation can nonetheless occur in finite volumes endowed with boundary conditions we establish the absense sic of such propagation particularly in two dimensions in finite volumes which are simply connected under a variety of conditions mainly for the potts model and the fortuin kasteleyn fk random cluster model allowing external fields for example for the fk model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes uniformly over all such volumes and all boundary conditions and implies a strong mixing property for such volumes with certain types of boundary conditions for the potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes which includes exponential decay of correlations in simply connected finite volumes uniformly over all such volumes and all boundary conditions

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem the numerical solution of wave scattering from large objects or from a large cluster of scatterers requires excessive computational resources and it becomes necessary to use approximate but fast methods such as the fast multipole method however since these methods are only approximate it is important to have an estimate for the error introduced in such calculations an analysis of the error for the fast multipole method is presented and estimates for truncation and numerical integration errors are obtained the error caused by polynomial interpolation in a multilevel fast multipole algorithm is also analyzed the total error introduced in a multilevel implementation is also investigated numerically

a mesh free approach to acoustic scattering from multiple spheres nested inside a large sphere by using diagonal translation operators a multiple scattering approach is presented to compute the solution of the helmholtz equation when a number of spherical scatterers are nested in the interior of an acoustically large enclosing sphere the solution is represented in terms of partial wave expansions and a linear system of equations is derived to enforce continuity of pressure and normal particle velocity across all material interfaces this approach yields high order accuracy and avoids some of the difficulties encountered when using integral equations that apply to surfaces of arbitrary shape calculations are accelerated by using diagonal translation operators to compute the interactions between spheres when the operators are numerically stable numerical results are presented to demonstrate the accuracy and efficiency of the method

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

optimal detection of symmetric mixed quantum states we develop a sufficient condition for the least squares measurement lsm or the square root measurement to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states using this condition we derive the optimal measurement for state sets with a broad class of symmetries we first consider geometrically uniform gu state sets with a possibly non abelian generating group and show that if the generator satisfies a weighted norm constraint then the lsm is optimal in particular for pure state gu ensembles the lsm is shown to be optimal for arbitrary gu state sets we show that the optimal measurement operators are gu with generator that can be computed very efficiently in polynomial time within any desired accuracy we then consider compound gu cgu state sets which consist of subsets that are gu when the generators satisfy a certain constraint the lsm is again optimal for arbitrary cgu state sets the optimal measurement operators are shown to be cgu with generators that can be computed efficiently in polynomial time

geometric coherence and quantum state discrimination the operational meaning of coherence measure lies at the very heart of the coherence theory in this paper we provide an operational interpretation for geometric coherence by proving that the geometric coherence of a quantum state is equal to the minimum error probability to discriminate a set of pure states with von neumann measurement on the other hand we also show that a task to ambiguously discriminate a set of linearly independent pure states can be regarded as a problem of calculating geometric coherence that is we reveal an equivalence relation between ambiguous quantum state discrimination and geometric coherence based on this equivalence moreover we improve the upper bound for geometric coherence and give the explicit expression of geometric coherence for a class of states besides we establish a complementarity relation of geometric coherence and path distinguishability with which the relationship between l 1 norm of coherence and geometric coherence is obtained finally with geometric coherence we study multiple copies of quantum state discrimination and give an example to show how to discriminate two pure states

a new fast multipole boundary element method for two dimensional acoustic problems abstract a new fast multipole boundary element method bem is presented in this paper for solving large scale two dimensional 2d acoustic problems based on the improved burton miller formulation this algorithm has several important improvements the fast multipole bem employs the improved burton miller formulation and successfully overcomes the non uniqueness difficulty associated with the conventional bem for exterior acoustic problems the improved burton miller formulation contains only weakly singular integrals and avoids the numerical difficulties associated to the evaluation of the hypersingular integral it leads to the numerical implementations more efficient and straightforward furthermore the fast multipole method fmm and the approximate inverse preconditioned generalized minimum residual method gmres iterative solver are adopted to greatly improve the overall computational efficiency the numerical examples with neumann boundary conditions are presented that clearly demonstrate the accuracy and efficiency of the developed fast multipole bem for solving large scale 2d acoustic problems in a wide range of frequencies

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems