the geometric measure of multipartite entanglement and the singular values of a hypermatrix it is shown that the geometric measure of entanglement of a pure multipartite state satisfies a polynomial equation generalizing the singular value equation of the matrix of coefficients of a bipartite state the equation is solved for a class of three qubit states

universal constructions in teichmuller theory we study a new model tess of a universal teichmuller space which is defined to be the collection tess of all ideal tesselations of the poincare disk together with a distinguished oriented edge modulo the natural action of the modbius group mob applying arguments from the decorated teichmuller theory of punctured surfaces which was developed elsewhere we find that tess is an infinite dimensional frehet manifold with canonical global coordinates and there are a universal sympletic form on tess which generalizes the classical weil petersson kahler two forms and a group g which generalizes the classical mapping class groups acting on tess and preserving in fact tess is isomorphic to the right coset space of mob in the group homeo of all orientation preserving homeomorphisms of the circle and one sees easily that tess naturally generalizes the other well known models of a universal teichmuller space namely bers universal teichmuller space and the right coset space of mob in the subgroup of c smooth elements of homeo which is of interest in physics the latter model supports the kirillov kostant two form and we show that the universal symplectic form pulls back to this two form implicit in our discussion so far is the fact that homeo is isomorphic to tess and one is led to ask for the characterization of various families of homeomorphisms in terms of the corresponding tesselation reporting on joint work with dennis sullivan we partially solve one such problem here using the ahlfors bers theory furthermore as in the classical decorated teichmuller theory there is an associated universal decorated teichmuller space tess which is actually of central concern in this work tess is a frechet manifold on which g acts and there is a g invariant two form just above for tess there is moreover a g invariant cell decomposition of tess which generalizes the cell decompositions of the classical decorated teichmullerspaces thus truly all of the structure of the decorated teichmuller spaces of punctured surfaces which is known from previous work arises as the restriction of a corresponding universal structure on tess

the geometric measure of entanglement for a symmetric pure state with non negative amplitudes in this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement for a symmetric pure state the closest product state in terms of the fidelity can be chosen as a symmetric product state we show that this conjecture is true for symmetric pure states whose amplitudes are all non negative in a computational basis the more general conjecture is still open

multipartite generalization of the schmidt decomposition we find a canonical form for pure states of a general multipartite system in which the constraints on the coordinates with respect to a factorizable orthonormal basis are simply that certain ones vanish and certain others are real for identical particles they are invariant under permutations of the particles as an application we find the dimension of the generic local equivalence class

invariance principles for random bipartite planar maps it is conjectured in the physics literature that properly rescaled random planar maps when conditioned to have a large number of faces should converge to a limiting surface whose law does not depend up to scaling factors on details of the class of maps that are sampled previous works on the topic starting with chassaing schaeffer have shown that the radius of a random quadrangulation with n faces converges in distribution once rescaled by n 1 4 to the diameter of the brownian snake up to a scaling constant using a bijection due to bouttier di francesco guitter between bipartite planar maps and a family of labeled trees we show the corresponding invariance principle for a class of random maps that follow a boltzmann distribution the radius of such maps conditioned to have n faces or n vertices and under a criticality assumption converges in distribution once rescaled by n 1 4 to a scaled version of the diameter of the brownian snake convergence results for the so called profile of maps are also provided as well as convergence of rescaled bipartite maps to the brownian map as introduced by marckert mokkadem the proofs of these results rely on a new invariance principle for two type spatial galton watson trees

branching processes in levy processes laplace functionals of snakes and superprocesses we use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism this construction depends on a path valued process called the levy snake which is of independent interest our method of proof involves a calculation of the laplace functional of the occupation field of the levy snake this calculation relies on an evaluation of the corresponding moment functionals which requires precise information about the underlying genealogical structure

bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication we show that entanglement guarantees difficulty in the discrimination of orthogonal multipartite states locally the number of pure states that can be discriminated by local operations and classical communication is bounded by the total dimension over the average entanglement a similar general condition is also shown for pure and mixed states these results offer a rare operational interpretation for three abstractly defined distancelike measures of multipartite entanglement

tensor principal component analysis via convex optimization this paper is concerned with the computation of the principal components for a general tensor known as the tensor principal component analysis pca problem we show that the general tensor pca problem is reducible to its special case where the tensor in question is super symmetric with an even degree in that case the tensor can be embedded into a symmetric matrix we prove that if the tensor is rank one then the embedded matrix must be rank one too and vice versa the tensor pca problem can thus be solved by means of matrix optimization under a rank one constraint for which we propose two solution methods 1 imposing a nuclear norm penalty in the objective to enforce a low rank solution 2 relaxing the rank one constraint by semidefinite programming interestingly our experiments show that both methods can yield a rank one solution for almost all the randomly generated instances in which case solving the original tensor pca problem to optimality to further cope with the size of the resulting convex optimization models we propose to use the alternating direction method of multipliers which reduces significantly the computational efforts various extensions of the model are considered as well

branching processes in levy processes the exploration process the main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure the exploration process appears as a simple local time functional of a levy process with no negative jumps whose laplace exponent coincides with the branching mechanism function this new relation between spectrally positive levy processes and continuous branching processes provides a unified perspective on both theories in particular we derive the adequate formulation of the classical ray knight theorem for such levy processes as a consequence of this theorem we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process

the uniform random tree in a brownian excursion to any brownian excursione with duration e and anyt 1 t p 0 e we associate a branching tree withp branches denoted byt p e t 1 t p which is closely related to the structure of the minima ofe our main theorem states that ife is chosen according to the ito measure and t 1 t p according to lebesgue measure on 0 e p the treet p e t 1 t p is distributed according to the uniform measure on the set of trees withp branches the proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree thus generalizing a well known representation theorem of bismut if we replace the ito measure by the law of the normalized excursion a simple conditioning argument leads to another remarkable result originally proved by aldous with a very different method

toward an understanding of entanglement for generalized n qubit w states we solve stationarity equations of the geometric measure of entanglement for multi qubit w type states in this way we compute analytically the maximal overlap of one parameter n qubit and two parameter four qubit w type states and their nearest product states possible extensions to arbitrary w type states and geometrical interpretations of these results are discussed in detail

self similar fragmentations derived from the stable tree i the basic object we consider is a certain model of continuum random tree called the stable tree we construct a fragmentation process f t t 0 out of this tree by removing the vertices located under height t thanks to a self similarity property of the stable tree we show that the fragmentation process is also self similar the semigroup and other features of the fragmentation are given explicitly asymptotic results are given as well as a couple of related results on continuous state branching processes

probabilistic and fractal aspects of levy trees we investigate the random continuous trees called levy trees which are obtained as scaling limits of discrete galton watson trees we give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted trees which is equipped with the gromov hausdorff distance to construct levy trees we make use of the coding by the height process which was studied in detail in previous work we then investigate various probabilistic properties of levy trees in particular we establish a branching property analogous to the well known property for galton watson trees conditionally given the tree below level a the subtrees originating from that level are distributed as the atoms of a poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root we study regularity properties of local times in the space variable and prove that the support of local time is the full level set except for certain exceptional values of a corresponding to local extinctions we also compute several fractal dimensions of levy trees including hausdorff and packing dimensions in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree we finally discuss some applications to super brownian motion with a general branching mechanism

a representation of exchangeable hierarchies by sampling from random real trees a hierarchy on a set s also called a total partition of s is a collection mathcal h of subsets of s such that s in mathcal h each singleton subset of s belongs to mathcal h and if a b in mathcal h then a cap b equals either a or b or varnothing every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h associated as follows with a random real tree mathcal t equipped with root element 0 and a random probability distribution p on the borel subsets of mathcal t given mathcal t p let t 1 t 2 ldots be independent and identically distributed according to p and let mathcal h comprise all singleton subsets of mathbb n and every subset of the form j t j in f x as x ranges over mathcal t where f x is the fringe subtree of mathcal t rooted at x there is also the alternative characterization every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h derived as follows from a random hierarchy mathscr h on 0 1 and a family u j of i i d uniform 0 1 random variables independent of mathscr h let mathcal h comprise all sets of the form j u j in b as b ranges over the members of mathscr h

matrix permanent and quantum entanglement of permutation invariant states we point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states this connection allows us to interpret the permanent as an angle between vectors by employing a recently introduced permanent inequality by carlen et al methods appl anal 13 1 2006 we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way

self similar fragmentations derived from the stable tree ii splitting at nodes we study a natural fragmentation process of the so called stable tree introduced by duquesne and le gall which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self similar with positive index explicit formulas for the semigroup are given and we provide asymptotic results we also give an alternative construction of this fragmentation using paths of levy processes hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the brownian continuum random tree or using brownian paths respectively due to aldous pitman and bertoin

scaling limits of random planar maps with large faces we discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index alpha in 1 2 when the number n of vertices of the map tends to infinity the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process which can be constructed from a centered stable process with no negative jumps and index alpha in particular the profile of distances in the map rescaled by the factor n 1 2 alpha converges to a random measure defined in terms of the distance process with the same rescaling of distances the vertex set viewed as a metric space converges in distribution as n to infty at least along suitable subsequences towards a limiting random compact metric space whose hausdorff dimension is equal to 2 alpha

sub gaussian tail bounds for the width and height of conditioned galton watson trees we study the height and width of a galton watson tree with offspring distribution satisfying e 1 0 var conditioned on having exactly n nodes under this conditioning we derive sub gaussian tail bounds for both the width largest number of nodes in any level and height greatest level containing a node the bounds are optimal up to constant factors in the exponent under the same conditioning we also derive essentially optimal upper tail bounds for the number of nodes at level k for 1 k n

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

counterexample to an additivity conjecture for output purity of quantum channels a conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel as measured by the p norm should be multiplicative with respect to the tensor product of channels we disprove this conjecture for p 4 79 the same example with p also disproves a conjecture for the multiplicativity of the injective norm of hilbert space tensor products

estimating entanglement measures in experiments we present a method to estimate entanglement measures in experiments we show how a lower bound on a generic entanglement measure can be derived from the measured expectation values of any finite collection of entanglement witnesses hence witness measurements are given a quantitative meaning without the need of further experimental data we apply our results to a recent multiphoton experiment m bourennane et al phys rev lett 92 087902 2004 giving bounds on the entanglement of formation and the geometric measure of entanglement in this experiment

a limit theorem for the contour process of condidtioned galton watson trees in this work we study asymptotics of the genealogy of galton watson processes conditioned on the total progeny we consider a fixed aperiodic and critical offspring distribution such that the rescaled galton watson processes converges to a continuous state branching process csbp with a stable branching mechanism of index 1 2 we code the genealogy by two different processes the contour process and the height process that le gall and le jan recently introduced we show that the rescaled height process of the corresponding galton watson family tree with one ancestor and conditioned on the total progeny converges in a functional sense to a new process the normalized excursion of the continuous height process associated with the stable csbp we deduce from this convergence an analogous limit theorem for the contour process in the brownian case 2 the limiting process is the normalized brownian excursion that codes the continuum random tree the result is due to aldous who used a different method

the real tree in this note it is shown that there exists a natural metric on the set formula of bounded subsets of r containing their infimum which endowstwith the structure of an r tree so that for everyt t the number of connected components oft t coincides with the cardinality p r of the set of subsets of r in addition the set of ends oftis explicitly determined and various further features oftare discussed too

random stable laminations of the disk we study large random dissections of polygons we consider random dissections of a regular polygon with n sides which are chosen according to boltzmann weights in the domain of attraction of a stable law of index 2 1 2 as n goes to infinity we prove that these random dissections converge in distribution towards a random compact set called the random stable lamination if 2 we recover aldous brownian triangulation however if 2 1 2 large faces remain in the limit and a dierent random compact set appears we show that the random stable lamination can be coded by the continuous time height function associated to the normalized excursion of a strictly stable spectrally positive levy process of index using this coding we establish that the hausdor dimension of the stable random lamination is almost surely 2 1

percolation on random triangulations and stable looptrees we study site percolation on angel and schramm s uniform infinite planar triangulation we compute several critical and near critical exponents and describe the scaling limit of the boundary of large percolation clusters in all regimes subcritical critical and supercritical we prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3 2 which was introduced in curien and kortchemski random stable looptrees arxiv 1304 1044 2014 we also give a conjecture linking looptrees of any index alpha in 1 2 with scaling limits of cluster boundaries in random triangulations decorated with o n models

scaling limits of random polya trees polya trees are rooted trees considered up to symmetry we establish the convergence of large uniform random polya trees with arbitrary degree restrictions to aldous continuum random tree with respect to the gromov hausdorff metric our proof is short and elementary and it is based on a novel decomposition it shows that the global shape of a random polya tree is essentially dictated by a large galton watson tree that it contains we also derive sub gaussian tail bounds for both the height and the width which are optimal up to constant factors in the exponent

limits of the boundary of random planar maps we discuss asymptotics for the boundary of critical boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter alpha in 1 2 first in the dense phase corresponding to alpha in 1 3 2 we prove that the scaling limit of the boundary is the random stable looptree with parameter 1 alpha 1 2 second we show the existence of a phase transition through local limits of the boundary in the dense phase the boundary is tree like while in the dilute phase corresponding to alpha in 3 2 2 it has a component homeomorphic to the half plane as an application we identify the limits of loops conditioned to be large in the rigid o n loop model on quadrangulations proving thereby a conjecture of curien and kortchemski

continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models given any regularly varying dislocation measure we identify a natural self similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths as an application we obtain continuum random tree limits of aldous s beta splitting models and ford s alpha models for phylogenetic trees this confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf

sub exponential tail bounds for conditioned stable bienayme galton watson trees we establish uniform sub exponential tail bounds for the width height and maximal outdegree of critical bienayme galton watson trees conditioned on having a large fixed size whose offspring distribution belongs to the domain of attraction of a stable law this extends results obtained for the height and width by addario berry devroye and janson in the finite variance case

the stable trees are nested we show that we can construct simultaneously all the stable trees as a nested family more precisely if 1 alpha alpha prime le 2 we prove that hidden inside any alpha stable tree we can find a version of an alpha prime stable tree rescaled by an independent mittag leffler type distribution this tree can be explicitly constructed by a pruning procedure of the underlying stable tree or by a modification of the fragmentation associated with it our proofs are based on a recursive construction due to marchal which is proved to converge almost surely towards a stable tree

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion