stability of the absolutely continuous spectrum of random schrodinger operators on tree graphs the subject of this work is random schrodinger operators on regular rooted tree graphs with stochastically homogeneous disorder the operators are of the form h t u v acting in l2 with t the adjacency matrix u a radially periodic potential and v a random potential this includes the only class of homogeneously random operators for which it was proven that the spectrum of h exhibits an absolutely continuous ac component a results established by a klein for weak disorder in case u 0 and v given by iid random variables on our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder the method yields the continuity of the ac spectral density of h at 0 the latter is shown to converge in the l 1 sense over closed borel sets in which h 0 has no singular spectrum the analysis extends to random potentials whose values at different sites need not be independent assuming only that their joint distribution is weakly correlated across different tree branches

the set of vertices with positive curvature in a planar graph with nonnegative curvature abstract in this paper we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature based on this we show that the automorphism group of a planar possibly infinite graph with nonnegative combinatorial curvature and positive total curvature is a finite group and give an upper bound estimate for the order of the group

note on basic features of large time behaviour of heat kernels large time behaviour of heat semigroups and more gener ally of positive selfadjoint semigroups is studied convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups this framework includes laplacians on manifolds metric graphs and discrete graphs

ricci curvature of finite markov chains via convexity of the entropy we study a new notion of ricci curvature that applies to markov chains on discrete spaces this notion relies on geodesic convexity of the entropy and is analogous to the one introduced by lott sturm and villani for geodesic measure spaces in order to apply to the discrete setting the role of the wasserstein metric is taken over by a different metric having the property that continuous time markov chains are gradient flows of the entropy using this notion of ricci curvature we prove discrete analogues of fundamental results by bakry emery and otto villani further we show that ricci curvature bounds are preserved under tensorisation as a special case we obtain the sharp ricci curvature lower bound for the discrete hypercube

on the lp spectrum of laplacians on graphs abstract we study the p independence of spectra of laplace operators on graphs arising from regular dirichlet forms on discrete spaces here a sufficient criterion is given solely by a uniform subexponential growth condition moreover under a mild assumption on the measure we show a one sided spectral inclusion without any further assumptions we study applications to normalized laplacians including symmetries of the spectrum and a characterization for positivity of the cheeger constant furthermore we consider laplacians on planar tessellations for which we relate the spectral p independence to assumptions on the curvature

isometries fock spaces and spectral analysis of schrodinger operators on trees abstract we construct conjugate operators for the real part of a completely non unitary isometry and we give applications to the spectral and scattering theory of a class of operators on complete fock spaces natural generalizations of the schrodinger operators on trees we consider c algebras generated by such hamiltonians with certain types of anisotropy at infinity we compute their quotient with respect to the ideal of compact operators and give formulas for the essential spectrum of these hamiltonians

a stationary planar random graph with singular stationary dual dyadic lattice graphs dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane we use them to give examples of random rooted graphs that are stationary for simple random walk but whose duals have only a singular stationary measure this answers a question of curien and shows behaviour different from the unimodular case the consequence is that planar duality does not combine well with stationary random graphs we also study harmonic measure on dyadic lattice graphs and show its singularity

singular continuous spectrum for the laplacian on certain sparse trees we present examples of rooted tree graphs for which the laplacian has singular continuous spectral measures for some of these examples we further establish fractional hausdorff dimensions the singular continuous components in these models have an interesting multiplicity structure the results are obtained via a decomposition of the laplacian into a direct sum of jacobi matrices

localization criteria for anderson models on locally finite graphs we prove spectral and dynamical localization for anderson models on locally finite graphs using the fractional moment method our theorems extend earlier results on localization for the anderson model on d we establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

volume growth spectrum and stochastic completeness of infinite graphs we study the connections between volume growth spectral properties and stochastic completeness of locally finite weighted graphs for a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum we then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite weighted graphs

displacement convexity of entropy and related inequalities on graphs we introduce the notion of an interpolating path on the set of probability measures on finite graphs using this notion we first prove a displacement convexity property of entropy along such a path and derive prekopa leindler type inequalities a talagrand transport entropy inequality certain hwi type as well as log sobolev type inequalities in discrete settings to illustrate through examples we apply our results to the complete graph and to the hypercube for which our results are optimal by passing to the limit we recover the classical log sobolev inequality for the standard gaussian measure with the optimal constant

generalization bounds of erm based learning processes for continuous time markov chains many existing results on statistical learning theory are based on the assumption that samples are independently and identically distributed i i d however the assumption of i i d samples is not suitable for practical application to problems in which samples are time dependent in this paper we are mainly concerned with the empirical risk minimization erm based learning process for time dependent samples drawn from a continuous time markov chain this learning process covers many kinds of practical applications e g the prediction for a time series and the estimation of channel state information thus it is significant to study its theoretical properties including the generalization bound the asymptotic convergence and the rate of convergence it is noteworthy that since samples are time dependent in this learning process the concerns of this paper cannot at least straightforwardly be addressed by existing methods developed under the sample i i d assumption we first develop a deviation inequality for a sequence of time dependent samples drawn from a continuous time markov chain and present a symmetrization inequality for such a sequence by using the resultant deviation inequality and symmetrization inequality we then obtain the generalization bounds of the erm based learning process for time dependent samples drawn from a continuous time markov chain finally based on the resultant generalization bounds we analyze the asymptotic convergence and the rate of convergence of the learning process

toward an efficient parallel eigensolver for dense symmetric matrices we describe a parallel algorithm for finding the eigenvalues and eigenvectors of a dense symmetric matrix with an emphasis on the dense linear algebra operations we follow the traditional three step process reduce to tridiagonal form solve the tridiagonal problem then backtransform the result since the different steps have different algorithmic characteristics this problem serves as a perfect vehicle for exploring some issues associated with parallel linear algebra calculations in particular we examine the effects of matrix distribution and blocking on the computational performance of tridiagonalization and backtransformation through experiments on an intel paragon we demonstrate that block storage of the matrix is not necessary for a highly efficient block algorithm the performance of our approach compares very favorably with that of the corresponding scalapack routines

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

an inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems we discuss an inverse free highly parallel spectral divide and conquer algorithm it can compute either an invariant subspace of a nonsymmetric matrix a or a pair of left and right deflating subspaces of a regular matrix pencil a lambda b this algorithm is based on earlier ones of bulgakov godunov and malyshev but improves on them in several ways this algorithm only uses easily parallelizable linear algebra building blocks matrix multiplication and qr decomposition but not matrix inversion similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function which requires matrix inversion and is faster but can be less stable than the new algorithm

ollivier ricci curvature for general graph laplacians heat equation laplacian comparison non explosion and diameter bounds abstract discrete time random walks on a finite set naturally translate via a one to one correspondence to discrete laplace operators typically ollivier curvature has been investigated via random walks we first extend the definition of ollivier curvature to general weighted graphs and then give a strikingly simple representation of ollivier curvature using the graph laplacian using the laplacian as a generator of a continuous time markov chain we connect ollivier curvature with the heat equation which is strongly related to continuous time random walks in particular we prove that a lower bound on the ollivier curvature is equivalent to a certain lipschitz decay of solutions to the heat equation this is a discrete analogue to a celebrated ricci curvature lower bound characterization by renesse and sturm our representation of ollivier curvature via the laplacian allows us to deduce a laplacian comparison principle by which we prove non explosion and improved diameter bounds

sectional curvature of polygonal complexes with planar substructures in this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature these complexes can be viewed as generalizations of 2 dimensional euclidean and hyperbolic buildings we focus on the case of non positive and negative combinatorial curvature as geometric results we obtain a hadamard cartan type theorem thinness of bigons gromov hyperbolicity and estimates for the cheeger constant we employ the latter to get spectral estimates show discreteness of the spectrum in the sense of a donnelly li type theorem and present corresponding eigenvalue asymptotics moreover we prove a unique continuation theorem for eigenfunctions and the solvability of the dirichlet problem at infinity

volume growth and spectrum for general graph laplacians we prove estimates relating exponential or sub exponential volume growth of weighted graphs to the bottom of the essential spectrum for general graph laplacians the volume growth is computed with respect to a metric adapted to the laplacian and use of these metrics produces better results than those obtained from consideration of the graph metric only conditions for absence of the essential spectrum are also discussed these estimates are shown to be optimal or near optimal in certain settings and apply even if the laplacian under consideration is an unbounded operator

bounding bar d distance by informational divergence a method to prove measure concentration there is a simple inequality by pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces we shall consider probability measures on sequences taken from countable alphabets and derive from pinsker s inequality bounds on the bar d distance by informational divergence such bounds can be used to prove the concentration of measure phenomenon for some nonproduct distributions

ab algorithm and its modifications for the spectral problems of linear pencils of matrices this paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular

transfer matrices hyperbolic geometry and absolutely continuous spectrum for some discrete schrodinger operators on graphs abstract we prove the existence of absolutely continuous spectrum for a class of discrete schrodinger operators on tree like graphs we consider potentials whose radial behaviour is subject only to an l bound in the transverse direction the potential must satisfy a condition such as periodicity the graphs we consider include binary trees and graphs obtained from a binary tree by adding edges possibly with weights our methods are motivated by the one dimensional transfer matrix method interpreted as a discrete dynamical system on the hyperbolic plane this is extended to more general graphs leading to a formula for the green s function bounds on the green s function then follow from the contraction properties of the transformations that arise in this generalization the bounds imply the existence of absolutely continuous spectrum

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

approximating l2 invariants and the atiyah conjecture let g be a torsion free discrete group and let denote the field of algebraic numbers in we prove that g fulfills the atiyah conjecture if g lies in a certain class of groups d which contains in particular all groups that are residually torsion free elementary amenable or are residually free this result implies that there are no nontrivial zero divisors in g the statement relies on new approximation results for l2 betti numbers over g which are the core of the work done in this paper another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial laplacian on l2 cochains on any normal covering space of a finite cw complex we establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the linnell class we also establish the absence of eigenvalues that are liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class 2003 wiley periodicals inc

the dual cheeger constant and spectra of infinite graphs abstract in this article we study the top of the spectrum of the normalized laplace operator on infinite graphs we introduce the dual cheeger constant and show that it controls the top of the spectrum from above and below in a similar way as the cheeger constant controls the bottom of the spectrum moreover we show that the dual cheeger constant at infinity can be used to characterize that the essential spectrum of the normalized laplace operator shrinks to one point

transportation cost information inequalities and applications to random dynamical systems and diffusions we first give a characterization of the l1 transportation cost information inequality on a metric space and next find some appropriate sufficient condition to transportation cost information inequalities for dependent sequences applications to random dynamical systems and diffusions are studied

curvature concentration and error estimates for markov chain monte carlo we provide explicit nonasymptotic estimates for the rate of convergence of empirical means of markov chains together with a gaussian or exponential control on the deviations of empirical means these estimates hold under a positive curvature assumption expressing a kind of metric ergodicity which generalizes the ricci curvature from differential geometry and on finite graphs amounts to contraction under path coupling

cheeger constants growth and spectrum of locally tessellating planar graphs in this article we study relations between the local geometry of planar graphs combinatorial curvature and global geometric invariants namely the cheeger constants and the exponential growth we also discuss spectral applications

the essential spectrum of the laplacian on rapidly branching tessellations in this paper we characterize absence of the essential spectrum of the laplacian under a hyperbolicity assumption for general graphs moreover we present a characterization for absence of the essential spectrum for planar tessellations in terms of curvature

riemannian ricci curvature lower bounds in metric measure spaces with finite measure in prior work 4 of the first two authors with savare a new riemannian notion of lower bound for ricci curvature in the class of metric measure spaces x d m was introduced and the corresponding class of spaces denoted by rcd k this notion relates the cd k n theory of sturm and lott villani in the case n to the bakry emery approach in 4 the rcd k property is defined in three equivalent ways and several properties of rcd k spaces including the regularization properties of the heat flow the connections with the theory of dirichlet forms and the stability under tensor products are provided in 4 only finite reference measures m have been considered the goal of this paper is twofold on one side we extend these results to general finite spaces on the other we remove a technical assumption appeared in 4 concerning a strengthening of the cd k condition this more general class of spaces includes euclidean spaces endowed with lebesgue measure complete noncompact riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower ricci curvature bounds

noise adaptive filtering of seismic shot records ambient noise can obscure reflections on deep crustal seismic data we use a spectral subtraction method to attenuate stationary noise our procedure called noise adaptive filtering is to fourier transform the noise before the first arrivals subtract the amplitude spectrum of the noise from the amplitude spectrum of the noisy data and inverse fourier transform the phase spectrum is not corrected but the method attenuates noise if the phase shift between the signal and noise is random the algorithm can be implemented as a frequency filter as a frequency wavenumber filter or as two separate frequency and wavenumber filters noise adaptive filtering is often superior to conventional frequency or frequency wavenumber filtering because it adapts to spatial variations in the noise without parameter testing noise adaptive filters can achieve noise rejection ratios of up to 45 db their dynamic range is about 25 db these filters work best when the input signal to noise ratio is on the order of 0 db and

a parallel implementation of the nonsymmetric qr algorithm for distributed memory architectures one approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the qr algorithm not long ago this was widely considered to be a hopeless task recent efforts have led to significant advances although the methods proposed up to now have suffered from scalability problems this paper discusses an approach to parallelizing the qr algorithm that greatly improves scalability a theoretical analysis indicates that the algorithm is ultimately not scalable but the nonscalability does not become evident until the matrix dimension is enormous experiments on the intel paragon system the ibm sp2 supercomputer the sgi origin 2000 and the intel asci option red supercomputer are reported

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

parallelizing the qr algorithm for the unsymmetric algebraic eigenvalue problem myths and reality over the last few years it has been suggested that the popular qr algorithm for the unsymmetric schur decomposition does not parallelize in this paper we present both positive and negative results on this subject in theory asymptotically perfect speedup can be obtained in practice reasonable speedup can be obtained on an mimd distributed memory computer for a relatively small number of processors however we also show theoretically that it is impossible for the standard qr algorithm to be scalable performance of a parallel implementation of the lapack dlahqr routine on the intel text paragon text tm system is reported

ricci curvature of markov chains on metric spaces abstract we define the coarse ricci curvature of metric spaces in terms of how much small balls are closer in wasserstein transportation distance than their centers are this definition naturally extends to any markov chain on a metric space for a riemannian manifold this gives back after scaling the value of ricci curvature of a tangent vector examples of positively curved spaces for this definition include the discrete cube and discrete versions of the ornstein uhlenbeck process moreover this generalization is consistent with the bakry emery ricci curvature for brownian motion with a drift on a riemannian manifold positive ricci curvature is shown to imply a spectral gap a levy gromov like gaussian concentration theorem and a kind of modified logarithmic sobolev inequality the bounds obtained are sharp in a variety of examples

the heat flow on metric random walk spaces abstract in this paper we study the heat flow on metric random walk spaces which unifies into a broad framework the heat flow on locally finite weighted connected graphs the heat flow determined by finite markov chains and some nonlocal evolution problems we give different characterizations of the ergodicity and prove that a metric random walk space with positive ollivier ricci curvature is ergodic furthermore we prove a cheeger inequality and as a consequence we show that a poincare inequality holds if and only if an isoperimetric inequality holds we also study the bakry emery curvature dimension condition and its relation with functional inequalities like the poincare inequality and the transport information inequalities

the unimportance of phase in speech enhancement the importance of fourier transform phase in speech enhancement is considered results indicate that a more accurate estimation of phase is unwarranted in speech enhancement at the s n ratios where the intelligibility scores of unprocessed speech range from 5 to 95 percent if the phase estimate is used to reconstruct speech by combining it with an independently estimated magnitude or to reconstruct speech using the phase only signal reconstruction algorithm

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

parallel algorithm for solving some spectral problems of linear algebra abstract an algorithm is described for computing the deflating subspaces of a regular linear matrix pencil b a more precisely the algorithm is intended to compute the projection matrices p and i p onto the deflating subspaces of matrix pencils corresponding to the eigenvalues inside and outside the unit circle this algorithm can be considered as an orthogonal version of the matrix sign function method it possesses a quadratic convergence rate and simultaneously with the projection operators operators computes a condition number of the problem we point out the extent to which this algorithm can be parallelized applications are made to solving the riccati equation

a quadratically convergent qr like method without shifts for the hermitian eigenvalue problem abstract we propose a new qr like algorithm symmetric squared qr ssqr method that can be readily parallelized using commonly available parallel computational primitives such as matrix matrix multiplication and qr decomposition the algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts we provide a rigorous convergence analysis of ssqr and derive structures for several of the important quantities generated by the algorithm we also discuss various practical implementation issues such as stopping criteria and deflation techniques we demonstrate the convergence behavior of ssqr using several numerical examples

geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature we apply alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature we obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point moreover we show the volume doubling property and the poincare inequality on such graphs the quadratic volume growth of these graphs implies the parabolicity finally we prove the polynomial growth harmonic function theorem analogous to the case of riemannian manifolds

mass transportation and rough curvature bounds for discrete spaces abstract we introduce and study rough approximate lower curvature bounds for discrete spaces and for graphs this notion agrees with the one introduced in j lott c villani ricci curvature for metric measure spaces via optimal transport ann of math 169 2009 in press and k t sturm on the geometry of metric measure spaces i acta math 196 2006 65 131 in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature k will have curvature k in the sense of j lott c villani ricci curvature for metric measure spaces via optimal transport ann of math 169 2009 in press k t sturm on the geometry of metric measure spaces i acta math 196 2006 65 131 moreover in the converse direction discretizations of metric measure spaces with curvature k will have rough curvature k we apply our results to concrete examples of homogeneous planar graphs

applications of median filtering to deconvolution pulse estimation and statistical editing of seismic data methodology for using median filtering as one step in cepstral deconvolution or seismic signature estimation the surprising conclusion from empirical studies on synthetic data is that in high noise situations cepstral based median filtering appears to perform exceptionally well as a deconvolver but poorly as a signature estimator modified journal abstract

diffusion determines the recurrent graph abstract we consider diffusion on discrete measure spaces as encoded by markovian semigroups arising from weighted graphs we study whether the graph is uniquely determined if the diffusion is given up to order isomorphism if the graph is recurrent then the complete graph structure and the measure space are determined up to an overall scaling as shown by counterexamples this result is optimal without the recurrence assumption the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass these investigations provide discrete counterparts to studies of diffusion on euclidean domains and manifolds initiated by arendt and continued by arendt biegert ter elst and arendt ter elst a crucial step in our considerations shows that order isomorphisms are actually unitary maps up to a scaling in our context

an analogue of the descartes euler formula for infinite graphs and higuchi s conjecture let r be a connected 2 manifold without boundary obtained from a possibly infinite collection of polygons by identifying them along edges of equal length let v be the set of vertices and for every v v let k v denote the gaussian curvature of v 2 minus the sum of incident polygon angles descartes showed that v k 4 whenever r may be realized as the surface of a convex polytope in r 3 more generally if r is made of finitely many polygons euler s formula is equivalent to the equation v 1 r where r is the euler characteristic of r our main theorem shows that whenever v 0 for every vertex v we apply our main theorem to deduce that r is made of finitely many polygons and is homeomorphic to either the 2 sphere or to the projective plane further we show that unless is a prism antiprism or the projective planar analogue of one of these that v 3444 this resolves a recent conjecture of higuchi

robust transport over networks we consider transportation over a strongly connected directed graph the scheduling amounts to selecting transition probabilities for a discrete time markov evolution which is designed to be consistent with initial and final marginal constraints on mass transport we address the situation where initially the mass is concentrated on certain nodes and needs to be transported over a certain time period to another set of nodes possibly disjoint from the first the evolution is selected to be closest to a prior measure on paths in the relative entropy sense such a construction is known as a schrodinger bridge between the two given marginals it may be viewed as an atypical stochastic control problem where the control consists in suitably modifying the prior transition mechanism the prior can be chosen to incorporate constraints and costs for traversing specific edges of the graph but it can also be selected to allocate equal probability to all paths of equal length connecting any two nodes i e a uniform distribution on paths this latter choice for prior transitions relies on the so called ruelle bowen random walker and gives rise to scheduling that tends to utilize all paths as uniformly as the topology allows thus this ruelle bowen law mathfrak m mathrm rb taken as prior leads to a transportation plan that tends to lessen congestion and ensures a level of robustness we also show that the distribution mathfrak m mathrm rb on paths which attains the maximum entropy rate for the random walker given by the topological entropy can itself be obtained as the time homogeneous solution of a maximum entropy problem for measures on paths also a schrodinger bridge problem albeit with prior that is not a probability measure finally we show that the paradigm of schrodinger bridges as a mechanism for scheduling transport on networks can be adapted to graphs that are not strongly connected as well as to weighted graphs in the latter case our approach may be used to design a transportation plan which effectively compromises between robustness and other criteria such as cost indeed we explicitly provide a robust transportation plan which assigns maximum probability to minimum cost paths and therefore compares favorably with optimal mass transportation strategies

li yau inequality on graphs abstractwe prove the li yau gradient estimate for the heat kernel on graphs the onlyassumption is a variant of the curvature dimension inequality which is purely local and can be considered as a new notion of curvature for graphs we compute thiscurvature for lattices and trees and conclude that it behaves more naturally than thealready existing notions of curvature moreover we show that if a graph has non negative curvature then it has polynomial volume growth we also derive harnack inequalities and heat kernel bounds from the gradient esti mate and show how it can be used to strengthen the classical buser inequality relatingthe spectral gap and the cheeger constant of a graph 1 introduction and main ideas in their celebrated work 15 li and yau proved an upper bound on the gradient ofpositive solutions of the heat equation in its simplest form for an n dimensional com pact manifold with non negative ricci curvature the li yau gradient estimate statesthat a positive solution uof the heat equation

localization and tensorization properties of the curvature dimension condition for metric measure spaces abstract this is an addendum to the paper k bacher k t sturm localization and tensorization properties of the curvature dimension condition for metric measure spaces j funct anal 259 2010 28 56 we prove the tensorization property for the curvature dimension condition add some detailed calculations including explicit dependence of constants and comment on assumptions and conjectures concerning the local to global statement in bacher and sturm 2010 1 and villani 2009 6 respectively

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

on modified logarithmic sobolev inequalities for bernoulli and poisson measures we show that for any positive functionfon the discrete cube 0 1 n ent np f pqe np1f df 2where npis the product measure of the bernoulli measure with probability of successp as well as related inequalities which may be shown to imply in the limit the classical gaussian logarithmic sobolev inequality as well as a logarithmic sobolev inequality for poisson measure we further investigate modified logarithmic sobolev inequalities to analyze integrability properties of lipschitz functions on discrete spaces in particular we obtain under modified logarithmic sobolev inequalities some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables

rigidity of the bonnet myers inequality for graphs with respect to ollivier ricci curvature abstract we introduce the notion of bonnet myers and lichnerowicz sharpness in the ollivier ricci curvature sense our main result is a classification of all self centered bonnet myers sharp graphs hypercubes cocktail party graphs even dimensional demi cubes johnson graphs j 2 n n the gosset graph and suitable cartesian products we also present a purely combinatorial reformulation of this result we show that bonnet myers sharpness implies lichnerowicz sharpness and classify all distance regular lichnerowicz sharp graphs under the additional condition 1 b 1 1 we also relate bonnet myers sharpness to an upper bound of bakry emery curvature which motivates a general conjecture about bakry emery curvature

analysis of the physical laplacian and the heat flow on a locally finite graph we study the physical laplacian and the corresponding heat flow on an infinite locally finite graph with possibly unbounded valence

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

hardy inequality and asymptotic eigenvalue distribution for discrete laplacians abstract in this paper we study in detail some spectral properties of the magnetic discrete laplacian we identify its form domain characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution

poincare modified logarithmic sobolev and isoperimetric inequalities for markov chains with non negative ricci curvature abstract we study functional inequalities for markov chains on discrete spaces with entropic ricci curvature bounded from below our main results are that when curvature is non negative but not necessarily positive the spectral gap the cheeger isoperimetric constant and the modified logarithmic sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space with respect to a suitable metric these estimates are discrete analogues of classical results of riemannian geometry obtained by li and yau buser and wang