some ideas about quantitative convergence of collision models to their mean field limit we consider a stochastic n particle model for the spatially homogeneous boltzmann evolution and prove its convergence to the associated boltzmann equation when n with non asymptotic estimates for any time t 0 we bound the distance between the empirical measure of the particle system and the measure given by the boltzmann evolution in a relevant hilbert space the control got is gaussian i e we prove that the distance is bigger than xn 1 2 with a probability of type o e x 2 the two main ingredients are a control of fluctuations due to the discrete nature of collisions and a kind of lipschitz continuity for the boltzmann collision kernel we study more extensively the case where our hilbert space is the homogeneous negative sobolev space smash dot h s then we are only able to give bounds for maxwellian models however numerical computations tend to show that our results are useful in practice

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

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

on quotients of spaces with ricci curvature bounded below abstract let m g be a smooth riemannian manifold and g a compact lie group acting on m effectively and by isometries it is well known that a lower bound of the sectional curvature of m g is again a bound for the curvature of the quotient space which is an alexandrov space of curvature bounded below moreover the analogous stability property holds for metric foliations and submersions the goal of the paper is to prove the corresponding stability properties for synthetic ricci curvature lower bounds specifically we show that such stability holds for quotients of rcd k n spaces under isomorphic compact group actions and more generally under metric measure foliations and submetries an rcd k n space is a metric measure space with an upper dimension bound n and weighted ricci curvature bounded below by k in a generalized sense in particular this shows that if m g has ricci curvature bounded below by k r and dimension n then the quotient space is an rcd k n space additionally we tackle the same problem for the cd cd and mcp curvature dimension conditions we provide as well geometric applications which include a generalization of kobayashi s classification theorem of homogeneous manifolds to rcd k n spaces with essential minimal dimension n n a structure theorem for rcd k n spaces admitting actions by large compact groups and geometric rigidity results for orbifolds such as cheng s maximal diameter and maximal volume rigidity theorems finally in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of super ricci flows

exact solution of the partially directed compact lattice animal model the authors calculate exactly the number generating function for partially directed compact lattice animals on the square lattice for the fully directed model they report results of new numerical studies

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

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

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

the mathematics of mixing things up how long should a markov chain monte carlo algorithm be run using examples from statistical physics ehrenfest urn ising model hard discs as well as card shuffling this tutorial paper gives an overview of a body of mathematical results that can give useful answers to practitioners viz seven shuffles suffice for practical purposes it points to new techniques path coupling geometric inequalities and harris recurrence the discovery of phase transitions in mixing times the cutoff phenomenon is emphasized

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

ricci ollivier curvature of the rooted phylogenetic subtree prune regraft graph abstract statistical phylogenetic inference methods use tree rearrangement operations such as subtree prune regraft spr to perform markov chain monte carlo mcmc across tree topologies the structure of the graph induced by tree rearrangement operations is an important determinant of the mixing properties of mcmc motivating the study of the underlying spr graph in greater detail in this paper we investigate the spr graph of rooted trees rspr graph in a new way by calculating the ricci ollivier curvature with respect to uniform and metropolis hastings random walks this value quantifies the degree to which a pair of random walkers from specified points move towards each other negative curvature means that they move away from one another on average while positive curvature means that they move towards each other in order to calculate this curvature we develop fast new algorithms for rspr graph computation we then develop formulas characterizing how the number of rspr neighbors of a tree changes after an rspr operation is applied to that tree these give bounds on the curvature as well as a flatness in the limit theorem indicating that paths of small topology changes are easy to traverse however we find that large topology changes i e moving a large subtree give pairs of trees with negative curvature we show using simulation that mean access time distributions depend on distance degree and curvature demonstrating the relevance of these results to stochastic tree search

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

complete nonorientable minimal surfaces with the highest symmetry group in this paper we present a family of complete nonorientable highly symmetrical minimal surfaces with arbitrary topology and one end for each topology we construct the most symmetrical example with one end furthermore if the euler characteristic of the closed associated surface is even our examples minimize the energy or the degree of the gauss map among the surfaces with their symmetry topology and one end finally we characterize these surfaces in terms of their topology symmetry total curvature and number of ends

enumeration of directed site animals on two dimensional lattices studies the problem of directed site animals on the square triangular and hexagonal lattices closed form expressions are proposed for a s the number of animals of size s on the square and triangular lattices these expressions have been checked for s or 33 and s or 10 for the square and triangular lattices respectively by explicit enumeration they imply that a s varies as lambda 2 s 0 for large s where lambda 3 for the square lattice and lambda 4 for the triangular and theta 1 2 for both for the hexagonal lattice a s is found for s or 48 the results are consistent with lambda 2 0252 or 0 0005 and theta 1 2

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

cutoff for conjugacy invariant random walks on the permutation group we prove a conjecture raised by the work of diaconis and shahshahani z wahrscheinlichkeitstheorie verwandte geb 57 2 159 179 1981 about the mixing time of random walks on the permutation group induced by a given conjugacy class to do this we exploit a connection with coalescence and fragmentation processes and control the kantorovich distance by using a variant of a coupling due to oded schramm as well as contractivity of the distance recasting our proof in the language of ricci curvature our proof establishes the occurrence of a phase transition which takes the following form in the case of random transpositions at time cn 2 the curvature is asymptotically zero for c le 1 and is strictly positive for c 1

enumeration of directed compact site animals in two dimensions the authors study the problem of directed compact site animals in two dimensions using exact enumeration methods the numerical analysis of the animal number series strongly indicates that the critical exponent theta equals zero they also estimate the non universal growth parameter lambda 2 66185 or 0 00050

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

strong contraction and influences in tail spaces we study contraction under a markov semi group and influence bounds for functions all of whose low level fourier coefficients vanish this study is motivated by the explicit construction of 3 regular expander graphs of mendel and naor though our results have no direct implication for the construction of expander graphs in the positive direction we prove an lp poincare inequality and moment decay estimates for mean 0 functions and for all 1 p proving the degree one case of a conjecture of mendel and naor in the negative direction we answer negatively two questions of hatami and kalai concerning extensions of the kahn kalai linial and harper theorems to tail spaces for example we construct a function f 1 1 1 1 whose fourier coefficients vanish up to level c log n with all influences bounded by c log n n for some constants 0 c c that is the kahn kalai linial theorem cannot be improved even if we assume that the first c log n fourier coefficients of the function vanish this implies there is a phase transition in the largest guaranteed influence of functions f 1 1 1 1 which occurs when the first g n log n fourier coefficients vanish and g n as n or g n is bounded as n

metric measure spaces with riemannian ricci curvature bounded from below in this paper we introduce a synthetic notion of riemannian ricci bounds from below for metric measure spaces x d m which is stable under measured gromov hausdorff convergence and rules out finsler geometries it can be given in terms of an enforcement of the lott sturm and villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow besides stability it enjoys the same tensorization global to local and local to global properties in these spaces which we call rcd k spaces we prove that the heat flow which can be equivalently characterized either as the flow associated to the dirichlet form or as the wasserstein gradient flow of the entropy satisfies wasserstein contraction estimates and several regularity properties in particular bakry emery estimates and the l lip feller regularization we also prove that the distance induced by the dirichlet form coincides with d that the local energy measure has density given by the square of cheeger s relaxed slope and as a consequence that the underlying brownian motion has continuous paths all these results are obtained independently of poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact as the infinite dimensional ones

directed lattice animals and the lee yang edge singularity the exponents nu perpendicular to gamma of directed animals in d dimensions are shown to equal those for the lee yang problem in d 1 dimensions for d 2 the author obtains the exact results nu perpendicular to theta 1 2 and for d or 7 the scaling relation theta d 1 nu perpendicular to

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

harnack inequalities for graphs with non negative ricci curvature we establish a harnack inequality for finite connected graphs with non negative ricci curvature as a consequence we derive an eigenvalue lower bound extending previous results for ricci flat graphs

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

evolution problems of leray lions type with nonhomogeneous neumann boundary conditions in metric random walk spaces abstract in this paper we study evolution problems of leray lions type with nonhomogeneous neumann boundary conditions in the framework of metric random walk spaces this covers cases with the p laplacian operator in weighted discrete graphs and nonlocal operators with nonsingular kernel in r n

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

mixing and concentration by ricci curvature abstract we generalise the coarse ricci curvature method of ollivier by considering the coarse ricci curvature of multiple steps in the markov chain this implies new spectral bounds and concentration inequalities we also extend this approach to the bounds for mcmc empirical averages obtained by joulin and ollivier we prove a recursive lower bound on the coarse ricci curvature of multiple steps in the markov chain making our method broadly applicable applications include the split merge random walk on partitions glauber dynamics with random scan and systemic scan for statistical physical spin models and random walk on a binary cube with a forbidden region

exact results for the generating function of directed column convex animals on the square lattice the mathematical properties of the generating function s x for the number sn of directed column convex lattice animals on a square lattice with a given directed site perimeter n are investigated in particular it is shown that s x can be expressed exactly in terms of algebraic hypergeometric functions a detailed investigation of the asymptotic behaviour of sn as n to infinity is carried out by applying the darboux method to the hypergeometric formula for s x it is also demonstrated that sn satisfies a four term recurrence relation finally it is noted that the techniques used to analyse the lattice animal generating function s x can be applied to any other generating function which satisfies a cubic algebraic equation

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

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

gradient flows of the entropy for finite markov chains let k be an irreducible and reversible markov kernel on a finite set x we construct a metric w on the set of probability measures on x and show that with respect to this metric the law of the continuous time markov chain evolves as the gradient flow of the entropy this result is a discrete counterpart of the wasserstein gradient flow interpretation of the heat flow in rn by jordan kinderlehrer and otto 1998 the metric w is similar to but different from the l2 wasserstein metric and is defined via a discrete variant of the benamou brenier formula

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

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

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

super ricci flows for weighted graphs abstract we present a notion of super ricci flow for time dependent finite weighted graphs a challenging feature is that these flows typically encounter singularities where the underlying graph structure changes our notion is robust enough to allow the flow to continue past these singularities as a crucial tool for this purpose we study the heat flow on such singular time dependent weighted graphs with changing graph structure we then give several equivalent characterizations of super ricci flows in terms of a discrete dynamic bochner inequality gradient and transport estimates for the heat flow and dynamic convexity of the entropy along discrete optimal transport paths the latter property can be used to show that our notion of super ricci flow is consistent with classical super ricci flows for manifolds or metric measure spaces in a discrete to continuum limit

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

a finite diameter theorem on rcd k infty spaces for positive k we prove a finite diameter theorem on rcd k infty spaces with a reasonable condition of the heat distributions the self improving property of the bakry emery condition proven by savare plays a key role in the proof

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

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

size and shape of directed lattice animals uses position space renormalisation and exact enumeration methods to study configurational properties of directed lattice animals clusters of directed bonds in which the tail of a new bond must be added to the head of existing bonds exponents are found for the divergence of correlation lengths parallel and perpendicular to the preferred axis in addition an exponent is found that characterises the singularity of the directed animal generating function and critical points are located