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

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

critical random walk in random environment on trees we study the behavior of random walk in random environment rwre on trees in the critical case left open in previous work representing the random walk by an electrical network we assume that the ratios of resistances of neighboring edges of a tree are i i d random variables whose logarithms have mean zero and finite variance then the resulting rwre is transient if simple random walk on is transient but not vice versa we obtain general transience criteria for such walks which are sharp for symmetric trees of polynomial growth in order to prove these criteria we establish results on boundary crossing by tree indexed random walks these results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean zero finite variance random walks in one dimension which are of independent interest

conformal invariance of domino tiling let u be a multiply connected region in r 2 with smooth boundary let p be a polyomino in z 2 approximating u as 0 we show that for certain boundary conditions on p the height distribution on a random domino tiling dimer covering of p is conformally invariant in the limit as tends to 0 in the sense that the distribution of heights of boundary components or rather the difference of the heights from their mean values only depends on the conformal type of u the mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way the mean height and all the moments are explicitly evaluated

indistinguishability of percolation clusters we show that when percolation produces infinitely many infinite clusters on a cayley graph one cannot distinguish the clusters from each other by any invariantly defined property this implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity a k a long range order we then derive applications concerning uniqueness in kazhdan groups and in wreath products and inequalities for p u

conformal invariance of voronoi percolation it is proved that in the voronoi model for percolation in dimension 2 and 3 the crossing probabilities are asymptotically invariant under conformal change of metric

random walks capacity and percolation on trees a collection of several different probabilistic processes involving trees is shown to have an unexpected unity this makes possible a fruitful interplay of these probabilistic processes the processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well our work has five specific aims first an exact correspondence between random walks and percolation on trees is proved extending and sharpening previous work of the author this is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes second we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation this capacitary formulation extends and sharpens work of fan on random interval coverings third we show how this formulation also applies to generalize work of evans on random labelling of trees fourth the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a s in particular we resolve a conjecture of griffeath on the necessity of the nash williams criterion fifth for this last purpose we establish several new basic results on branching processes in varying environments

isoperimetric inequalities and markov chains soit a i j i j n la matrice stochastique d une chaine de markov reversible sur n 1 2 soit n 2 tel que f r c f d f c 0 n ou r 2n n 2 ou c 0 est independant de f et ou r indique la norme dans l r n d pour la mesure i i alors on a sup i j j 1 a i j 0 n 2

critical and oscillatory behavior of a system of smart preys and predators it is shown that a system of smart preys and predators exhibits irreversible phase transitions between a regime of prey predator coexistence and an state where predator extinction is observed within the coexistence regime the system exhibits a transition between a regime where the densities of species remain constant and another with self sustained oscillations respectively this transition is located by means of a combined treatment involving finite size scaling and fourier transforms furthermore it is shown that the transition can be rationalized in terms of the standard percolation theory the existence of an oscillatory regime in the thermodynamic limit which is in contrast to previous findings of boccara et al phys rev e 50 4531 1994 may be due to subtle differences between the studied models

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

towards a theory of negative dependence the fkg theorem says that the positive lattice condition an easily checkable hypothesis which holds for many natural families of events implies positive association a very useful property thus there is a natural and useful theory of positively dependent events there is as yet no corresponding theory of negatively dependent events there is however a need for such a theory this paper unfortunately contains no substantial theorems its purpose is to present examples that motivate a need for such a theory give plausibility arguments for the existence of such a theory outline a few possible directions such a theory might take and state a number of specific conjectures which pertain to the examples and to a wish list of theorems

geometry of the uniform spanning forest transitions in dimensions 4 8 12 the uniform spanning forest usf in d is the weak limit of random uniformly chosen spanning trees in n n d pemantle 11 proved that the usf consists a s of a single tree if and only if d 4 we prove that any two components of the usf in d are adjacent a s if 5 d 8 but not if d 9 more generally let n x y be the minimum number of edges outside the usf in a path joining x and y in d then max n x y x y in mathbb z d left lfloor d 1 4 right rfloor textrm a s

harmonic functions on planar and almost planar graphs and manifolds via circle packings the circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant harmonic bounded dirichlet function if p is a bounded circle packing in bbb r 2 whose contacts graph is a bounded valence triangulation of a disk then with probability 1 the simple random walk on p converges to a limit point moreover in this situation any continuous function on the limit set of p extends to a continuous harmonic function on the closure of the contacts graph of p that is this dirichlet problem is solvable we define the notions of almost planar graphs and manifolds and show that under the assumptions of transience and bounded local geometry these possess non constant harmonic bounded dirichlet functions let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

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

a sparse approach to mixed weak type inequalities in this paper we provide some quantitative mixed weak type estimates assuming conditions that imply that uv in a infty for calderon zygmund operators rough singular integrals and commutators the main novelty of this paper lies in the fact that we rely upon sparse domination results pushing an approach to endpoint estimates that was introduced in domingo salazar et al bull lond math soc 48 1 63 73 2016 and extended in lerner et al adv math 319 153 181 2017 and li et al j geom anal 2018

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

quasi symmetries of determinantal point processes the main result of this paper is that determinantal point processes on mathbb r corresponding to projection operators with integrable kernels are quasi invariant in the continuous case under the group of diffeomorphisms with compact support theorem 1 4 in the discrete case under the group of all finite permutations of the phase space theorem 1 6 the radon nikodym derivative is computed explicitly and is given by a regularized multiplicative functional theorem 1 4 applies in particular to the sine process as well as to determinantal point processes with the bessel and the airy kernels theorem 1 6 to the discrete sine process and the gamma kernel process the paper answers a question of grigori olshanski

loop erased self avoiding random walk in two and three dimensions if n is the position of the self avoiding random walk in d obtained by erasing loops from simple random walk then it is proved that the mean square displacemente n 2 grows at least as fast as the flory predictions for the usual saw i e at least as fast asn3 2 ford 2 andn6 5 ford 3 in particular if the mean square displacement of the usual saw grows liken1 18 ind 3 as expected then the loop erased process is in a different universality class

random walks and harmonic functions on infinite planar graphs using square tilings we study a wide class of transient planar graphs through a geometric model given by a square tiling of a cylinder for many graphs the geometric boundary of the tiling is a circle and is easy to describe in general the simple random walk on the graph converges with probability 1 to a point in the geometric boundary we obtain information on the harmonic measure and estimates on the rate of convergence this allows us to extend results we previously proved for triangulations of a disk

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large