first passage percolation network flows and electrical resistances we show that the first passage times of first passage percolation on 2 are such that p 0n n decay geometrically as n where may represent any of the four usual first passage time processes the former estimate requires no moment condition on the time coordinates but there exists a geometrically decaying estimate for the latter quantity if and only if the time coordinate distribution has finite moment generating function near the origin here is the time constant and 0 we study the line to line first passage times and describe applications to the maximum network flow through a randomly capacitated subsection of 2 and to the asymptotic behaviour of the electrical resistance of a subsection of 2 when the edges of the subsection are wires in an electrical network with random resistances in the latter case we show for example that if each edge resistance equals 1 or ohms with probabilities p and 1 p respectively then the effective resistance rn across opposite faces of an n by n box satisfies the following a if p 1 2 then p rn 1 as n b if p 1 2 then there exists p such that p p 1 underset raise0 3em hbox smash scriptscriptstyle leqslant mathop lim inf r n limits n to infty underset raise0 3em hbox smash scriptscriptstyle leqslant mathop lim sup r n limits n to infty underset raise0 3em hbox smash scriptscriptstyle leqslant v p 1

domination by product measures we consider families of 0 1 valued random variables indexed by the vertices of countable graphs with bounded degree first we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site the probability of seeing a 1 at this site is at least a value p which is large enough then this random field dominates a product measure with positive density moreover the density of this dominated product measure can be made arbitrarily close to 1 provided that p is close enough to 1 next we address the issue of obtaining the critical value of p defined as the threshold above which the domination by positive density product measures is assured for the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than k units this critical value is shown to be 1 k k k 1 k 1 and a discontinuous transition is shown to occur similar critical values of p are found for other classes of probability measures on 0 1 z for the class of k dependent measures the critical value is again 1 k k k 1 k 1 with a discontinuous transition for the class of two block factors the critical value is shown to be 1 2 and a continuous transition is shown to take place in this case thus both the critical value and the nature of the transition are different in the two block factor and 1 dependent cases

on a competitive model of laplacian growth we introduce a competitive model of laplacian growth in both stochastic and deterministic versions this defines two different aggregation laws with probabilities and 1 the parameter varying from 0 to 1 is used to weight a ratio between the inner and outer harmonic measures that leads to a competition between the eden like process and the dla solved with site sticking conditions we perform numerical and qualitative analysis of the competitive growth

overall properties of a discrete membrane with randomly distributed defects a prototype for variational percolation problems with surface energies is considered the description of the macroscopic properties of a two dimensional discrete membrane with randomly distributed defects in the spirit of the weak membrane model of blake and zisserman quadratic springs that may break at a critical length of the elongation after introducing energies depending on suitable independent identically distributed random variables this is done by exhibiting an equivalent continuum energy computed using convergence geometric measure theory and percolation arguments we show that below a percolation threshold the effect of the defects is negligible and the continuum description is given by the dirichlet integral while above that threshold an additional griffith fracture term appears in the energy which depends only on the defect probability through the chemical distance on the weak cluster of defects

the speed of a biased random walk on a percolation cluster at high density we study the speed of a biased random walk on a percolation cluster on z d in function of the percolation parameter p we obtain a first order expansion of the speed at p 1 which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice

growth pattern formation in benard flows abstract the effects of benard flows on diffusion limited aggregation dla is investigated by a growth pattern formation gpf computational model the gpf introduced consists of a dla process driven by a thermal convection field in a plane viscous incompresible flow the flow regime is governed by the grashoff prandtl and schmidt dimensionless parameters moving from low to high grashoff numbers the particle motions obtained change from primarily stochastic brownian trajectories with hausdorff dimension 2 0 to mainly deterministic curvilinear trajectories with hausdorff dimension 1 0 correspondingly the resulting clusters change their hausdorff dimension from the dla fractal to 1 0

laplacian growth with infinite noise reduction abstract four models of laplacian growth with infinite noise reduction on a square lattice are investigated these are the four possible combinations of dla and dbm with counters on bonds or sites which we call sdla bdla sdbm and bdbm all four give rise to compact patterns with nontrivial scaling exponents which we roughly estimate while bdla and bdbm are nearly identical and sdla is very similar to both sdbm produces a different pattern resembling the effect of a finite surface tension also the scaling exponent for this case could be different from the rest

variational problems with percolation dilute spin systems at zero temperature we study the asymptotic behavior of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of convergence and techniques mixing geometric measure theory and percolation while scaling to zero the lattice spacing the limit is not trivial above a percolation threshold since the lattice energies are not equi coercive a suitable notion of limit magnetization must be defined which can be characterized by two phases separated by an interface the macroscopic surface tension at this interface is characterized through a first passage percolation formula which highlights interesting connections between variational problems and percolation issues a companion result on the asymptotic description on energies defined on paths in a dilute environment is also given

estimation of positive definite m matrices and structure learning for attractive gaussian markov random fields abstract consider a random vector with finite second moments if its precision matrix is an m matrix then all partial correlations are non negative if that random vector is additionally gaussian the corresponding markov random field gmrf is called attractive we study estimation of m matrices taking the role of inverse second moment or precision matrices using sign constrained log determinant divergence minimization we also treat the high dimensional case with the number of variables exceeding the sample size the additional sign constraints turn out to greatly simplify the estimation problem we provide evidence that explicit regularization is no longer required to solve the resulting convex optimization problem we propose an algorithm based on block coordinate descent in which each sub problem can be recast as non negative least squares problem illustrations on both simulated and real world data are provided

remarks on scaling a model of witten sander type we try to prove rigorously that the perimeter of the large witten sander cluster does not scale as the square root of its area by making a forced comparison with the ill posed hele shaw problem of fluid dynamics the attempt is not completely successful nevertheless some interesting consequences of the comparison are derived

mean field diffusion limited aggregation a density model for viscous fingering phenomena we explore a universal density formalism to describe nonequilibrium growth processes specifically the immiscible viscous fingering in hele shaw cells usually referred to as the saffman taylor problem for that we develop an alternative approach to the viscous fingering phenomena whose basic concepts have been recently published in a rapid communication phys rev e 63 045305 r 2001 this approach uses the diffusion limited aggregation dla paradigm as a core we introduce a mean field dla generalization in stochastic and deterministic formulations the stochastic model a quasicontinuum dla simulates monte carlo patterns which demonstrate a striking resemblance to natural hele shaw fingers and for steady state growth regimes follow precisely the saffman taylor analytical solutions in channel and sector configurations the relevant deterministic theory a complete set of differential equations for a time development of density fields is derived from that stochastic model as a principal conclusion we prove an asymptotic equivalency of both the stochastic and deterministic mean field dla formulations to the classic saffman taylor hydrodynamics in terms of an interface evolution

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

path selection in a poisson field a criterion for path selection for channels growing in a poisson field is presented we invoke a generalization of the principle of local symmetry we then use this criterion to grow channels in a confined geometry the channel trajectories reveal a self similar shape as they reach steady state analyzing their paths we identify a cause for branching that may result in a ramified structure in which the golden ratio appears

extremal bipartite matrices abstract let g be an undirected graph on vertices 1 n let m g be the convex cone of all positive semidefinite hermitian matrices a satisfying aij 0 if i j is not an edge of g in the case that g is a complete bipartite graph we characterize all extreme rays of m g in doing so we shall also find the maximum rank that an extreme matrix in m g can have

typical and generic ranks in matrix completion abstract we consider the problem of exact low rank matrix completion from a geometric viewpoint given a partially filled matrix m we keep the positions of specified and unspecified entries fixed and study the minimal completion rank if the entries of the matrix are complex and the known entries are chosen randomly according to a continuous distribution then for a fixed pattern of locations of specified and unspecified entries there is a unique minimum completion rank which occurs with probability one we call this rank the generic completion rank over the real numbers there can be multiple ranks that occur with positive probability we call them typical completion ranks we introduce these notions formally and provide a number of inequalities and exact results on typical and generic ranks for different families of patterns of known and unknown entries

maximum likelihood estimation for linear gaussian covariance models summary we study parameter estimation in linear gaussian covariance models which are p dimensional gaussian models with linear constraints on the covariance matrix maximum likelihood estimation for this class of models leads to a non convex optimization problem which typically has many local maxima using recent results on the asymptotic distribution of extreme eigenvalues of the wishart distribution we provide sufficient conditions for any hill climbing method to converge to the global maximum although we are primarily interested in the case in which n p the proofs of our results utilize large sample asymptotic theory under the scheme n p 1 remarkably our numerical simulations indicate that our results remain valid for p as small as 2 an important consequence of this analysis is that for sample sizes n 14p maximum likelihood estimation for linear gaussian covariance models behaves as if it were a convex optimization problem

needle crystal formation in two dimensions a simple model based on conformal mapping is constructed to simulate the two dimensional growth of needle crystals growing out regularly from a common point the resulting patterns are very similar to those which were obtained by other methods the radius of gyration exponent beta is measured and found to be in the range 0 58 0 65 independent of the number of needles n it is shown that for large patterns beta asymptotically approaches a value of 2 3 to compare the patterns with the dla aggregates the average slope alpha was measured for the density density correlation function plotted logarithmically log r to log c r the relation between alpha and beta which is characteristic of dla fractal aggregates is satisfied only in the case n 8

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

diffusion limited aggregation near the percolation threshold abstract diffusion limited aggregation dla and dielectric breakdown db models have been used to simulate growth controlled by a laplacian field on a square lattice network a fraction near the percolation threshold c of the bonds had a high conductivity equal to 1 while the others had a low conductivity equal to r we used 10 5 r r 10 8 for db we find crossover from growth on an incipient percolation cluster with fractal dimensionality d 1 3 for small length scales to that on a uniform substrate d 1 7 for a large length scales the crossover length behaves as l r a with the crossover exponent a 0 25 the results were using a scaling theory

viscous fingers on fractals we investigate the problem of flow in porous media near the percolation threshold by studying the generelized model of viscous fingering vf on fractal structures we obtain analytic expressions for the fractal dimensions of the resulting structures which are in excellent agreement with existing experimental results and exact relations for the exponent dt which describes the scaling of the time it takes the fluid to cross the sample with the sample size in terms of geometrical exponents for various experimental situations lastly we discuss the relation between the continuous viscous fingers model and stochastic processes such as dielectric breakdown model dbm and diffusion limited aggregation dla

laplacian growth with separately controlled noise and anisotropy conformal mapping models are used to study the competition of noise and anisotropy in laplacian growth for this purpose a family of models is introduced with the noise level and directional anisotropy controlled independently fractalization is observed in both anisotropic growth and growth with varying noise the fractal dimension is determined from the cluster size scaling with cluster area for isotropic growth d 1 7 at both high and low noise for anisotropic growth with reduced noise the dimension can be as low as d 1 5 and apparently is not universal also we study the fluctuations of particle areas and observe in agreement with previous studies that exceptionally large particles may appear during growth leading to pathologically irregular clusters this difficulty is circumvented by using an acceptance window for particle areas

multiscale finite difference diffusion monte carlo method for simulating dendritic solidification abstract we present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field the first key feature of this method is an efficient multiscale diffusion monte carlo dmc algorithm which allows off lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid liquid interface as a result the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution the second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface inside this layer and in the solid the diffusion equation is solved using a standard finite difference algorithm that is interfaced with the dmc algorithm using the local conservation law for the diffusing quantity here we combine this algorithm with a previously developed phase field formulation of the interface dynamics and demonstrate that it can accurately simulate three dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

large deviations for the chemical distance in supercritical bernoulli percolation the chemical distance d x y is the length of the shortest open path between two points x and y in an infinite bernoulli percolation cluster in this work we study the asymptotic behavior of this random metric and we prove that for an appropriate norm depending on the dimension and the percolation parameter the probability of the event 0 x d 0 x x 1 e 1 e exponentially decreases when x 1 tends to infinity from this bound we also derive a large deviation inequality for the corresponding asymptotic shape result

large deviations for the contact process and two dimensional percolation summary the following results are proved 1 for the upper invariant measure of the basic one dimensional supercritical contact process the density of l s has the usual large deviation behavior the probability of a large deviation decays exponentially with the number of sites considered 2 for supercritical two dimensional nearest neighbor site or bond percolation the density ya of sites inside a square a which belong to the infinite cluster has the following large deviation properties the probability that ya deviates from its expected value by a positive amount decays exponentially with the area of a while the probability that it deviates from its expected value by a negative amount decays exponentially with the perimeter of a these two problems are treated together in this paper because similar techniques renormalization are used for both

geometry of maximum likelihood estimation in gaussian graphical models author s uhler caroline advisor s sturmfels bernd abstract algebraic statistics exploits the use of algebraic techniques to develop new paradigms and algorithms for data analysis the development of computational algebra software provides a powerful tool to analyze statistical models in part i of this thesis we use methods from computational algebra and algebraic geometry to study gaussian graphical models algebraic methods have proven to be useful for statistical theory and applications alike we describe a particular application to computational biology in part ii part i of this thesis investigates geometric aspects of maximum likelihood estimation in gaussian graphical models more generally we study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection in chapter 2 we examine these problems at the interface of statistics and optimization from the perspective of convex algebraic geometry and characterize the cone of all sufficient statistics for which the maximum likelihood estimator mle exists in chapter 3 we develop an algebraic elimination criterion which allows us to find exact lower bounds on the number of observations needed to ensure that the mle exists with probability one this is applied to bipartite graphs grids and colored graphs we also present the first instance of a graph for which the mle exists with probability one even when the number of observations equals the treewidth computational algebra software can be used to study graphs with a limited number of vertices and edges in chapter 4 we study the problem of existence of the mle from an asymptotic point of view by fixing a class of graphs and letting the number of vertices grow to infinity we prove that for very large cycles already two observations are sufficient for the existence of the mle with probability one part ii of this thesis describes an application of algebraic statistics to association studies rapid research progress in genotyping techniques have allowed large genome wide association studies existing methods often focus on determining associations between single loci and a specific phenotype however a particular phenotype is usually the result of complex relationships between multiple loci and the environment we develop a method for finding interacting genes i e epistasis using markov bases we test our method on simulated data and compare it to a two stage logistic regression method and to a fully bayesian method showing that we are able to detect the interacting loci when other methods fail to do so finally we apply our method to a genome wide dog data set and identify epistasis associated with canine hair length

self organized segregation on the grid we consider an agent based model with exponentially distributed waiting times in which two types of agents interact locally over a graph and based on this interaction and on the value of a common intolerance threshold tau decide whether to change their types this is equivalent to a zero temperature ising model with glauber dynamics an asynchronous cellular automaton with extended moore neighborhoods or a schelling model of self organized segregation in an open system and has applications in the analysis of social and biological networks and spin glasses systems some rigorous results were recently obtained in the theoretical computer science literature and this work provides several extensions we enlarge the intolerance interval leading to the expected formation of large segregated regions of agents of a single type from the known size epsilon 0 to size approx 0 134 namely we show that for 0 433 tau 1 2 and by symmetry 1 2 tau 0 567 the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood we further extend the interval leading to expected large segregated regions to size approx 0 312 considering almost segregated regions namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows in this case we show that for 0 344 tau le 0 433 and by symmetry for 0 567 le tau 0 656 the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood this behavior is reminiscent of supercritical percolation where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster the exponential bounds that we provide also imply that complete segregation where agents of a single type cover the whole grid does not occur with high probability for p 1 2 and the range of intolerance considered

on the chemical distance for supercritical bernoulli percolation we prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process we improve earlier results of gartner and molchanov and grimmett and marstrand and answer affirmatively a conjecture of kozlov

robust estimators for nondecomposable elliptical graphical models robust estimators of the restricted covariance matrices associated with elliptical graphical models are studied general asymptotic results which apply to both decomposable and nondecomposable graphical models are presented for robust plug in type estimators these extend results previously established only for the decomposable case furthermore a class of graphical m estimators for the restricted covariance matrices is introduced and compared with the corresponding plug in m estimators the two approaches are shown to be asymptotically equivalent under random sampling from an elliptical distribution a simulation study demonstrates the superiority of the graphical m estimators for small samples

dynamics of a complex interface we study the motion of the interface between two fluids in a pressure field a more viscous fluid surrounds a finite region filled with a less viscous fluid whose pressure is constant the more viscous fluid is incompressible and moves with a velocity proportional to the gradient of its pressure the pressure jump across the interface between the fluids is proportional to the curvature of the interface the proportionality constant is the surface tension in the two dimensional case the interface can be described by a complex function which is analytic in the exterior of the unit circle the dynamics are governed by a nonlocal nonlinear equation for this function if all the singularities of the function are situated near the origin then the nonlocal interactions contribute little to the evolution of the singularities in this case the nonlocal evolution equation can be approximated by a local nonlinear one we prove that this equation has solutions with certain uniform behavior as the surface tension is allowed to become vanishingly small

homogenization of surface and length energies for spin systems abstract we study the homogenization of lattice energies related to ising systems of the form e e u i j c i j e u i u j with u i a spin variable indexed on the portion of a cubic lattice e z d by computing their limit in the framework of surface energies in a bv setting we introduce a notion of homogenizability of the system c i j e that allows to treat periodic almost periodic and random statistically homogeneous models the latter in dimension two when the coefficients are positive ferromagnetic energies in which case the limit energy is finite on bv 1 and takes the form f u u 1 d h d 1 is the normal to u 1 where is characterized by an asymptotic formula in the random case can be expressed in terms of first passage percolation characteristics the result is extended to coefficients with varying sign under the assumption that the areas where the energies are antiferromagnetic are well separated finally we prove a dual result for discrete curves

linear estimating equations for exponential families with application to gaussian linear concentration models abstract in many families of distributions maximum likelihood estimation is intractable because the normalization constant for the density which enters into the likelihood function is not easily available the score matching estimator 35 provides an alternative where this normalization constant is not required for an exponential family e g a gaussian linear concentration model the corresponding estimating equations become linear 2 36 and the score matching estimator is shown to be consistent and asymptotically normally distributed as the number of observations increase to infinity although not necessarily efficient for linear concentration models that are also linear in the covariance 37 we show that the score matching estimator is identical to the maximum likelihood estimator hence in such cases it is also efficient gaussian graphical models and graphical models with symmetries 32 form particularly interesting subclasses of linear concentration models and we investigate the potential use of the score matching estimator for this case

simulating hydrodynamics a pedestrian model a hele shaw cell contains two fluids seperated by an interface because the fluids are held in a narrow regions between two plates the cell can be described by a set of two dimensional hydrodynamic equations which determine the velocity fields in the fluids as well as the motion of the interface between them a discretized version of these equations can be implemented in terms of the motion of random walkers the walkers have the effect of carrying pieces of the fluid from one place to another they simulate a discrete version of the laplace equation and obey the appropriate boundary conditions for the fluid the walker hydrodynamic connection is explored in the limiting situation in which the viscosity of one of the fluids vanishes an algorithm is constructed and a few exemplary simulations are shown

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

fractal viscous fingering experiments and models abstract highly ramified fingering structures occur when water is forced under pressure to displace a high viscosity shear thinning fluid in a hele shaw cell we found that water fingers grow by splitting of the leading finger tips giving rise to arborescent structures which obey fractal scaling their hausdorff dimension d f is 1 70 for a wide range of experimental conditions we show that diffusion limited growth models can accurately simulate the growth of these fractal patterns

flory approximant for the fractal dimension of the viscous finger pattern transient and asymptotic behaviors we present a flory type mean field theory to calculate the fractal dimension d of the radial viscous finger pattern formed by the less viscous fluid in a hele shaw cell we predict a the relation between the interfacial tension s between the two fluids and the onset of the asymptotic behavior and b the variation of the effective fractal dimension with time in the transient regime for a fixed

the correlation length for the high density phase of bernoulli percolation l auteur examine 2 types standards de fonctions de connectivite dans la phase de densite elevee de filtration de bernoulli du plus proche voisin il demontre que les 2 quantites decroissent exponentiellement avec le meme taux constant la reciproque de cette constante definit donc une longueur de correlation pour laquelle on n arrive pas a montrer qu elle est finie au dessus d une densite critique associee a la transition de phases des proprietes de semicontinuite et de convexite sont montrees pour cette longueur de correlation et on obtient des resultats analogues a ceux concernant le modele d ising en deux dimensions

graphical gaussian models with edge and vertex symmetries summary we introduce new types of graphical gaussian models by placing symmetry restrictions on the concentration or correlation matrix the models can be represented by coloured graphs where parameters that are associated with edges or vertices of the same colour are restricted to being identical we study the properties of such models and derive the necessary algorithms for calculating maximum likelihood estimates we identify conditions for restrictions on the concentration and correlation matrices being equivalent this is for example the case when symmetries are generated by permutation of variable labels for such models a particularly simple maximization of the likelihood function is available

dynamics of finger arrays in a diffusion limited growth model with a drift abstract we propose a phenomenological model of fingering dynamics in the presence of an external drift motivated by recent experiments on quasi two dimensional electrodeposition with hydrodynamic convection we study the dynamical transition between a laplacian growth regime and a weak finger competition regime the model proposed defines a wavelength selection problem of the finger array which is coupled in a nontrivial way to the single finger selection problem in a channel the wavelength selection mechanism proposed is associated to the crossover from a diffusive to a ballistic behavior of the aggregating particles as the coarsening spatial scale of the pattern matches the appropriate diffusion length this yields specific predictions for the selected wavelength the dynamics of the model are studied numerically with boundary integral methods in the quasistatic approximation and complementarily using a biased random walk monte carlo scheme on a lattice results are in qualitative agreement with the theoretical predictions