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1
Slowman, Alexander Barrett. "Nonequilibrium emergent interactions between run-and-tumble random walkers." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/28989.
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Nonequilibrium statistical physics involves the study of many-particle systems that break time reversibility|also known as detailed balance|at some scale. For states in thermal equilibrium, which must respect detailed balance, the comprehensive theory of statistical mechanics was developed to explain how their macroscopic properties arise from interactions between their microscopic constituent particles; for nonequilibrium states no such theory exists. The study of active matter, made up of particles that individually transduce free energy to produce systematic movement, provides a paradigm in which to develop an understanding of nonequilibrium behaviours. In this thesis, we are interested in particular in the microscopic interactions that generate the clustering of active particles that has been widely observed in simulations, and may have biological relevance to the formation of bacterial assemblages known as biofilms, which are an important source of human infection. The focus of this thesis is a microscopic lattice-based model of two random walkers interacting under mutual exclusion and undergoing the run-and-tumble dynamics that characterise the motion of certain species of bacteria, notably Escherichia coli. I apply perturbative and exact analytic approaches from statistical physics to three variants of the model in order to find the probability distributions of their nonequilibrium steady states and elucidate the emergent interactions that manifest. I first apply a generating function approach to the model on a one-dimensional periodic lattice where the particles perform straight line runs randomly interspersed by instantaneous velocity reversals or tumbles, and find an exact solution to the stationary probability distribution. The distribution can be interpreted as an effective non-equilibrium pair potential that leads to a finite-range attraction in addition to jamming between the random walkers. The finite-range attraction collapses to a delta function in the limit of continuous space and time, but the combination of this jamming and attraction is suffciently strong that even in this continuum limit the particles spend a finite fraction of time next to each other. Thus, although the particles only interact directly through repulsive hard-core exclusion, the activity of the particles causes the emergence of attractive interactions, which do not arise between passive particles with repulsive interactions and dynamics respecting detailed balance. I then relax the unphysical assumption of instantaneous tumbling and extend the interacting run-and-tumble model to incorporate a finite tumbling duration, where a tumbling particle remains stationary on its site. Here the exact solution for the nonequilibrium stationary state is derived using a generalisation of the previous generating function approach. This steady state is characterised by two lengthscales, one arising from the jamming of approaching particles, familiar from the instant tumbling model, and the other from one particle moving when the other is tumbling. The first of these lengthscales vanishes in a scaling limit where continuum dynamics is recovered. However, the second, entirely new, lengthscale remains finite. These results show that the feature of a finite tumbling duration is relevant to the physics of run-and-tumble interactions. Finally, I explore the effect of walls on the interacting run-and-tumble model by applying a perturbative graph-theoretic approach to the model with reflecting boundaries. Confining the particles in this way leads to a probability distribution in the low tumble limit with a much richer structure than the corresponding limit for the model on a periodic lattice. This limiting probability distribution indicates that an interaction over a finite distance emerges not just between the particles, but also between the particles and the reflecting boundaries. Together, these works provide a potential pathway towards understanding the clustering of self-propelled particles widely observed in active matter from a microscopic perspective.
2
Nedrebø, Per Mathias. "A Parallel Implementation of Mortal Random Walkers in the Pore Network of a Sandstone." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9806.
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<p>Simulations of the nuclear magnetic resonance relaxation method is an important part of a digital laboratory developed by Numerical Rocks. The laboratory is used to model petrophysical properties and simulating fluid flow in the pore scale of reservoir rocks. The nuclear magnetic resonance relaxation method can be simulated on a computer using a method involving random walkers. This computer simulation can be parallelized to reduce computational time. The aim of this study has been to examine how overlapping boundaries affects speed-up and communication in a parallel simulation of random walkers. Several parallel algorithms have been proposed and implemented. It was found that an overlapping partitioning of the problem is recommended, and that communication decreases exponentially with increasing overlap. However, increased overlap resulted only in a small negative impact on memory usage and speed-up.</p>
3
Bowditch, Adam. "Biased randomly trapped random walks and applications to random walks on Galton-Watson trees." Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/97359/.
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In this thesis we study biased randomly trapped random walks. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive. This application was initially considered model in its own right. We prove conditions under which the biased randomly trapped random walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. We also study the regime in which the walk is sub-ballistic; in this case we prove convergence to a stable subordinator. Furthermore, we study the fluctuations of the walk in the ballistic but sub-diffusive regime. In this setting we show that the walk can be properly centred and rescaled so that it converges to a stable process. The biased random walk on the subcritical GW-tree conditioned to survive fits suitably into the randomly trapped random walk model; however, due to a lattice effect, we cannot obtain such strong limiting results. We prove conditions under which the walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. In these cases the trapping is weak enough that the lattice effect does not have an influence; however, in the sub-ballistic regime it is only possible to obtain converge along specific subsequences. We also study biased random walks on infinite supercritical GW-trees with leaves. In this setting we determine critical upper and lower bounds on the bias such that the walk satisfies a quenched invariance principle.
4
Huang, Tsongjy. "Random walks on randomly partitioned lattices with applications toward protein fluctuations." Diss., The University of Arizona, 1995. http://hdl.handle.net/10150/187406.
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Random walks on state space partitions provide an abstract generic picture for the description of macroscopic fluctuations in complex systems like proteins. We first determine the average residence probability and the average distribution of residence times in a particular macroscopic state for the ensemble of random partitions of a one-dimensional state space. We then extend our study to the Bethe lattice and also the 2-, 3- and higher dimensional lattices. Our treatment involves both extensive analytical and numerical analyses. Finally, we compare the solution of our model on the Bethe lattice with the experimental data and find excellent agreement.
5
Windisch, David. "Random walks, disconnection and random interlacements /." [S.l.] : [s.n.], 2009. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=18343.
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6
Gnacik, Michal. "Quantum random walks." Thesis, Lancaster University, 2014. http://eprints.lancs.ac.uk/69946/.
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In this thesis we investigate the convergence of various quantum random walks to quantum stochastic cocycles defined on a Bosonic Fock space. We prove a quantum analogue of the Donsker invariance principle by invoking the so-called semigroup representation of quantum stochastic cocycles. In contrast to similar results by other authors our proof is relatively elementary. We also show convergence of products of ampliated random walks with different system algebras; in particular, we give a sufficient condition to obtain a cocycle via products of cocycles. The CCR algebra, its quasifree representations and the corresponding quasifree stochastic calculus are also described. In particular, we study in detail gauge-invariant and squeezed quasifree states. We describe repeated quantum interactions between a `small' quantum system and an environment consisting of an infinite chain of particles. We study different cases of interaction, in particular those which occur in weak coupling limits and low density limits. Under different choices of scaling of the interaction part we show that random walks, which are generated by the associated unitary evolutions of a repeated interaction system, strongly converge to unitary quantum stochastic cocycles. We provide necessary and sufficient conditions for such convergence. Furthermore, under repeated quantum interactions, we consider the situation of an infinite chain of identical particles where each particle is in an arbitrary faithful normal state. This includes the case of thermal Gibbs states. We show that the corresponding random walks converge strongly to unitary cocycles for which the driving noises depend on the state of the incoming particles. We also use conditional expectations to obtain a simple condition, at the level of generators, which suffices for the convergence of the associated random walks. Limit cocycles, for which noises depend on the state of the incoming particles, are also obtained by investigating what we refer to as `compressed' random walks. Lastly, we show that the cocycles obtained via the procedure of repeated quantum interactions are quasifree, thus the driving noises form a representation of the relevant CCR algebra. Both gauge-invariant and squeezed representations are shown to occur.
7
Forghani, Behrang. "Transformed Random Walks." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32538.
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We consider transformations of a given random walk on a countable group determined by Markov stopping times. We prove that these transformations preserve the Poisson boundary. Moreover, under some mild conditions, the asymptotic entropy (resp., rate of escape) of the transformed random walks is equal to the asymptotic entropy (resp., rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov's formula from ergodic theory.
8
Buckley, Stephen Philip. "Problems in random walks in random environments." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:06a12be2-b831-4c2a-87b1-f0abccfb9b8b.
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Recent years have seen progress in the analysis of the heat kernel for certain reversible random walks in random environments. In particular the work of Barlow(2004) showed that the heat kernel for the random walk on the infinite component of supercritical bond percolation behaves in a Gaussian fashion. This heat kernel control was then used to prove a quenched functional central limit theorem. Following this work several examples have been analysed with anomalous heat kernel behaviour and, in some cases, anomalous scaling limits. We begin by generalizing the first result - looking for sufficient conditions on the geometry of the environment that ensure standard heat kernel upper bounds hold. We prove that these conditions are satisfied with probability one in the case of the random walk on continuum percolation and use the heat kernel bounds to prove an invariance principle. The random walk on dynamic environment is then considered. It is proven that if the environment evolves ergodically and is, in a certain sense, geometrically d-dimensional then standard on diagonal heat kernel bounds hold. Anomalous lower bounds on the heat kernel are also proven - in particular the random conductance model is shown to be "more anomalous" in the dynamic case than the static. Finally, the reflected random walk amongst random conductances is considered. It is shown in one dimension that under the usual scaling, this walk converges to reflected Brownian motion.
9
Nadal, Céline. "Matrices aléatoires et leurs applications à la physique statistique et quantique." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00633266.
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Cette thèse est consacrée à l'étude des matrices aléatoires et à quelques unes de leurs applications en physique, en particulier en physique statistique et en physique quantique.C'est un travail essentiellement analytique complété par quelques simulations numériques Monte Carlo. Dans un premier temps j'introduis la théorie des matrices aléatoires de façon assez générale : je définis les principaux ensembles de matrices aléatoires (en particulier gaussiens) et décris leurs propriétés fondamentales (distribution des valeurs propres, densité, etc). Dans un second temps je m'intéresse à des systèmes physiques d'interfaces à l'équilibre qui peuvent être modélisés par des marcheurs ''vicieux'', c'est-à-dire des marcheurs aléatoires conditionnés à ne pas se croiser. On peut montrer que la distribution des positions des marcheurs à un temps donné est exactement celle des valeurs propres d'une matrice aléatoire. J'étudie ensuite un problème physique qui relève d'un domaine très différent, celui de l'information quantique, mais qui est également étroitement relié aux matrices aléatoires: celui de l'intrication pour des états aléatoires dans un système quantique bipartite (fait de deux sous-parties) de grande taille. Enfin je m'intéresse à certaines propriétés des matrices aléatoires comme la distribution du nombre de valeurs propres positives ou encore la distribution de la valeur propre maximale (loi de Tracy-Widom près de la moyenne et grandes déviations loin de la moyenne).
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Oosthuizen, Joubert. "Random walks on graphs." Thesis, Stellenbosch : Stellenbosch University, 2014. http://hdl.handle.net/10019.1/86244.
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Thesis (MSc)--Stellenbosch University, 2014.<br>ENGLISH ABSTRACT: We study random walks on nite graphs. The reader is introduced to general
Markov chains before we move on more specifically to random walks on graphs.
A random walk on a graph is just a Markov chain that is time-reversible. The
main parameters we study are the hitting time, commute time and cover time.
We nd novel formulas for the cover time of the subdivided star graph and
broom graph before looking at the trees with extremal cover times.
Lastly we look at a connection between random walks on graphs and electrical
networks, where the hitting time between two vertices of a graph is expressed
in terms of a weighted sum of e ective resistances. This expression in turn
proves useful when we study the cover cost, a parameter related to the cover
time.<br>AFRIKAANSE OPSOMMING: Ons bestudeer toevallige wandelings op eindige gra eke in hierdie tesis. Eers
word algemene Markov kettings beskou voordat ons meer spesi ek aanbeweeg
na toevallige wandelings op gra eke. 'n Toevallige wandeling is net 'n Markov
ketting wat tyd herleibaar is. Die hoof paramaters wat ons bestudeer is die
treftyd, pendeltyd en dektyd. Ons vind oorspronklike formules vir die dektyd
van die verdeelde stergra ek sowel as die besemgra ek en kyk daarna na die
twee bome met uiterste dektye.
Laastens kyk ons na 'n verband tussen toevallige wandelings op gra eke en
elektriese netwerke, waar die treftyd tussen twee punte op 'n gra ek uitgedruk
word in terme van 'n geweegde som van e ektiewe weerstande. Hierdie uitdrukking
is op sy beurt weer nuttig wanneer ons die dekkoste bestudeer, waar
die dekkoste 'n paramater is wat verwant is aan die dektyd.
11
Phetpradap, Parkpoom. "Intersections of random walks." Thesis, University of Bath, 2011. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.548100.
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We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk. We call this the range of the random walk. We derive a large deviation principle for the probability that the range of simple random walk deviates from its mean. Our result describes the behaviour for deviation below the typical value. This is a result analogous to that obtained by van den Berg, Bolthausen, and den Hollander for the volume of the Wiener sausage. In the second part of the thesis, we are interested in the number of lattice sites visited by two independent simple random walks starting at the origin. We call this the intersection of ranges. We derive a large deviation principle for the probability that the intersection of ranges by time n exceeds a multiple of n. This is also an analogous result of the intersection volume of two independent Wiener sausages.
12
Montgomery, Aaron. "Topics in Random Walks." Thesis, University of Oregon, 2013. http://hdl.handle.net/1794/13335.
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We study a family of random walks defined on certain Euclidean lattices that are related to incidence matrices of balanced incomplete block designs. We estimate the return probability of these random walks and use it to determine the asymptotics of the number of balanced incomplete block design matrices. We also consider the problem of collisions of independent simple random walks on graphs. We prove some new results in the collision problem, improve some existing ones, and provide counterexamples to illustrate the complexity of the problem.
13
Ngoc, Anh Do Hoang. "Anomalous diffusion and random walks on random fractals." Doctoral thesis, Universitätsbibliothek Chemnitz, 2010. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201000205.
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The purpose of this research is to investigate properties of diffusion processes
in porous media. Porous media are modelled by random Sierpinski carpets,
each carpet is constructed by mixing two different generators with the same
linear size. Diffusion on porous media is studied by performing random
walks on random Sierpinski carpets and is characterized by the random walk
dimension $d_w$.
In the first part of this work we study $d_w$ as a function of the ratio of constituents
in a mixture. The simulation results show that the resulting $d_w$ can
be the same as, higher or lower than $d_w$ of carpets made by a single constituent
generator. In the second part, we discuss the influence of static external
fields on the behavior of diffusion. The biased random walk is used to model
these phenomena and we report on many simulations with different field
strengths and field directions. The results show that one structural feature
of Sierpinski carpets called traps can have a strong influence on the observed
diffusion properties. In the third part, we investigate the effect of diffusion
under the influence of external fields which change direction back and forth
after a certain duration. The results show a strong dependence on the period
of oscillation, the field strength and structural properties of the carpet.
14
Dou, Carl C. Z. (Carl Changzhu). "Studies of random walks on groups and random graphs." Thesis, Massachusetts Institute of Technology, 1992. http://hdl.handle.net/1721.1/13243.
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Gabucci, Ilenia. "Random walks classici e quantistici." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/17759/.
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Questa tesi si propone di fornire uno studio sul random walk. Partendo da un approfondimento sulla teoria della probabilita` alla base di tale processo ed in particolare le distribuzioni Binomiale e Gaussiana, si `e potuto studiare il caso del random walk classico, sia nel caso del reticolo monodimensionale che per reticoli a piu` dimensioni. Sempre nell’ambito classico si sono analizzati anche i processi stocastici dipendenti dal tempo, detti Processi di Markov, e il moto browniano, per cui si sono ricavate le equazioni del moto della particella browniana, ovvero le equazioni di Langevin. Si sono in seguito definiti i quattro postulati della meccanica quantistica, introducendo il concetto di quantum bit, per cui si `e fornito un confronto con il bit classico. Si `e inoltre trattato il calcolo quantistico, definendo il concetto di porta logica e affrontando l’esempio importante del gate di Hadamard, utile nel caso del random walk quantistico. Quest’ultimo `e stato introdotto separando il caso nel discreto e nel continuo, cercando di sottolineare le differenze che presenta con il caso classico.
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Codling, Edward Alexander. "Biased random walks in biology." Thesis, University of Leeds, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275673.
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Janse, van Rensburg Esaias Johannes. "Field theory and random walks." Thesis, University of Cambridge, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.328723.
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Carigi, Giulia. "On the recurrence of random walks in Lévy random environments." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/10088/.
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This thesis investigates one-dimensional random walks in random environment whose transition probabilities might have an infinite variance. The ergodicity of the dynamical system ''from the point of view of the particle'' is proved under the assumptions of transitivity and existence of an absolutely continuous steady state on the space of the environments. We show that, if the average of the local drift over the environments is summable and null, then the RWRE is recurrent. We provide an example satisfying all the hypotheses.
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Tokushige, Yuki. "Random Walks on random trees and hyperbolic groups: trace processes on boundaries at infinity and the speed of biased random walks." Kyoto University, 2019. http://hdl.handle.net/2433/242580.
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Dykiel, Patrik. "Asymptotic properties of coalescing random walks." Thesis, Uppsala University, Department of Mathematics, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121369.
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He, Mu. "The Torsion Angle of Random Walks." TopSCHOLAR®, 2013. http://digitalcommons.wku.edu/theses/1242.
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In this thesis, we study the expected mean of the torsion angle of an n-stepequilateral random walk in 3D. We consider the random walk is generated within a confining sphere or without a confining sphere: given three consecutive vectors →e1 , →e2 , and →e3 of the random walk then the vectors →e1 and →e2 define a plane and the vectors →e2 and →e3 define a second plane. The angle between the two planes is called the torsion angle of the three vectors. Algorithms are described to generate random walks which are used in a particular space (both without and with confinement). The torsion angle is expressed as a function of six variables for a random walk in both cases: without confinement and with confinement, respectively. Then we find the probability density functions of these six variables of a random walk and demonstrate an explicit integral expression for the expected mean torsion value. Finally, we conclude that the expected torsion angle obtained by the integral agrees with the numerical average torsion obtained by a simulation of random walks with confinement.
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Bertacchi, D., and Andreas Cap@esi ac at. "Random Walks on Diestel--Leader Graphs." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1004.ps.
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Lapeyre, Gerald John. "Random walks on a fluctuating lattice." Diss., The University of Arizona, 2001. http://hdl.handle.net/10150/298790.
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In recent years, studies of diffusion in random media have been extended to include the effects of media in which the defects fluctuate randomly in time. Typically, the diffusive motion of particles in a static medium persists when the medium is allowed to fluctuate, with the diffusivity (diffusion constant) D depending on the character of the fluctuations. In the present work, we study random walks on lattices in which the bonds connecting vertices open and close randomly in time, and the walker is not allowed to cross a closed bond. Variations of the model studied here have been used to model the diffusion of CO through myoglobin, the transport of ions in polymer solutions, and conduction in hydrogenated amorphous silicon. The major objective in analyzing these systems is to find efficient methods for computing the diffusivity. In this dissertation, we focus mainly on methods of computing the diffusivity in our model. In addition, we study the critical behavior of the model and present a demonstration, valid for a restricted range of model parameters, that the distribution of the displacement converges in time to a Gaussian with width D. To compute the diffusivity, we use a numerical renormalization group (RG) method, power series expansions in model parameters, and Monte Carlo simulations. We choose a model with two parameters characterizing the bond fluctuations--the time scale of fluctuations tau and the mean open-bond density p. We calculate a series expansion of the diffusivity to about 10th order in the parameter nu = exp(.1/τ) on the hypercubic lattice Zᵈ for d = 1, 2, 3, as well as on the Bethe lattice. We compute the same power series expansion to 3rd order in ν for arbitrary d. We compute estimates of the diffusivity on the Bethe lattice using the RG methods and show by comparison to Monte Carlo data that the RG provides excellent quantitative predictions of D when τ is not too large.
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Jiang, Jianping. "Random Walks and Their Scaling Limits." Diss., The University of Arizona, 2015. http://hdl.handle.net/10150/556605.
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This dissertation deals with two-dimensional random walks and their conformally invariant scaling limits. More precisely, we study two kinds of random walks: sum of independently identically distributed random variables (with Brownian motion as the scaling limit) and self-interacting random walks (with Schramm-Loewner Evolution processes as their scaling limits). We organize our main results in three parts. In the first part, we study two types of probability measures on Brownian paths: fixed time ensemble and fixed endpoints ensemble. We prove a relationship between those two ensembles. The relationship is that if we take a curve from the fixed time ensemble, weight it by a suitable power of the distance of the two endpoints and then apply the conformal map that takes those two endpoints to the endpoints from the other ensemble, then the resulting curve is distributed as the other ensemble. The second part deals with exploration processes and their scaling limits. We define two radial exploration processes in a domain D, i.e., self-interacting random walks between a boundary point and an interior point. We prove those processes satisfy the reversibility property. The reversibility property enables us to prove the distribution of the last hitting point with the boundary of any radial SLE₆ is the harmonic measure. We also define an exploration process in the full-plane and prove its scaling limit is the full-plane SLE₆. A by-product of these result is that the time-reversal of a radial SLE₆ trace in D (aiming at 0, say) after the last visit to ∂D is the full-plane SLE₆ trace (starting at 0) up to the first visit of ∂D. The last part is devoted to Dirichlet problems in a domain D. Let f be the solution to a continuous Dirichlet problem. We approximate the continuous Laplacian by the generator for a continuous-state random walk with each step uniformly distributed in a disk of radius h. Let f_h be the corresponding solution. We find the limit of (f_h-f)/h as h approaches 0.
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Deligiannidis, Georgios. "Some results associated with random walks." Thesis, University of Nottingham, 2010. http://eprints.nottingham.ac.uk/13104/.
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In this thesis we treat three problems from the theory and applications of random walks. The first question we tackle is from the theory of the optimal stopping of random walks. We solve the infinite-horizon optimal stopping problem for a class of reward functions admitting a representation introduced in Boyarchenko and Levendorskii [1], and obtain closed expressions for the expected reward and optimal stopping time. Our methodology is a generalization of an early paper by Darling et al. [2] and is based on probabilistic techniques: in particular a path decomposition related to the Wiener-Hopf factorization. Examples from the literature and perturbations are treated to demonstrate the flexibility of our approach. The second question is related to the path structure of lattice random walks. We obtain the exact asymptotics of the variance of the self- intersection local time Vn which counts the number of times the paths of a random walk intersect themselves. Our approach extends and improves upon that of Bolthausen [3], by making use of complex power series. In particular we state and prove a complex Tauberian lemma, which avoids the assumption of monotonicity present in the classical Tauberian theorem. While a bound of order 0(n2) has previously been claimed in the literature ([3], [4]) we argue that existing methods only show the tipper bound O(n2 log n), unless extra conditions are imposed to ensure monotonicity of the underlying sequence. Using the complex Tauberian approach we show that Var (Vn ) Cn2, thus settling a long-standing misunderstanding. Finally, in the last chapter, we prove a functional central limit theorem for one-dimensional random walk in random scenery, a result conjectured in 1979 by Kesten and Spitzer [5]. Essentially random walk in random scenery is the process defined by the partial suins of a collection of random variables (the random scenery), sampled by a random walk. We show that for Z-valued random walk attracted to the symmetric Cauchy law, and centered random scenery with second moments, a functional central limit theorem holds, thus proving the Kesten and Spitzer [5] conjecture which had remained open since 1979. Our proof makes use of tile asymptotic results obtained in the Chapter 3.
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Xu, Chang. "Convex hulls of planar random walks." Thesis, University of Strathclyde, 2017. http://digitool.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=28164.
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For the perimeter length Ln and the area An of the convex hull of the first n steps of a planar random walk, this thesis study n â mean and variance asymptotics and establish distributional limits. The results apply to random walks both with drift (the mean of random walk increments) and with no drift under mild moments assumptions on the increments. Assuming increments of the random walk have finite second moment and non zero mean, Snyder and Steele showed that nâ1Ln converges almost surely to a deterministic limit, and proved an upper bound on the variance Var[Ln] = O(n).We show that nâ1Var[Ln] converges and give a simple expression for the limit,which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for Ln in the non-degenerate case. Then we focus on the perimeter length with no drift and area with both drift and zero-drift cases. These results complement and contrast with previous work and establish non-Gaussian distributional limits. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are non-zero.
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fr, kaimanov@univ-rennes1. "Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1085.ps.
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Sylvester, John A. "Random walks, effective resistance and neighbourhood statistics in binomial random graphs." Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/106467/.
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The binomial random graph model G(n; p), along with its near-twin sibling G(n; m), were the starting point for the entire study of random graphs and even probabilistic combinatorics as a whole. The key properties of these models are woven into the fabric of the field and their behaviour serves as a benchmark to compare any other model of random structure. In this thesis we contribute to the already rich literature on G(n; p) in a number of directions. Firstly, vertex to vertex hitting times of random walks in G(n; p) are considered via their interpretation as potential differences in an electrical network. In particular we show that in a graph satisfying certain connectivity properties the effective resistance between two vertices is typically determined, up to lower order terms, by the degrees of these vertices. We apply this to obtain the expected values of hitting times and several related indices in G(n; p), and to prove that these values are concentrated around their mean. We then study the statistics of the size of the r-neighbourhood of a vertex in G(n; p). We show that the sizes of these neighbourhoods satisfy a central limit theorem. We also bound the probability a vertex in G(n; p) has an r-neighbourhood of size k from above and below by functions of n; p and k which match up to constants. Finally, in the last chapter the extreme values of the r-degree sequence are studied. We prove a novel neighbourhood growth estimate which states that with high probability the size of a vertex's r neighbourhood is determined, up to lower order terms, by the size of its first neighbourhood. We use this growth estimate to bound the number of vertices attaining a smallest r-neighbourhood.
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Bui, Thi Thuy. "Limit theorems for branching random walks and products of random matrices." Thesis, Lorient, 2020. https://tel.archives-ouvertes.fr/tel-03261556.
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L’objectif du sujet de ma thèse est d’établir des théorèmes limites pour des marches aléatoires avec branchement gouvernées par des produits de matrices aléatoires, en profitant des progrès récents sur les produits de matrices aléatoires et en y établissant de nouveaux résultats adaptés au besoin. La première partie concerne le modèle classique d'une marche aléatoire avec branchement sur la droite réelle. Nous établissons une borne Berry-Esseen et une asymptotique précise de déviation modérée de type Cramér pour la mesure de comptage qui compte le nombre de particules de n-ième génération situées dans une région donnée. La deuxième partie est consacrée à l'étude des produits $G_n = A_n \ldots A_1$ de matrices aléatoires réelles $A_i$ de type $d \times d$, indépendantes et identiquement distribuées. Dans cette partie, avec une motivation pour des applications aux marches aléatoires avec branchement gouvernées par des produits de matrices aléatoires, nous améliorons et étendons le théorème central limite et le théorème limite local établis par Le Page (1982). Dans la troisième partie, on considère un modèle de marches aléatoires avec branchement, où les mouvements des individus sont gouvernés par des produits de matrices aléatoires de type $d \times d$. A l'aide des résultats établis à la deuxième partie pour les produits de matrices aléatoires, on établit un théorème central limite et une expansion asymptotique à grande déviation de type Bahadur-Rao pour la mesure de comptage $ Z_n^x $ qui compte le nombre de particules de n-ième génération situées dans une région donnée avec normalisation appropriée. La quatrième partie est une suite de la troisième partie. Dans cette partie, on établit la borne de type Berry-Esseen à propos de la vitesse de convergence dans le théorème central limite et une asymptotique précise de déviation modérée de type Cramér pour $ Z_n^x $<br>The main objective of my thesis is to establish limit theorems for a branching random walk with products of random matrices by taking advantage of recent advances in products of random matrices and establishing new results as needed. The first part concerns the classic branching random walk on the real line. We establish a Berry- Esseen bound and a Cramér type moderate deviation expansion for the counting measure which counts the number of particles of nth generation situated in a given region. The second part is devoted to the study of the products $G_n = A_n \ldots A_1$ of real random matrices $A_i$ of type $ d \times d$, independent and identically distributed. In this part, with a motivation for applications to branching random walks governed by products of random matrices, we improve and extend the central limit theorem and the local limit theorem established by Le Page (1982). In the third part, we consider a branching random walk model, where the movements of individuals are governed by products of random matrices of type $ d \times d $. Using the results established in the second part for the products of random matrices, we establish a central limit theorem and a large deviation asymptotic expansion of the Bahadur-Rao type for the counting measure $ Z_n^x $ which counts the number n-th generation particles located in a given region with suitable norming. The fourth part is a continuation of the third part. In this part, we establish the Berry-Esseen bound which gives the speed of convergence in the central limit theorem and a precise Cramér- type moderate deviation asymptotic for $ Z_n^x $
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Djurdjevac, Nataša [Verfasser]. "Methods for analyzing complex networks using random walker approaches / Nataša Djurdjevac." Berlin : Freie Universität Berlin, 2012. http://d-nb.info/1028497512/34.
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Nakashima, Makoto. "Almost sure central limit theorem for branching random walks in random environment." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/157736.
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Wallén, Daniel. "Cover times of random walks on graphs." Thesis, Uppsala University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-125278.
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Schulzky, Christian Berthold. "Anomalous Diffusion and Random Walks on Fractals." Doctoral thesis, Universitätsbibliothek Chemnitz, 2000. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200000705.
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In dieser Arbeit werden verschieden Ansätze diskutiert, die zum Verständnis und zur Beschreibung anomalen Diffusionsverhaltens beitragen, wobei insbesondere zwei unterschiedliche Aspekte hervorgehoben werden. Zum einen wird das Entropieproduktions-Paradoxon beschrieben, welches bei der Analyse der Entropieproduktion bei der anomalen Diffusion, beschrieben durch fraktionale Diffusionsgleichungen auftritt. Andererseits wird ein detaillierter Vergleich zwischen Lösungen verallgemeinerter Diffusionsgleichungen mit numerischen Daten präsentiert, die durch Iteration der Mastergleichung auf verschiedenen Fraktalen produziert worden sind.
Die Entropieproduktionsrate für superdiffusive Prozesse wird berechnet und zeigt einen unerwarteten Anstieg beim Übergang von dissipativer Diffusion zur reversiblen Wellenausbreitung. Dieses Entropieproduktions-Paradoxon ist die direkte Konsequenz einer anwachsenden intrinsischen Rate bei Prozessen mit zunehmendem Wellencharakter. Nach Berücksichtigung dieser Rate zeigt die Entropie den erwarteten monotonen Abfall. Diese Überlegungen werden für generalisierte Entropiedefinitionen, wie die Tsallis- und Renyi-Entropien, fortgeführt.
Der zweite Aspekt bezieht sich auf die anomale Diffusion auf Fraktalen, im Besonderen auf Sierpinski-Dreiecke und -Teppiche. Die entsprechenden Mastergleichungen werden iteriert und die auf diese Weise numerisch gewonnenen Wahrscheinlichkeitsverteilungen werden mit den Lösungen vier verschiedener verallgemeinerter Diffusionsgleichungen verglichen.
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Sbihi, Amine M. (Amine Mohammed). "Covering times for random walks on graphs." Thesis, McGill University, 1990. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74538.
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This thesis is a contribution to the covering times problems for random walks on graphs. By considering uniform random walks on finite connected graphs, the covering time is defined as the time (number of steps) taken by the random walk to visit every vertex. The motivating problem of this thesis is to find bounds for the expected covering times. We provide explicit bounds that are uniformly valid over all starting points and over large classes of graphs. In some cases the asymptotic distribution of the suitably normalized covering time is given as well.
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Schubert, Sven. "Random walks in complex systems anomalous relaxation /." Doctoral thesis, [S.l. : s.n.], 1999. http://deposit.ddb.de/cgi-bin/dokserv?idn=956664911.
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Bertacchi, Daniela, Fabio Zucca, and Andreas Cap@esi ac at. "Classification on the Average of Random Walks." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1026.ps.
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at, Klaus Schmidt@univie ac. "Growth and Recurrence of Stationary Random Walks." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1071.ps.
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Folz, Matthew Bryan. "Adapted metrics and random walks on graphs." Thesis, University of British Columbia, 2013. http://hdl.handle.net/2429/44947.
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This thesis discusses various aspects of continuous-time simple random walks on measure weighted graphs, with a focus on behaviors related to large-scale geometric properties of the underlying graph. In contrast to previous work in this area, the majority of the results presented here are applicable to random walks with unbounded generators. A recurring theme in this research is the use of novel distance functions for graphs known as adapted metrics, which are demonstrated to be a powerful tool for studying random walks on graphs.
Chapter 2 provides an overview of the relevant probabilistic and analytic theory, and provides multiple constructions of the heat kernel and a brief introduction to the theory of Dirichlet forms. Chapter 3 introduces adapted metrics, which play a central role in the following chapters, and which are especially useful in understanding random walks with unbounded generators. Chapter 4 discusses heat kernel estimates, and presents an overview of on-diagonal heat kernel estimates for graphs, as well as techniques for obtaining various off-diagonal estimates of the heat kernel. The off-diagonal estimates were proved by the author, and are notable for their use of adapted metrics. Chapter 5 introduces metric graphs, a continuous analogue of graphs which possess many desirable analytic properties, and analyzes the problem of constructing a Brownian motion on a metric graph which behaves similarly to a given continuous-time simple random walk. Chapter 6 analyzes recurrence and transience of graphs, and proves an original estimate relating adapted volume growth to recurrence of graphs. Chapter 7 discusses bounds for the bottom of the essential spectrum in terms of geometric inequalities such as volume growth estimates and isoperimetric inequalities. The main result of this chapter was proved by the author, and establishes sharp estimates for the bottom of the spectrum in terms of the adapted volume growth. Chapter 8 considers stochastic completeness of graphs and proves sharp criteria relating volume growth to stochastic completeness. The results of this chapter were proved by the author, using the machinery of metric graphs.
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Maddalena, Daniela. "Stationary states in random walks on networks." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/10170/.
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In this thesis we dealt with the problem of describing a transportation network in which the objects in movement were subject to both finite transportation capacity and finite accomodation capacity. The movements across such a system are realistically of a simultaneous nature which poses some challenges when formulating a mathematical description. We tried to derive such a general modellization from one posed on a simplified problem based on asyncronicity in particle transitions. We did so considering one-step processes based on the assumption that the system could be describable through discrete time Markov processes with finite state space. After describing the pre-established dynamics in terms of master equations we determined stationary states for the considered processes. Numerical simulations then led to the conclusion that a general system naturally evolves toward a congestion state when its particle transition simultaneously and we consider one single constraint in the form of network node capacity. Moreover the congested nodes of a system tend to be located in adjacent spots in the network, thus forming local clusters of congested nodes.
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Volkov, Oleksii. "Random Walks on Products of Hyperbolic Groups." Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/41955.
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The subject area of this thesis is the theory of random walks on groups. First, we
study random walks on products of hyperbolic groups and show that the Poisson
boundary can be identified with an appropriate geometric boundary (the skeleton).
Second, we show that in the particular case of free and free-product factors, the Hausdorff dimension of the conditional measures on product fibers of the Poisson boundary is related to the asymptotic entropy and the rate of escape of the corresponding conditional random walks via a generalized entropy-dimension formula.
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Böhm, Walter, and Kurt Hornik. "On Two-Periodic Random Walks with Boundaries." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 2008. http://epub.wu.ac.at/936/1/document.pdf.
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Two-periodic random walks are models for the one-dimensional motion of particles in which the jump probabilities depend on the parity of the currently occupied state. Such processes have interesting applications, for instance in chemical physics where they arise as embedded random walk of a special queueing problem. In this paper we discuss in some detail first passage time problems of two-periodic walks, the distribution of their maximum and the transition functions when the motion of the particle is restricted by one or two absorbing boundaries. As particular applications we show how our results can be used to derive the distribution of the busy period of a chemical queue and give an analysis of a somewhat weird coin tossing game.<br>Series: Research Report Series / Department of Statistics and Mathematics
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Schwalbe, Lehtihet André, and Bulancea Oscar Lindvall. "Quantum Random Walks with Perturbing Potential Barriers." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-210864.
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With a recent interest in quantum computers, the properties of quantum mechanicalcounterparts to classical algorithms have been studied in the hope of providing efficientalgorithms for quantum computers. Because of the success of classical random walks inproviding good algorithms on classical computers, attention has been turned to quantumrandom walks, since they may similarly be used to construct efficient probabilisticalgorithms on quantum computers. In this thesis we examine properties of the quantumwalk on the line, in particular the standard deviation and the shape of the probabilitydistribution, and the effect of potentials perturbing the walk. We model these potentialsas rectangular barriers between the walker’s positions and introduce a probability of thewalker failing to perform the step procedure, similar to that of Wong in Ref. [14]. We findthat a potential localized around the starting position leads to an increased standard deviationand makes the walk increasingly ballistic. We also find that uniformly distributedrandom potentials have the general effect of localizing the distribution, similar to that ofAnderson localization.
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Izsak, Alexander. "The second eigenvalue and random walks in random regular graphs with increasing girth." Thesis, University of British Columbia, 2009. http://hdl.handle.net/2429/12649.
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The goal of this thesis is to upper bound the expected value of the second largest eigenvalue in magnitude of random regular graphs with a given minimum girth.
Having a small upper bound implies such random graphs are likely to be expanders and thus have several combinatorial properties useful in
various fields of computer science. The best possible upper bound asymptotically on the second eigenvalue has already been proven for random regular graphs
without conditions on the girth. Finding this upper bound though required long and complicated analysis due to tangles, which are certain small subgraphs that contain cycles. This thesis thus hypothesizes that specifying a minimum girth large enough will prevent tangles from occurring in random graphs and thus proving an optimal upper bound on the second eigenvalue can avoid the difficult analysis required in order to handle tangles.
To find such an upper bound on random regular graphs with specified minimum girth we consider the probability that a random walk in such a random graph returns to the first vertex of the walk in the k-th step of the walk. We prove for 2-regular graphs that the random walk is more likely to visit any given vertex not in the walk
than the starting vertex of the walk on the k-th step, and bound how much more likely this event is. We also analyze the d-regular case and we believe our findings will
lead to a similar result in this case.
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Ahn, Sung Won. "Oscillation of quenched slowdown asymptotics of random walks in random environment in Z." Thesis, Purdue University, 2016. http://pqdtopen.proquest.com/#viewpdf?dispub=10170588.
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<p> We consider a one dimensional random walk in a random environment (RWRE) with a positive speed lim<i><sub>n</sub></i><sub>→∞</sub> (<i>X<sub>n</sub>/</i>) = υ<sub>α</sub> > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities <i>P</i><sub> ω</sub>(<i>X<sub>n</sub></i> < <i>xn</i>) with <i> x</i>∈ (0,υ<sub>α</sub>) decay approximately like exp{-<i> n</i><sup>1-1/</sup><i><sup>s</sup></i>} for a deterministic <i> s</i> > 1. More precisely, they showed that <i>n</i><sup> -γ</sup> log <i>P</i><sub>ω</sub>(<i>X<sub>n </sub></i> < <i>xn</i>) converges to 0 or -∞ depending on whether γ > 1 - 1/<i>s</i> or γ < 1 - 1/<i> s</i>. In this paper, we improve on this by showing that <i>n</i><sup> -1+1/</sup><i><sup>s</sup></i> log <i>P</i><sub> ω</sub>(X<sub>n</sub> < <i>xn</i>) oscillates between 0 and -∞ , almost surely.</p>
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Lacroix-A-Chez-Toine, Bertrand. "Extreme value statistics of strongly correlated systems : fermions, random matrices and random walks." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLS122/document.
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La prévision d'événements extrêmes est une question cruciale dans des domaines divers allant de la météorologie à la finance. Trois classes d'universalité (Gumbel, Fréchet et Weibull) ont été identifiées pour des variables aléatoires indépendantes et de distribution identique (i.i.d.).La modélisation par des variables aléatoires i.i.d., notamment avec le modèle d'énergie aléatoire de Derrida, a permis d'améliorer la compréhension des systèmes désordonnés. Cette hypothèse n'est toutefois pas valide pour de nombreux systèmes physiques qui présentent de fortes corrélations. Dans cette thèse, nous étudions trois modèles physiques de variables aléatoires fortement corrélées : des fermions piégés,des matrices aléatoires et des marches aléatoires. Dans la première partie, nous montrons plusieurs correspondances exactes entre l'état fondamental d'un gaz de Fermi piégé et des ensembles de matrices aléatoires. Le gaz Fermi est inhomogène dans le potentiel de piégeage et sa densité présente un bord fini au-delà duquel elle devient essentiellement nulle. Nous développons une description précise des statistiques spatiales à proximité de ce bord, qui va au-delà des approximations semi-classiques standards (telle que l'approximation de la densité locale). Nous appliquons ces résultats afin de calculer les statistiques de la position du fermion le plus éloigné du centre du piège, le nombre de fermions dans un domaine donné (statistiques de comptage) et l'entropie d'intrication correspondante. Notre analyse fournit également des solutions à des problèmes ouverts de valeurs extrêmes dans la théorie des matrices aléatoires. Nous obtenons par exemple une description complète des fluctuations de la plus grande valeur propre de l'ensemble complexe de Ginibre.Dans la deuxième partie de la thèse, nous étudions les questions de valeurs extrêmes pour des marches aléatoires. Nous considérons les statistiques d'écarts entre positions maximales consécutives (gaps), ce qui nécessite de prendre en compte explicitement le caractère discret du processus. Cette question ne peut être résolue en utilisant la convergence du processus avec son pendant continu, le mouvement Brownien. Nous obtenons des résultats analytiques explicites pour ces statistiques de gaps lorsque la distribution de sauts est donnée par la loi de Laplace et réalisons des simulations numériques suggérant l'universalité de ces résultats<br>Predicting the occurrence of extreme events is a crucial issue in many contexts, ranging from meteorology to finance. For independent and identically distributed (i.i.d.) random variables, three universality classes were identified (Gumbel, Fréchet and Weibull) for the distribution of the maximum. While modelling disordered systems by i.i.d. random variables has been successful with Derrida's random energy model, this hypothesis fail for many physical systems which display strong correlations. In this thesis, we study three physically relevant models of strongly correlated random variables: trapped fermions, random matrices and random walks.In the first part, we show several exact mappings between the ground state of a trapped Fermi gas and ensembles of random matrix theory. The Fermi gas is inhomogeneous in the trapping potential and in particular there is a finite edge beyond which its density vanishes. Going beyond standard semi-classical techniques (such as local density approximation), we develop a precise description of the spatial statistics close to the edge. This description holds for a large universality class of hard edge potentials. We apply these results to compute the statistics of the position of the fermion the farthest away from the centre of the trap, the number of fermions in a given domain (full counting statistics) and the related bipartite entanglement entropy. Our analysis also provides solutions to open problems of extreme value statistics in random matrix theory. We obtain for instance a complete description of the fluctuations of the largest eigenvalue in the complex Ginibre ensemble.In the second part of the thesis, we study extreme value questions for random walks. We consider the gap statistics, which requires to take explicitly into account the discreteness of the process. This question cannot be solved using the convergence of the process to its continuous counterpart, the Brownian motion. We obtain explicit analytical results for the gap statistics of the walk with a Laplace distribution of jumps and provide numerical evidence suggesting the universality of these results
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Jamshidpey, Arash. "Population Dynamics in Random Environment, Random Walks on Symmetric Group, and Phylogeny Reconstruction." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/34623.
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This thesis concerns applications of some probabilistic tools to phylogeny reconstruction and population genetics. Modelling the evolution of species by continuous-time random walks on the signed permutation groups, we study the asymptotic medians of a set of random permutations sampled from simple random walks at time 0.25cn, for c> 0. Running k independent random walks all starting at identity, we prove that the medians approximate the ancestor (identity permutation) up to time 0.25n, while there exists a constant c>1 after which the medians loose credibility as an estimator. We study the median of a set of random permutations on the symmetric group endowed with different metrics. In particular, for a special metric of dissimilarity, called breakpoint, where the space is not geodesic, we find a large group of medians of random permutations using the concept of partial geodesics (or geodesic patches). Also, we study the Fleming-Viot process in random environment (FVRE) via martingale and duality methods. We develop the duality method to the case of time-dependent and quenched martingale problems. Using a family of dual processes we prove the convergence of the Moran processes in random environments to FVRE in Skorokhod topology. We also study the long-time behaviour of FVRE and prove the existence of equilibrium for the joint annealed-environment process and prove an ergodic theorem for the latter.
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Devore, Lucas Clay. "Random Walks with Elastic and Reflective Lower Boundaries." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/134.
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Carlsund, Anna. "Cover Times, Sign-dependent Random Walks, and Maxima." Doctoral thesis, KTH, Mathematics, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3624.
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Jones, Elinor Mair. "Large deviations of random walks and levy processes." Thesis, University of Manchester, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.491853.
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Katzenbeisser, Walter, and Wolfgang Panny. "The Maximal Height of Simple Random Walks Revisited." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 1998. http://epub.wu.ac.at/1126/1/document.pdf.
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In a recent paper Katzenbeisser and Panny (1996) derived distributional results for a number of so called simple random walk statistics defined on a simple random walk in the sense of Cox and Miller (1968) starting at zero and leading to state 1 after n steps, where 1 is arbitrary, but fix. In the present paper the random walk statistics Dn = the one-sided maximum deviation and Qn = the number of times where the maximum is achieved, are considered and distributional results are presented, when it is irrespective, where the random walk terminates after n steps. Thus, the results can be seen as generalizations of some well known results about (purely) binomial random walk, given e.g. in Revesz (1990). (author's abstract)<br>Series: Forschungsberichte / Institut für Statistik