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The looperased random walk and the uniform spanning tree on the fourdimensional discrete torus
, 2008
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On rough isometries of Poisson processes on the line
, 2007
"... Benjamini and Szegedy asked (independently) whether two independent Poisson point processes on the line with the same intensity are rough isometric (quasiisometric) a.s.. Szegedy conjectured the answer is positive. We prove that this question is equivalent to the following question, given two indep ..."
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Benjamini and Szegedy asked (independently) whether two independent Poisson point processes on the line with the same intensity are rough isometric (quasiisometric) a.s.. Szegedy conjectured the answer is positive. We prove that this question is equivalent to the following question, given two independent Bernoulli percolations A and B on the natural numbers with 0 adjoined to each of them, there exist constants and probability p> 0 such that for any n the first n points of A are rough isometric to an initial segment of B with these constants, with 0 mapping to 0 and with probability at least p. We then make some progress towards the conjecture by showing that if the constants of the rough isometry are allowed to grow with n then constants of order √ log n will suffice (this quantitative variant was introduced by Benjamini). It appears that this is the first result to improve upon the trivial construction which has constants of order log n. Furthermore, the rough isometry we construct is (weakly) monotone and we include a discussion of monotone rough isometries, their properties and an interesting lattice structure inherent in them. 1
LOOPERASED RANDOM WALK ON FINITE GRAPHS AND THE RAYLEIGH PROCESS
, 2007
"... Let (Gn) ∞ n=1 be a sequence of finite graphs, and let Yt be the length of a looperased random walk on Gn after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which Gn is the ddimensional torus of sizelength n for d≥4, the process (Yt) ∞ t=0, ..."
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Let (Gn) ∞ n=1 be a sequence of finite graphs, and let Yt be the length of a looperased random walk on Gn after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which Gn is the ddimensional torus of sizelength n for d≥4, the process (Yt) ∞ t=0, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used looperased random walks to show that the uniform spanning tree on large finite graphs converges to the Brownian continuum random tree of Aldous.
CONVERGENCE OF THE LENGTH OF THE LOOPERASED RANDOM WALK ON FINITE GRAPHS TO THE RAYLEIGH PROCESS
, 2006
"... Abstract. Let (Gn) ∞ n=1 be a sequence of finite graphs, and let Yt be the length of a looperased random walk on Gn after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which Gn is the ddimensional torus of sizelength n for d≥4, the process (Yt ..."
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Abstract. Let (Gn) ∞ n=1 be a sequence of finite graphs, and let Yt be the length of a looperased random walk on Gn after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which Gn is the ddimensional torus of sizelength n for d≥4, the process (Yt) ∞ t=0, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used looperased random walks to show that the uniform spanning tree on large finite graphs converges to the Brownian continuum random tree of Aldous. 1.
The Uniform Spanning Tree and related models
, 2009
"... The following type of question is characteristic of discrete probability: given a large finite set of objects, defined by some combinatorial property, what does a typical element ”look like”, that is, can we describe some of its statistical features. Simple as it is to state such questions, they can ..."
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The following type of question is characteristic of discrete probability: given a large finite set of objects, defined by some combinatorial property, what does a typical element ”look like”, that is, can we describe some of its statistical features. Simple as it is to state such questions, they can lead to surprisingly
1.1 Algebraic complexity
"... My main area of research is theoretical computer science, with a focus on algebraic complexity theory. I am also interested in other fields of mathematics, such as probability theory and combinatorics. ..."
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My main area of research is theoretical computer science, with a focus on algebraic complexity theory. I am also interested in other fields of mathematics, such as probability theory and combinatorics.
2.1 Encompassing a Point............................ 10
, 809
"... Lawler, Schramm and Werner showed that the scaling limit of the looperased random walk on Z 2 is SLE2. We consider scaling limits of the looperasure of random walks on other planar graphs (graphs embedded into C so that edges do not cross one another). We show that if the scaling limit of the rand ..."
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Lawler, Schramm and Werner showed that the scaling limit of the looperased random walk on Z 2 is SLE2. We consider scaling limits of the looperasure of random walks on other planar graphs (graphs embedded into C so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its looperasure is SLE2. Our main contribution is showing that for such graphs, the discrete Poisson kernel can be approximated by the continuous one. One example is the infinite component of supercritical percolation on Z 2. Berger and Biskup showed that the scaling limit of the random walk on this graph is planar Brownian motion. Our results imply that the scaling limit of the looperased
unknown title
, 803
"... Abstract We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the looperased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in ε = 4 − d. Up to two loop, the FR ..."
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Abstract We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the looperased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in ε = 4 − d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d = 4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d = 4, which can be used as a further test of the conjecture.
By
, 2007
"... The looperased random walk (LERW) was first studied in 1980 by Lawler as an attempt to analyze selfavoiding walk (SAW) which provides a model for the growth of a linear polymer in a good solvent. The selfavoiding walk is simply a path on a lattice that does not visit the same site more than once. ..."
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The looperased random walk (LERW) was first studied in 1980 by Lawler as an attempt to analyze selfavoiding walk (SAW) which provides a model for the growth of a linear polymer in a good solvent. The selfavoiding walk is simply a path on a lattice that does not visit the same site more than once. Proving things about the collection of all such paths is a formidable challenge to rigorous mathematical methods. Eventually, it was discovered that SAW and LERW are in different universality classes. LERW is a model for a random simple path with important applications in combinatorics, computer science and quantum field theory. This model has continued to receive attention in recent years, in part because of connections with uniform spanning trees. This report contains some of the most important results about looperased random walk and its scaling limit in several dimensions. Although there is an extensive body of work concerning LERW, we will rather give a summary of the key properties and results. The first two chapter of this report provide the preliminaries for the looperased random walk. In Chapter 1, the necessary background and history about looperased random walk and its scaling limit is presented. Chapter 2 introduces i some aspects and definition of LERW for d ≥ 3. In Chapter 3, it is shown that LERW converges to Brownian motion for d ≥ 5. In Chapter 4, the same result as in Chapter 3 holds. It is shown that LERW for d = 4 converges to Brownian motion, but a logarithmic correction to the scale is needed. A promising paper by Kozma is introduced in Chapter 5 which shows that the scaling limit of LERW in d = 3 exists and is invariant under rotations and dilatations. Chapter 6 presents some important concepts and facts from complex analysis including the Loewner equation, and gives an introduction to the stochastic Loewner evolution (SLE). At the end of this chapter, it is shown that the scaling limit of LERW for d = 2 is equal to the radial SLE2 path. The final chapter reviews Wilson’s algorithm which generates uniform spanning trees (UST) using LERW. ii