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Abstract: The rotor router is a derandomized model for a random walk. Each vertex sends visiting particles to its neighbours in a periodic manner. In some ways this system behaves similarly to simple random walks, while in others it differs. I will describe the behaviour of this system on trees, exhibiting (among other properties) a first order transition for transience.
This is joint work with Ander Holroyd.
Abstract: Consider the permutation-valued process obtained by performing random transpositions at rate 1, started from the identity permutation. In joint work with Rick Durrett it was observed that this can be coupled with an Erdös-Rényi random graph in such a way that the cycles of the permutation are at any given time subsets of the connected components of the graph. In 2005 Oded Schramm proved a beautiful result, conjectured by David Aldous, that the distribution of cycle sizes within the giant component has Poisson-Dirichlet asymptotics, for any supercritical time. In particular, giant cycles emerge at the same time as a giant component for the Erdös-Rényi random graph. I will present a very short and non-computational proof of this fact, which can also be generalised to other models. I will discuss related conjectures and ideas, and (time-permitting) explain briefly how this is connected to a problem on mixing times of random walks (this last bit being joint work with Oded Schramm and Ofer Zeitouni).
Abstract: We present a new version of the contraction method for the asymptotic behavior of solutions of convolution equations of the type appearing for instance in self-avoiding random walks. The method appeared first in a paper with Christine Ritzmann. The new version is simpler, and probably more flexible.
This is work in progress with Christine Ritzmann and Felix Rubin.
Abstract: In invasion percolation, the edges of a graph are given i.i.d. uniform edge weights, and an infinite cluster is grown by inductively adding the boundary edge of minimal weight. By considering the edges whose weight is greater than all subsequent accepted weights, the invasion cluster is divided into a chain of ponds linked by outlets. Working on a regular tree, we show that the sizes of the ponds grow exponentially, in a law of large numbers, central limit theorem and large deviations sense, and give asymptotics for the size of a fixed pond. We compare these results with known results in Z2, and explore why they can be expected to hold in more general graphs.
Abstract: Condition supercritical percolation on the square lattice so that the origin is enclosed by a dual circuit whose interior traps an area of n2. The Wulff problem concerns the shape of the circuit. We study the circuit's fluctuation. A well-known measure of this fluctuation is maximum local roughness (MLR), which is the greatest distance from a point on the circuit to the boundary of circuit's convex hull. Another is maximum facet length (MFL), the length of the longest line segment of which this convex hull is comprised. In a forthcoming article, I prove that, for various models including supercritical percolation, under the conditioned measure, MLR = Θ(n1/3 (log n)2/3) and MFL = Θ(n2/3 (log n)1/3). The essential hypotheses that are needed for these results are translation invariance, exponential decay of dual connectivity, bounded energy and the ratio-weak-mixing assumption (which I'll discuss if there is enough time). An important tool is a result establishing the profusion of regeneration sites in the circuit. The talk will focus on deriving the main results with this tool.
Abstract: I will talk on a model, q-lattice animals. The model has one parameter q. It coincides with ordinary lattice animals when q=1, and coincides with ordinary percolation when q=1-p (p is the bond occupation probability of percolation). In this sense, the model tries to interpolate between lattice animals and percolation (we consider q between 1-p and 1). We analyze this model using lace expansion and correlation inequality of van den Berg-Kesten type. Basic results presented include: existence of critical phenomena, and mean-field type critical behavior in “high” dimensions. Although there is no rigorous proof, our analysis suggests that the upper critical dimension of the model is eight, same as lattice animals.
This is based on a joint work with Keita Tamenaga.
Abstract: Statistical-mechanical models on Zd exhibit mean-field behaviour in dimension d>dc, where dc is the critical dimension. It is believed -and partially proved- that dc does not change as long as the coupling function has finite variance. If, on the other hand, the coupling function has infinite variance, then the value of dc decreases; in this case we call the model ‘long-range’. I aim to present results for these long-range models, indicate the technical difficulties in obtaining these results, and explain how to resolve them. I use self-avoiding walk as primary example, but shall also present recent results for percolation, oriented percolation, and the Ising model.
Abstract: We discuss recent progress on the (near-)critical behavior of percolation.
We study both finite range models, in which the displacement along a bond is uniformly bounded, and certain long-range models, in which the probability of a bond between x and y decays as a power of |x-y|, where the power is such that the expected number of bonds per vertex is bounded. We prove mean-field behavior for d>dc, either when the dimension is sufficiently large or the model is sufficiently spread-out, where we heuristically identify the upper critical dimension in terms of the number of spatial moments of the bond occupation probability function.
We further prove that the IIC exists in this generality, and study random walks on the IIC. Interestingly, the Alexander-Orbach conjecture holds true for all these models, while, assuming that the one-arm exponent exists and takes on its mean-field value, we see different behavior of exit times of Euclidean balls for the long-range model compared to the finite-range one.
This talk is based on joint work with Markus Heydenreich, Tim Hulshof and Akira Sakai.
Abstract: Suppose that at each site in the d-dimensional square lattice, a single cookie is present with probability p, independently of other sites. Now run a random walk in this random cookie environment such that the walker has a drift to the right when it eats a cookie (“excitement”), and a drift to the left (“tide”) otherwise. We prove that in high dimensions the first coordinate of the speed of the random walk is continuous and strictly increasing in the right drift parameter and in the percolation parameter p. As a consequence we get that in high dimensions, for every tide parameter (in [0,1]), one can tune the other parameters (also in [0,1]) so that the resulting speed of the random walk is zero.
Abstract: The Abelian sandpile was proposed in the physics literature as a simple model for self-organized criticality. The model describes the stochastic evolution of ‘grain’-configurations that consist of two types of activity: ‘grains’ are added at discrete time steps, and after each addition, the configuration is relaxed according to a simple rule. An interesting feature of the model is that relaxation turns out to be a very non-local process. In this talk, I will give an introduction to the model and discuss the construction of an infinite version on transitive graphs under certain conditions.
Abstract: In this talk, we summarize recent work on the behavior of random walks on random media. We estimate the following quantities and observe 'anomalous' behavior of the random walks: (i) Average of the exit time from the ball of radius R, (ii) Heat kernel and spectral dimension. Examples of the random media include critical percolation clusters, random walk traces and critical random graphs.
Abstract: We study the evolution of the susceptibility in the subcritical random graph $G(n,p)$ as $n$ tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal.
This is joint work with Svante Janson.
Abstract: We study the probability that the origin is connected to the sphere of radius r (an arm event) in critical percolation in high dimensions, namely when the dimension d is large enough or when d > 6 and the lattice is sufficiently spread out. We prove that this probability decays like 1/r2. Furthermore, we show that the probability of having k disjoint arms to distance r emanating from the vicinity of the origin is 1/r2k.
This is joint work with Gady Kozma.
Abstract: Consider random walk and various statistical-mechanical models (such as percolation and the Ising model) that are defined by the Zd-symmetric 2-body interaction D. Suppose that the function D(x) decays as |x|-d-a with a>0, so that the variance diverges if a < 2 in particular. We show that, for random walk with d > min{a,2} and for the other models with d bigger than the model-dependent upper-critical dimension (e.g., = 2min{a,2} for the Ising model and self-avoiding walk), the critical two-point function (= Green's function) decays as |x|min{a,2}-d. This for a > 2 reproves the results by Hara (2008) and Hara, van der Hofstad and Slade (2003).
The talk is based on joint work in progress with L.-C. Chen.
Abstract: Let x and y be points chosen uniformly at random from the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order n2 (log n)1/6, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on the four-dimensional discrete torus is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for d ≥ 5, in combination with results of Lawler concerning intersections of four-dimensional random walks.
Abstract: We discuss work in progress with David Brydges. Our first goal is to prove |x|-2 decay of the critical two-point function for certain models of self-avoiding walks on Z4. The walk two-point function is identified as the two-point function of a supersymmetric field theory with quartic self-interaction, and the field theory is then analysed using renormalisation group methods.
Abstract: We will discuss the trajectory of a simple random walk on a random d-regular graph of n vertices, with d > 2. More precisely, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. We show that this so-called `vacant set' exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* ∈ (0,∞) such that, if u < u*, with high probability as n grows, the largest component of the vacant set has a volume of order n, and if u > u*, then it has a volume of order log n. The critical value u* coincides with the critical intensity of a random interlacement process on an infinite d-regular tree.
This talk is based on a joint work with Jiří Černý and David Windisch, which can be found here.
Abstract: We investigate the asymptotic behaviour of the self-repelling Brownian polymer model initiated by Durrett and Rogers in 1992. This is a continuous space-time analogue of the so-called myopic (or true) self-repelling random walk. We identify a natural stationary (in time) and ergodic distribution of the environment (essentially, gradient of smeared out occupation time measure of the process), as seen from the moving particle. We prove that in three and more dimensions, in this stationary (and ergodic) regime, the displacement of the moving particle scales diffusively and its finite dimensional distributions converge to those of a Wiener process. The main tool is the nonreversible version of the Kipnis-Varadhan-type CLT for additive functionals of ergodic Markov processes and a relaxation of Varadhan's Sector Condition.
This is joint work with Illés Horváth and Bálint Vetô (Budapest).