Schedule for: 16w5085 - Random Structures in High Dimensions

A welcome drink will be served at the hotel.

What can be said about the connective constant of a graph? The celebrated work of Hara and Slade on self-avoiding walks is directed largely at the effect of dimension. This talk is devoted to recent work (with Zhongyang Li) based more on considerations of algebra and combinatorics than of geometry or analysis. We will report on inequalities for connective constants, and will present a locality theorem. Cayley graphs of finitely generated groups are examples of special interest, and we shall discuss the relevance of amenability. The Cayley graph of the Grigorchuk group (which has a compressed exponential growth rate) poses special challenges.

In recent years there have been important experiments involving the pulling of polymers from a wall. These are carried out with atomic force microscopes and other devices to determine properties of polymers, including biological polymers such as DNA. We have studied a simple model of this system, comprising two-dimensional self-avoiding walks, anchored to a wall at one end and then pulled from the wall at the other end. In addition, we allow for binding of monomers in contact with the wall. The geometry is shown in the following figure: There are two parameters in the model, the strength of the interaction of monomers with the surface (wall), and the force, normal to the wall, pulling the polymer. We have constructed (numerically) the complete phase diagram, and can prove the locus of certain phase boundaries in that phase diagram, and also the order of certain phase transitions as the phase boundaries are crossed. A schematic of the phase diagram is shown below. Most earlier work focussed on simpler models of random, directed and partially directed walk models. There has been little numerical work on the more realistic SAW model. A recent rigorous treatment by van Rensburg and Whittington established the existence of a phase boundary between an adsorbed phase and a ballistic phase when the force is applied normal to the surface. We give the first proof that this phase transition is first-order. As well as finding the phase boundary very precisely, we also estimate various critical points and exponents to high precision, or, in some cases exactly (conjecturally). We use exact enumeration and series analysis techniques to identify this phase boundary for SAWs on the square lattice. Our results are derived from a combination of three ingredients: (i) Rigorous results. (ii) Faster algorithms giving extended series data. (iii) New numerical techniques to extract information from the data. A second calculation considers polymers squeezed towards a surface by a second wall parallel to the surface wall. In this problem we ignore the interaction between surface monomers and the wall. We find, remarkably, that in this geometry there arises an unexpected stretched exponential term in the asymptotic expression for the number of configurations. We show explicitly that this can occur even if one uses simple random walks as the polymer model, rather than the more realistic self-avoiding walks. Aspects of this work have been carried out with Nick Beaton, Iwan Jensen, Greg Lawler and Stu Whittington.

We study random walks on random planar maps with the half plane topology. In the parabolic case we prove recurrence, and in the hyperbolic case positive speed away from the boundary. Joint works with Gourab Ray and Asaf Nachmias.

We consider three random walk models on several two-dimensional lattices - the usual nearest neighbor random walk, the nearest neighbor random walk without backtracking and the smart kinetic walk (a type of self-avoiding walk). For all these models the distribution of the point where the walk exits a simply connected domain in the plane converges weakly to the harmonic measure on the boundary as the lattice spacing goes to zero. We study the first order correction, i.e., the limit of the difference divided by the lattice spacing. Monte Carlo simulations lead us to conjecture that this measure has density $c f(z)$ where the function $f(z)$ only depends on the domain and the constant $c$ only depends on the model and the lattice. So there is a form of universality for this first order correction. For a particular random walk model with continuously distributed steps we can prove the conjecture.

We prove a CLT under diffusive scaling for the displacement of a random walk on $Z^d$ in stationary and ergodic doubly stochastic random environment, under the $H_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. Joint work with Gady Kozma.

A major breakthrough in percolation was the 1990 result by Hara and Slade proving mean-field behavior of percolation in high-dimensions, showing that at criticality there is no percolation and identifying several percolation critical exponents. The main technique used is the lace expansion, a perturbation technique that allowed Hara and Slade to compare percolation paths to random walks based on the idea that faraway pieces of percolation paths are almost independent in high dimensions. In this talk, we describe these seminal 1990 results, as well as a number of novel results for high-dimensional percolation that have been derived since and that build on the shoulders of these giants. Time permitting, I intend to highlight the following topics: (1) Critical percolation on the tree and critical branching random walk to fix ideas and to obtain insight in the kind of results that can be proved in high-dimensional percolation; (2) The recent computer-assisted proof, with Robert Fitzner, that identifies the critical behavior of nearest-neighbor percolation above 11 dimensions using the so-called Non-Backtracking Lace Expansion (NoBLE) that builds on the unpublished work by Hara and Slade proving mean-field behavior above 18 dimension; (3) The identification of arm exponents in high-dimensional percolation in two works by Asaf Nachmias and Gady Kozma, using a clever and novel difference inequality argument, and its implications for the incipient infinite cluster and random walks on them; (4) Super-process limits of large critical percolation clusters and the incipient infinite cluster. We assume no prior knowledge about percolation.

The lace expansion was initiated by Brydges and Spencer in 1985. Since then, it has been a powerful tool to rigorously prove mean-field (MF) results for various statistical-mechanical models in high dimensions. For example, Hara and Slade succeeded in showing the MF behavior for nearest-neighbor self-avoiding walk on $\mathbb{Z}^{d \geq 5}$. Recently, van der Hofstad and Fitzner managed to prove the MF results for nearest-neighbor percolation on $\mathbb{Z}^{d \geq 11}$ by using the so-called NoBLE (Non-Backtracking Lace Expansion). For sufficiently spread-out percolation, however, the MF results are known to hold for all $d$ above the percolation upper-critical dimension 6, without using the NoBLE. Our goal is to show the MF behavior for the nearest-neighbor models, for all $d$ above the model-dependent upper-critical dimension, in a simpler and more accessible way. To achieve this goal, we consider the nearest-neighbor models on the $d$-dimensional BCC (Body-Centered Cube) lattice. (This is just like working on the triangular or hexagonal lattice instead of the square lattice in two dimensions.) Because of the nice properties of the BCC lattice, we can simplify the analysis and more easily prove the mean-field results for $d$ close to the corresponding upper-critical dimension, currently $d \geq 6$ for self-avoiding walk and $d \geq 10$ for percolation. This talk is based on joint work with Lung-Chi Chen, Satoshi Handa and Yoshinori Kamijima for self-avoiding walk, and on joint work with the above three colleagues and Markus Heydenreich for percolation.

By incipient infinite cluster we denote critical percolation conditioned on the cluster of the origin to be infinite. This conditional measure, which is achieved as a suitable limiting scheme, is singular w.r.t. (ordinary) critical percolation. We define the backbone $B$ as the set of those vertices $x$, for which $\{x connected to the origin\}$ and $\{x connected to infinity\}$ occur disjointly. Our main result is that $B$, properly rescaled, converges to a Brownian motion path in sufficiently high dimension. One interpretation of this result is that spatial dependencies of the backbone vanish in the scaling limit. The result is achieved through a lace expansion of events of the form $P(x and y are connected and there are m pivotal bonds between x and y)$. This extends the original Hara-Slade expansion for percolation and gives rise to some new diagrammatic estimates. The talk is based on joint work with R. van der Hofstad, T. Hulshof, and G. Miermont.

We consider the problem of constructing the Langevin dynamic of a lattice U(1) gauge theory in two spatial dimensions. The model consists of a vector field and a scalar field interacting on a 2D lattice, and we study the continuum limit of its natural dynamic for short time. This dynamic is not a priorly parabolic, but we can turn it into a parabolic system with a time-dependent family of U(1) gauge transformations; we then apply Hairer's theory of regularity structures to the parabolic equations.

Motivated by phase transitions in quantum spin models, we study random permutations of vertices (induced by products of uniform independent random transpositions on edges) in the case of high-dimensional hypercubes. We establish the existence of a transition accompanied by emergence of cycles of diverging lengths. (Joint work with Piotr Miłoś and Daniel Ueltschi.)

Go to Hotel Hacienda los Laureles at 8:30 am. to board bus or buses. Price: $300.00 Mexican Pesos per person and the payment will be directly with the company when staff of Turismo el Convento arrive to the hotel and you can pay in cash or credit card. This price includes: Passenger insurance Certified Guide Licensed Driver Bottle water Admission Round transportation from the hotel

Kurt Symanzik and others recognized since the 1960s that the study of the (lattice) fields associated with certain now-classical models of statistical mechanics and Euclidean field theory leads naturally to consider random loop models. These loop models are interesting in their own right, and have recently been the focus of renewed attention. In this talk, I will brielfy introduce the Symanzik polymer representation of Euclidean field theory, and use it as a starting point to define new random fields with interesting properties, thus completing the loop. (Partly based on joint work with Marcin Lis, and with Alberto Gandolfi and Matthew Kleban.)

We prove rigorous upper and lower bounds for some critical exponents in Abelian sandpiles in dimensions d >= 2: these concern the toppling probability, the avalanche radius and the avalanche cluster size. In d > 4, we establish the mean-field exponent for the radius apart from a logarithmic factor. (Joint work with Jack Hanson and Sandeep Bhupatiraju.)

Under the usual formulation of weak convergence of branching particle systems to super-Brownian motion, the state of the process at a fixed time is a measure on $R^d$. As a result, the weak convergence statement does not encode the genealogy present in e.g. the voter model and lattice trees. In joint work-in-progress with Ed Perkins we consider weak convergence of the so-called historical processes (where the state of the process at a fixed time is a measure on genealogical paths in $R^d$) for these models.

We construct the bi-Laplacian Gaussian field on $R^4$ as a scaling limit of a field in $Z^4$ constructed using a wired spanning forest. The proof requires improving the known results about four dimensional loop-erased random walk. There are similar (and somewhat easier) results for higher dimensions. This is joint work with Xin Sun and Wei Wu.

In joint work with Vincent Tassion and Wei Wu, we have studied the minimal spanning forests on the nearest neighbor slabs with vertex sets such as $Z^2 \times \{0,1,...k\}^{d-2}$. For $Z^d$ itself, it is known that the forest is a single tree for $d = (1 and) 2$ but nothing is known for $d>2$ except it is conjectured that the $d=2$ behavior continues until some $d_c$ (probably 6 or 8) above which there are infinitely many trees in the forest. Our result is that, in slabs, there is only a single tree. The work is related to that of Duminil-Copin, Sidoravicius and Tassion who proved that there is no infinite cluster in critical Bernoulli percolation in such slabs. We also get new results for that critical percolation setting.

I will attempt to explain the recent progress in our understanding of the shape of the large peaks in a typical sample of the two-dimensional Discrete Gaussian Free Field over a large but finite domain in the square lattice. As a consequence, I will give ideas from the construction of the supercritical Liouville Quantum Gravity measure, as well as a proof of the so called freezing phenomenon associated with this process. Based on joint work with Oren Louidor.

In recent years, interest in time changes of stochastic processes according to irregular measures has arisen from various sources. Fundamental examples of such time-changed processes include the so-called Fontes-Isopi-Newman (FIN) diffusion, the introduction of which was motivated by the study of localization and aging properties of physical spin systems, and the two-dimensional Liouville Brownian motion, which is the diffusion naturally associated with planar Liouville quantum gravity. The FIN diffusion is known to be the scaling limit of the one-dimensional Bouchaud trap model, and the two-dimensional Liouville Brownian motion is conjectured to be the scaling limit of simple random walk on random planar maps. We will provide a general framework for studying such time changed processes and their discrete approximations in the case when the underlying stochastic process is strongly recurrent, in the sense that it can be described by a resistance form, as introduced by J. Kigami. In particular, this includes the case of Brownian motion on tree-like spaces and low-dimensional self-similar fractals. If time permits, we also discuss heat kernel estimates for the relevant time-changed processes. This is a joint work with D. Croydon (Warwick) and B.M. Hambly (Oxford).

We present a time change construction of affine processes on $R_+^m \times R^n$. These processes were systematically studied in (Duffie, Filipovi\'c and Schachermayer, 2003), since they contain interesting classes of processes such as L\'evy processes, continuous branching processes with immigration, and processes of the Ornstein-Uhlenbeck type. The construction is based on a (basically) continuous functional of a multidimensional L\'evy process, which implies that limit theorems for L\'evy processes (both almost sure and in distribution) can be inherited to affine processes. The construction can be interpreted as a multiparameter time change scheme or as a (random) ordinary differential equation driven by discontinuous functions. In particular, we propose approximation schemes for affine processes based on the Euler method for solving the associated discontinuous ODEs, which are shown to converge.

There has been much interest in the embedding complexity of curves and surfaces in lattices, including the differences in exponential growth rates for embeddings subject to different topological constraints. This includes questions about knotting and linking for simple closed curves and graphs in $Z^3$, to model the entanglement complexity of flexible polymer molecules, and questions about embeddings of random surfaces in $Z^d$ and the effects of genus and the number of boundary components on their exponential growth rate. Despite much study, there are a number of conjectures about the complexity of these embeddings that remain unproved. Restricting the geometry by confining the curves or surfaces to a tube (or prism) in $Z^d$, however, makes the system quasi-one-dimensional and potentially more tractable. Restricting to a tube is also of interest for exploring the effects of geometrical constraints, such as when modelling polymers under confinement. In this talk, I will review recent results about the topological complexity of polygons and closed 2-manifolds embedded in tubes in $Z^d$. For the case of polygons in a $2 x 1 x \infty$ sublattice of $Z^3$, knot theory results of Shimokawa and Ishihara lead to a proof that polygons with fixed knot type have the same exponential growth rate as unknotted polygons. For closed 2-manifolds in a tube in $Z^d$, if the embeddings are orientable with fixed genus $d \neq 4$, we prove with Sumners and Whittington that the exponential growth rate is independent of the genus and obtain a similar result for the non-orientable case when $d>4$. More generally, transfer matrix arguments can be used to prove pattern theorems and we establish, for example, that: the typical genus of a closed 2-manifold embedding increases with the size of the manifold; orientable manifolds are exponentially rare when $d>4$; and for $d=4$ all except exponentially few 2-manifolds contain a local knotted (4,2)-ball pair.

In a connected graph with random positive edge weights, pairs of vertices can be joined to obtain an a.s. unique path of minimal total weight. It is natural to ask about the typical total weight of such optimal paths, and about the number of edges they contain. To this end we consider the first passage percolation exploration process, which tracks the flow of fluid travelling across edges at unit speed and therefore discovers optimal paths in order of length. On the complete graph, adding exponential edge weights results in optimal paths with logarithmically many edges - the same "small world" path lengths that are typical of many random graphs. However, by changing the edge weight distribution, we can obtain paths that are asymptotically shorter or longer than logarithmic. This talk will explain how tail properties of the edge weight distribution can be translated quite precisely into scaling properties of optimal paths.

The renormalisation group has been Gordon Slade's main focus of research for the past decade. I will explain some of the ideas and results.