# Schedule for: 16w5085 - Random Structures in High Dimensions

Arriving in Oaxaca, Mexico on Sunday, June 26 and departing Friday July 1, 2016

Sunday, June 26 | |
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14:00 - 23:59 | Check-in begins (Front desk at your assigned hotel) |

19:30 - 22:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

20:30 - 21:30 |
Informal gathering ↓ A welcome drink will be served at the hotel. (Hotel Hacienda Los Laureles) |

Monday, June 27 | |
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07:30 - 08:45 | Breakfast (Restaurant at your assigned hotel) |

08:45 - 09:00 | Introduction and Welcome (Conference Room San Felipe) |

09:00 - 09:50 |
Geoffrey Grimmett: An algebraic approach to counting self-avoiding walks ↓ 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. (Conference Room San Felipe) |

10:00 - 10:50 |
Tony Guttmann: Random and self-avoiding walks subject to tension and compression ↓ 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. (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe)) |

11:30 - 12:20 |
Omer Angel: Random walks on half planar maps ↓ 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. (Conference Room San Felipe) |

12:30 - 12:40 | Group Photo (Hotel Hacienda Los Laureles) |

13:30 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:00 - 15:50 |
Tom Kennedy: The first order correction to the exit distribution for some random walks ↓ 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. (Conference Room San Felipe) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:20 |
Balint Toth: Central limit theorem for random walks in doubly stochastic random environment ↓ 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. (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Tuesday, June 28 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:50 |
Remco Van der Hofstad: Progress in high-dimensional percolation ↓ 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. (Conference Room San Felipe) |

10:00 - 10:50 |
Akira Sakai: The lace expansion for the nearest-neighbor models on the BCC lattice ↓ 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. (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe) |

11:30 - 12:20 |
Markus Heydenreich: The backbone scaling limit of high-dimensional incipient infinite cluster ↓ 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. (Conference Room San Felipe) |

13:30 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:00 - 15:50 |
Hao Shen: A stochastic PDE with U(1) gauge symmetry ↓ 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. (Conference Room San Felipe) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:20 |
Roman Kotecky: Emergence of long cycles for random interchange process on hypercubes ↓ 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.) (Conference Room San Felipe) |

19:00 - 21:00 | Dinner + Reception (Restaurant Hotel Hacienda Los Laureles) |

Wednesday, June 29 | |
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07:30 - 08:30 | Breakfast (Restaurant at your assigned hotel) |

08:30 - 12:30 |
Tour to Monte Alban ↓ 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 (--) |

13:00 - 14:30 | Lunch: Note earlier time than usual (Restaurant Hotel Hacienda Los Laureles) |

14:30 - 15:20 |
Federico Camia: Random loops in statistical mechanics and Euclidean field theory ↓ 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.) (Conference Room San Felipe) |

15:30 - 16:20 |
Antal Jarai: Inequalities for critical exponents in d-dimensional sandpiles ↓ 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.) (Conference Room San Felipe) |

16:30 - 17:00 | Coffee Break (Conference Room San Felipe) |

17:00 - 17:50 |
Mark Holmes: Weak convergence of historical processes ↓ 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. (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Thursday, June 30 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:50 |
Greg Lawler: Uniform Spanning Forests and Bi-Laplacian Gaussian Field ↓ 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. (Conference Room San Felipe) |

10:00 - 10:50 |
Charles Newman: Minimal Spanning Tree on a Slab ↓ 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. (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe) |

11:00 - 11:30 |
Marek Biskup: Structure of extreme local maxima of 2D Discrete Gaussian Free Field ↓ 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. (Conference Room San Felipe) |

13:30 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

15:00 - 15:50 |
Takashi Kumagai: Time changes of stochastic processes associated with resistance forms ↓ 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). (Conference Room San Felipe) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:20 |
Geronimo Uribe Bravo: Affine processes and multiparameter time changes ↓ 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. (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Friday, July 1 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 09:50 |
Christine Soteros: The embedding complexity of closed curves (polygons) and surfaces (closed 2-manifolds) in tubes in $Z^d$ ↓ 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. (Conference Room San Felipe) |

10:00 - 10:50 |
Jesse Goodman: Long and short paths in first passage percolation on complete graphs. ↓ 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. (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe) |

11:30 - 12:20 |
Roland Bauerschmidt: The renormalisation group ↓ 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. (Conference Room San Felipe) |

13:30 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |