Titlebook: Coarse Geometry and Randomness; école d’été de Proba Itai Benjamini Book 2013 Springer International Publishing Switzerland 2013 82B43,82B4

多產(chǎn)子

Elisabeth Lewis Corley,Joseph MegelIn this section we present an example of a bounded degree graph with a positive Cheeger constant (i.e. nonamenable graph) which is Liouville, that is, it admits no non constant bounded harmonic functions. This example shows that the theorem proved in Sect. 12 cannot be extended to general graphs.

里程碑

Spinal-Fusion

On the Structure of Vertex Transitive Graphs,This short section contains several facts and open problems regarding vertex transitive graphs, starting with the following theorem from [BS92] which refines an earlier result of Aldous.

專心

Percolation on Graphs,In this section we introduce and discuss some basic properties of percolation, a fundamental random process on graphs. For background on percolation see [Gri99].

紳士

Random Planar Geometry,What is a typical random surface? This question has arisen in the theory of two-dimensional quantum gravity where discrete triangulations have been considered as a discretization of a random continuum Riemann surface. As we will see the typical random surface has a geometry which is very different from the one of the Euclidean plane.

heckle

Growth and Isoperimetric Profile of Planar Graphs,In this section we review a joint work with Panos Papasoglu, see [.], in which the following is proved:

點(diǎn)燃

Critical Percolation on Non-Amenable Groups,For a given graph ., let . (or just .(.) when . is clear from the context). From the definition of .. we know that .(.) = 0 for any . < .., and .(.) > 0 whenever . > ... A major and natural question that arises is: Does .(..)= 0 or .(..) > 0?.

向外

Percolation on Expanders,This section is devoted to percolation on finite graphs. More precisely we will try to understand percolation on a sequence of finite graphs, whose number of vertices tends to infinity. Detailed proofs of the material appearing in this section and additional extensions can be found at [ABS04].

Ancestor

Harmonic Functions on Graphs,The main goal of this section is to present the Kaimanovich-Vershik entropic criterion for the existence of harmonic function on Cayley graphs. Note that this section requires more background in probability compared to previous sections. We begin with some definition and simple facts.

禁令