Every planar graph with the Liouville property is amenable

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic functions proved by Benjamini and Schramm. As a corollary we obtain that every planar non-amenable graph admits Dirichlet harmonic functions

Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.Comment: 23 pages. Together with arXiv:1305.6260, this version replaces the old. The main results have been strengthened and an earlier error in the statement corrected. To appear in J. Theoret. Proba

Palm pairs and the general mass-transport principle

We consider a lcsc group G acting properly on a Borel space S and measurably on an underlying sigma-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form.Comment: 26 page

Critical percolation of free product of groups

In this article we study percolation on the Cayley graph of a free product of groups. The critical probability $p_c$ of a free product $G_1*G_2*...*G_n$ of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups $G_1,G_2,...,G_n$. For finite groups these equations are polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that $p_c$ for the Cayley graph of the modular group $\hbox{PSL}_2(\mathbb Z)$ (with the standard generators) is $.5199...$, the unique root of the polynomial $2p^5-6p^4+2p^3+4p^2-1$ in the interval $(0,1)$. In the case when groups $G_i$ can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of $G_1*G_2*...*G_n$ and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, $p_{\mathrm{exp}}$ of the free product is just the minimum of $p_{\mathrm{exp}}$ for the factors

Another derivation of the geometrical KPZ relations

We give a physicist's derivation of the geometrical (in the spirit of Duplantier-Sheffield) KPZ relations, via heat kernel methods. It gives a covariant way to define neighborhoods of fractals in 2d quantum gravity, and shows that these relations are in the realm of conformal field theory

What does the local structure of a planar graph tell us about its global structure?

The local k-neighborhood of a vertex v in an unweighted graph G = (V,E) with vertex set V and edge set E is the subgraph induced by all vertices of distance at most k from v. The rooted k-neighborhood of v is also called a k-disk around vertex v. If a graph has maximum degree bounded by a constant d, and k is also constant, the number of isomorphism classes of k-disks is constant as well. We can describe the local structure of a bounded-degree graph G by counting the number of isomorphic copies in G of each possible k-disk. We can summarize this information in form of a vector that has an entry for each isomorphism class of k-disks. The value of the entry is the number of isomorphic copies of the corresponding k-disk in G. We call this vector frequency vector of k-disks. If we only know this vector, what does it tell us about the structure of G? In this paper we will survey a series of papers in the area of Property Testing that leads to the following result (stated informally): There is a k = k(ε,d) such that for any planar graph G its local structure (described by the frequency vector of k-disks) determines G up to insertion and deletion of at most εd n edges (and relabelling of vertices)

Postural Changes in Blood Pressure Associated with Interactions between Candidate Genes for Chronic Respiratory Diseases and Exposure to Particulate Matter

BACKGROUND. Fine particulate matter [aerodynamic diameter ≤ 2.5 μm (PM2.5)] has been associated with autonomic dysregulation. OBJECTIVE. We hypothesized that PM2.5 influences postural changes in systolic blood pressure (ΔSBP) and in diastolic blood pressure (ΔDBP) and that this effect is modified by genes thought to be related to chronic lung disease. METHODS. We measured blood pressure in participants every 3-5 years. ΔSBP and ΔDBP were calculated as sitting minus standing SBP and DBP. We averaged PM2.5 over 48 hr before study visits and analyzed 202 single nucleotide polymorphisms (SNPs) in 25 genes. To address multiple comparisons, data were stratified into a split sample. In the discovery cohort, the effects of SNP x PM2.5 interactions on ΔSBP and ΔDBP were analyzed using mixed models with subject-specific random intercepts. We defined positive outcomes as p < 0.1 for the interaction; we analyzed only these SNPs in the replicate cohort and confirmed them if p < 0.025 with the same sign. Confirmed associations were analyzed within the full cohort in models adjusted for anthropometric and lifestyle factors. RESULTS. Nine hundred forty-five participants were included in our analysis. One interaction with rs9568232 in PHD finger protein 11 (PHF11) was associated with greater ΔDBP. Interactions with rs1144393 in matrix metalloprotease 1 (MMP1) and rs16930692, rs7955200, and rs10771283 in inositol 1,4,5-triphosphate receptor, type 2 (ITPR2) were associated with significantly greater ΔSBP. Because SNPs associated with ΔSBP in our analysis are in genes along the renin-angiotensin pathway, we then examined medications affecting that pathway and observed significant interactions for angiotensin receptor blockers but not angiotensin-converting enzyme inhibitors with PM2.5. CONCLUSIONS. PM2.5 influences blood pressure and autonomic function. This effect is modified by genes and drugs that also act along this pathway.National Institute of Environmental Health Sciences (T32 ES07069, ES0002, ES015172-01, ES014663, P01 ES09825); United States Environmental Protection Agency (R827353, R832416); National Institutes of Health/National Institute of Aging (AG027014); United States Department of Veterans Affairs; Massachusetts Veterans Epidemiology Research and Information Cente

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components

Trapping in the random conductance model

We consider random walks on $\Z^d$ among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time $2n$. We show that in the situations when the heat kernel exhibits subdiffusive decay --- which is known to occur in dimensions $d\ge4$ --- the walk gets trapped for a time of order $n$ in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.Comment: 21 pages, version to appear in J. Statist. Phy