Independence ratio and random eigenvectors in transitive graphs

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $\lambda_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a $3$-regular transitive graph is at least $$q=\frac{1}{2}-\frac{3}{4\pi}\arccos\biggl(\frac{1-\lambda _{\min}}{4}\biggr).$$ The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least $q-o(1)$. We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.Comment: Published at http://dx.doi.org/10.1214/14-AOP952 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Tertiary-Quaternary subduction processes and related magmatism in the Alpine-Mediterranean region

During Tertiary to Quaternary times, convergence between Eurasia and Africa resulted in a variety of collisional orogens and different styles of subduction in the Alpine-Mediterranean region. Characteristic features of this area include arcuate orogenic belts and extensional basins, both of which can be explained by roll-back of subducted slabs and retreating subduction zones. After cessation of active subduction, slab detachment and post-collisional gravitational collapse of the overthickened lithosphere took place. This complex tectonic history was accompanied by the generation of a wide variety of magmas. Most of these magmas (e.g. low-K tholeiitic, calc-alkaline, shoshonitic and ultrapotassic types) have trace element and isotopic fingerprints that are commonly interpreted to reflect enrichment of their source regions by subduction-related fluids. Thus, they can be considered as ‘subduction-related’ magmas irrespective of their geodynamic relationships. Intraplate alkali basalts are also found in the region generally postdated the ‘subduction-related’ volcanism. These mantle-derived magmas have not been, or only slightly, influenced by subduction-related enrichment. This paper summarises the geodynamic setting of the Tertiary-Quaternary “subduction-related” magmatism in the different segments of the Alpine-Mediterranean region (Betic-Alboran-Rif province, Central Mediterranean, the Alps, Carpathian-Pannonian region, Dinarides and Hellenides, Aegean and Western Anatolia), and discusses the main characteristics and compositional variation of the magmatic rocks. Radiogenic and stable isotope data indicate the importance of continental crustal material in the genesis of these magmas. Interaction with crustal material probably occurred both in the upper mantle during subduction (‘source contamination’) and in the continental crust during ascent of mantle-derived magmas (either by mixing with crustal melts or by crustal contamination). The 87Sr/86Sr and 206Pb/204Pb isotope ratios indicate that an enriched mantle component, akin to the source of intraplate alkali mafic magmas along the Alpine foreland, played a key role in the petrogenesis of the ‘subduction-related’ magmas of the Alpine-Mediterranean region. This enriched mantle component could be related to mantle plumes or to long-term pollution (deflection of the central Atlantic plume and recycling of crustal material during subduction) of the shallow mantle beneath Europe since the late Mesozoic. In the first case, subduction processes could have had an influence in generating asthenospheric flow by deflecting nearby mantle plumes due to slab roll-back or slab break-off. In the second case, the variation in the chemical composition of the volcanic rocks in the Mediterranean region can be explained by “statistical sampling” of the strongly inhomogeneous mantle followed by variable degrees of crustal contamination

Correlation bound for distant parts of factor of IID processes

We study factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$, where $k$ denotes the distance of the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.Comment: 18 pages, 5 figure