Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Free Random Variables

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let $$\Gamma $$
 be a random $$d$$
 -regular graph on $$n$$
 vertices, and let $$\lambda $$
 be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon (Combinatorica 6(2), 83–96, 1986) that a random $$d$$
 -regular graph is “almost Ramanujan”, in the following sense: for every $$\varepsilon >0$$
 , $$\lambda <2\sqrt{d-1}+\varepsilon $$
 asymptotically almost surely. Friedman famously presented a proof of this conjecture in Friedman (Memoirs of the AMS 910, 2008). Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random $$d$$
 -regular graph satisfies $$\lambda <2\sqrt{d-1}+1$$
 a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For $$d$$
 even, a $$d$$
 -regular graph on $$n$$
 vertices is an $$n$$
 -covering space of a bouquet of $$d/2$$
 loops. More generally, fixing an arbitrary base graph $$\Omega $$
 , we study the spectrum of $$\Gamma $$
 , a random $$n$$
 -covering of $$\Omega $$
 . Let $$\lambda $$
 be the largest absolute value of a non-trivial eigenvalue of $$\Gamma $$
 . Extending Alon’s conjecture to this more general model, Friedman (Duke Math J 118(1),19–35, 2003) conjectured that for every $$\varepsilon >0,$$
 a.a.s. 
 $$\lambda <\rho +\varepsilon $$
 , where $$\rho $$
 is the spectral radius of the universal cover of $$\Omega $$
 . When $$\Omega $$
 is regular we get a bound of $$\rho +0.84$$
 , and for an arbitrary $$\Omega $$
 , we prove a nearly optimal upper bound of $$\sqrt{3}\rho $$
 . This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

Expansion of random graphs: new proofs, new results

We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group $G$, a word $w$ in the free group on $k$ generators induces a word map from $G^k$ to $G$. We say that $w$ is measure preserving with respect to $G$ if given uniform distribution on $G^k$, the image of this word map distributes uniformly on $G$. It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. Here we prove this conjecture. The main ingredients of the proof include random coverings of Stallings graphs, algebraic extensions of free groups, and Mobius inversions. Our methods yield the stronger result that a subgroup of $F_k$ is measure preserving if and only if it is a free factor. 
As an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of $F_k$ form a closed set in this topology.

Measure Preserving Words are Primitive

A word w 2 Fk, the free group on k generators, is called primitive if it belongs to some basis of Fk. Associated with w and a finite group G is the word map w : G×...×G ! G defined on the direct product of k copies of G. We call w measure preserving if given uniform measure on G×...×G, the image of this word map induces uniform measure on G (for every finite group G). It is easy to see that every primitive word is measure preserving, and several authors have conjectured that the two properties are, in fact, equivalent. Here we prove this conjecture. The main ingredients of the proof include random coverings of Stallings graphs, algebraic extensions of free groups and Mobius inversions. Our methods yield the stronger result that a subgroup of Fk is measure preserving iff it is a free factor. As an interesting corollary of this result we resolve a question on the profinite topology of free groups and show that the primitive elements of Fk form a closed set in this topology.

/pdf/measure-preserving-words-are-primitive-2voqhyhonk.pdf

Measure preserving words are primitive

We study here the spectra of random lifts of graphs Let G be a finite connected graph, and let the infinite tree T be its universal cover space If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman (Graphs Duke Math J 118 (2003), 19–35), in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O(λ 11-2 ρ1-2) Here we improve this bound to O(λ 11-3 ρ2-3) It is conjectured in (Friedman, Graphs Duke Math J 118 (2003) 19–35) that the statement holds with the bound ρ + o(1) which, if true, is tight by (Greenberg, PhD thesis, 1995) For G a bouquet with d-2 loops, our arguments yield a simple proof that almost every d-regular graph has second eigenvalue O(d2-3) For the bouquet, Friedman (2008) has famously proved the (nearly?) optimal bound of $ 2\sqrt{d - 1} + o(1) $

Central to our work is a new analysis of formal words Let w be a formal word in letters g 1±1 ,…,g k±1  The word map associated with w maps the permutations σ1,…,σk ∈ Sn to the permutation obtained by replacing for each i, every occurrence of gi in w by σi We investigate the random variable X w(n) that counts the fixed points in this permutation when the σi are selected uniformly at random The analysis of the expectation E(X w(n) ) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs imprimitive words A major ingredient of a our work is a second categorization of formal words with the same property We establish some results and make a few conjectures about the relation between the two categorizations These conjectures suggest a possible approach to (a slightly weaker version of) Friedman's conjecture

As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A Nica (Nica, Random Struct Algorithms 5 (1994), 703–730), which determines, for every fixed w, the limit distribution (as n →∞) of X w(n)  A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w = ud for some word u © 2010 Wiley Periodicals, Inc Random Struct Alg, 2010

/pdf/word-maps-and-spectra-of-random-graph-lifts-4uw5vrurtm.pdf

Word maps and spectra of random graph lifts

Let Fk be the free group on k generators. A word w ∈ Fk is called primitive if it belongs to some basis of Fk. We investigate two criteria for primitivity, and consider more generally subgroups of Fk which are free factors.

Doron Puder

Papers

Expansion of random graphs: new proofs, new results

Measure Preserving Words are Primitive

Measure preserving words are primitive

Word maps and spectra of random graph lifts

Primitive words, free factors and measure preservation