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

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

Thermodynamics and Statistical Mechanics. By P.T. Landsberg (Oxford University Press: Oxford, 1979.) £9.75.

/pdf/thermodynamics-and-statistical-mechanics-2fmyb73mkl.pdf

Thermodynamics and statistical mechanics

These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the Hammersley-Welsh bound on the number of self-avoiding walks and its consequences for the growth rates of bridges and self-avoiding poly- gons. A detailed proof that the connective constant on the hexagonal lattice equals p 2 + p 2 is then provided. The lace expansion for self-avoiding walks is described, and its use in understanding the critical behaviour in dimensions d > 4 is discussed. Functional integral representations of the self-avoiding walk model are discussed and developed, and their use in a renormalisation group analysis in dimension 4 is sketched. Problems and solutions from tutorials are included.

/pdf/lectures-on-self-avoiding-walks-4b0cmlal0n.pdf

Lectures on self-avoiding walks

We consider random d-regular graphs on N vertices, with degree d at least (log N)4. We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by Wigner's semicircle law, down to the optimal scale given by the typical eigenvalue spacing (up to a logarithmic correction). Aside from well-known consequences for the local eigenvalue distribution, this result implies the complete (isotropic) delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity.© 2017 Wiley Periodicals, Inc.

Local Semicircle Law for Random Regular Graphs

We prove that the susceptibility of the continuous-time weakly self-avoiding walk on $${\mathbb{Z}^d}$$
 , in the critical dimension d = 4, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent $${\frac{1}{4}}$$
 for the logarithm. The susceptibility has been well understood previously for dimensions d ≥ 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.

/pdf/logarithmic-correction-for-the-susceptibility-of-the-4-76touhg3cp.pdf

Logarithmic Correction for the Susceptibility of the 4-Dimensional Weakly Self-Avoiding Walk: A Renormalisation Group Analysis

For the two-dimensional one-component Coulomb plasma, we derive an asymptotic expansion of the free energy up to order $N$, the number of particles of the gas, with an effective error bound $N^{1-\kappa}$ for some constant $\kappa > 0$. This expansion is based on approximating the Coulomb gas by a quasi-free Yukawa gas. Further, we prove that the fluctuations of the linear statistics are given by a Gaussian free field at any positive temperature. Our proof of this central limit theorem uses a loop equation for the Coulomb gas, the free energy asymptotics, and rigidity bounds on the local density fluctuations of the Coulomb gas, which we obtained in a previous paper.

The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem

We prove $${|x|^{-2}}$$
 decay of the critical two-point function for the continuous-time weakly self-avoiding walk on $${\mathbb{Z}^{d}}$$
 , in the upper critical dimension d = 4. This is a statement that the critical exponent $${\eta}$$
 exists and is equal to zero. Results of this nature have been proved previously for dimensions $${d \ge 5}$$
 using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.

/pdf/critical-two-point-function-of-the-4-dimensional-weakly-self-1bu41r0tdd.pdf

Roland Bauerschmidt

Papers

Lectures on self-avoiding walks

Local Semicircle Law for Random Regular Graphs

Logarithmic Correction for the Susceptibility of the 4-Dimensional Weakly Self-Avoiding Walk: A Renormalisation Group Analysis

The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem

Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk