Local semicircle law for random regular graphs

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

Three-dimensional tricritical spins and polymers

We consider two intimately related statistical mechanical problems on $\mathbb{Z}^3$: (i) the tricritical behaviour of a model of classical unbounded $n$-component continuous spins with a triple-well single-spin potential (the $|\varphi|^6$ model), and (ii) a random walk model of linear polymers with a three-body repulsion and two-body attraction at the tricritical theta point (critical point for the collapse transition) where repulsion and attraction effectively cancel. The polymer model is exactly equivalent to a supersymmetric spin model which corresponds to the $n=0$ version of the $|\varphi|^6$ model. For the spin and polymer models, we identify the tricritical point, and prove that the tricritical two-point function has Gaussian long-distance decay, namely $|x|^{-1}$. The proof is based on an extension of a rigorous renormalisation group method that has been applied previously to analyse the $|\varphi|^4$ and weakly self-avoiding walk models on $\mathbb{Z}^4$.Comment: Accepted versio

Structural stability of a dynamical system near a non-hyperbolic fixed point

We prove structural stability under perturbations for a class of discrete-time dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well-posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. The class of dynamical systems we study, and the boundary conditions we impose, arise in a renormalisation group analysis of the 4-dimensional weakly self-avoiding walk and the 4-dimensional n-component $|\varphi|^4$ spin model.Comment: 31 pages, to appear in Ann. Henri Poincar

A renormalisation group method. III. Perturbative analysis

This paper is the third in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. In this paper, we motivate and present a general approach towards second-order perturbative renormalisation, and apply it to a specific supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$. Our focus is on the critical dimension $d=4$. The results include the derivation of the perturbative flow of the coupling constants, with accompanying estimates on the coefficients in the flow. These are essential results for subsequent application to the 4-dimensional weakly self-avoiding walk, including a proof of existence of logarithmic corrections to their critical scaling. With minor modifications, our results also apply to the 4-dimensional $n$-component $|\varphi|^4$ spin model.Comment: 40 pages, revised version, will appear in J. Stat. Phy

Critical two-point function of the 4-dimensional weakly self-avoiding walk

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 $d \geq 5$ 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.Comment: 26 pages, revised version, will appear in Commun. Math. Phy

Renormalisation group analysis of 4D spin models and self-avoiding walk

We give an overview of results on critical phenomena in 4 dimensions, obtained recently using a rigorous renormalisation group method. In particular, for the $n$-component $|\varphi|^4$ spin model in dimension 4, with small coupling constant, we prove that the susceptibility diverges with a logarithmic correction to the mean-field behaviour with exponent $(n+2)/(n+8)$. This result extends rigorously to $n=0$, interpreted as a supersymmetric version of the model that represents exactly the continuous-time weakly self-avoiding walk. We also analyse the critical two-point function of the weakly self-avoiding walk, the specific heat and pressure of the $|\varphi|^4$ model, as well as scaling limits of the spin field close to the critical point.Comment: Proceedings for ICMP, Santiago de Chile, July 201

Fluctuations in a kinetic transport model for quantum friction

We consider a linear Boltzmann equation that arises in a model for quantum friction. It describes a particle that is slowed down by the emission of bosons. We study the stochastic process generated by this Boltzmann equation and we show convergence of its spatial trajectory to a multiple of Brownian motion with exponential scaling. The asymptotic position of the particle is finite in mean, even though its absolute value is typically infinite. This is contrasted to an approximation that neglects the influence of fluctuations, where the mean asymptotic position is infinite

Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis

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 1/4 for the logarithm. The susceptibility has been well understood previously for dimensions $d \geq 5$ 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.Comment: 63 pages, revised version, will appear in Commun. Math. Phy

Finite-order correlation length for 4-dimensional weakly self-avoiding walk and ∣φ∣4|\varphi|^4 spins

We study the 4-dimensional $n$-component $|\varphi|^4$ spin model for all integers $n \ge 1$, and the 4-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case $n=0$ interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order $p$, and prove the existence of a logarithmic correction to mean-field scaling, with power $\frac 12\frac{n+2}{n+8}$, for all $n \ge 0$ and $p>0$. The proof is based on an improvement of a rigorous renormalisation group method developed previously.Comment: 27 page

Lectures on Self-Avoiding Walks

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 polygons. A detailed proof that the connective constant on the hexagonal lattice equals $\sqrt{2+\sqrt{2}}$ 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.Comment: 73 pages, 15 figures. Lecture notes for course given at XIV Brazilian School of Probability and Clay Mathematics Institute 2010 Summer School "Probability and Statistical Physics in Two and more Dimensions", B\'uzios - Rio de Janeiro (BR), 2-7 August 2010. Final version to appear on Clay Mathematics Proceedings 15 (2012