Renormalization group analysis of the random first-order transition

We consider the approach describing glass formation in liquids as a progressive trapping in an exponentially large number of metastable states. To go beyond the mean-field setting, we provide a real-space renormalization group (RG) analysis of the associated replica free-energy functional. The present approximation yields in finite dimensions an ideal glass transition similar to that found in the mean field. However, we find that along the RG flow the properties associated with metastable glassy states, such as the configurational entropy, are only defined up to a characteristic length scale that diverges as one approaches the ideal glass transition. The critical exponents characterizing the vicinity of the transition are the usual ones associated with a first-order discontinuity fixed point.     

Renormalization group study of quantum fluctuations near classical critical points of Hamiltonian systems

A general framework is considered for treating quantum corrections to the classical limit in the Wigner function formalism. We discuss the quantal effect on the classical phenomena such as period doubling and the breakup of KAM tori. By using an exact renormalization group method, the scaling factor for Planck's constant is derived as an eigenvalue of the linearized renormalization transformation.     

Renormalization group for network models of quantum Hall transitions

We analyze in detail the renormalization group flows which follow from the recently proposed all orders β functions for the Chalker–Coddington network model. The flows in the physical regime reach a true singularity after a finite scale transformation. Other flows are regular and we identify the asymptotic directions. One direction is in the same universality class as the disordered XY model. The all orders β function is computed for the network model of the spin quantum Hall transition and the flows are shown to have similar properties. It is argued that fixed points of general current–current interactions in 2d should correspond to solutions of the Virasoro master equation. Based on this we identify two coset conformal field theories osp(2N|2N)₁/u(1)₀ and osp(4N|4N)₁/su(2)₀ as possible fixed points and study the resulting multifractal properties. We also obtain a scaling relation between the typical amplitude exponent α₀ and the typical point contact conductance exponent X_t which is expected to hold when the density of states is constant.     

Renormalization group study of compressible Ising systems

The mean field renormalization group approach is used to study compressible Ising models. The phase diagram as well as estimates of critical exponents are obtained for systems where shear forces are neglected.     

Renormalization group study of a critical lattice model

We start a nonperturbative study of the Wilson-Kadanoff renormalization group (RG) in weakly coupled massless lattice models. Nonlocal hierarchical models are introduced to mimic the infrared behaviour of the \(\tfrac{1}{2}(\nabla \phi )^2 + \lambda (\nabla \phi )^4\) model and the like. The RG is shown to drive these to the line of fixed points corresponding to the massless \(\tfrac{1}{2}c_\infty (\lambda )(\nabla \phi )^2\) models.     

Renormalization of random Jacobi operators

We describe an approach, based on Baldi's large deviation theorem, to carry out the statistical mechanics of a class of infinite dimensional dynamical systems.     

Renormalization group equations and the continuum limit of some renormalizable quantum field theories

The renormalization group equations for asymptotically and non-asymptotically free theories are discussed by exploiting the general properties of their integral curves. Some constraints are derived on the existence of a consistent continuum limit for pure Yang-Mills theories in an infinite quantization volume. An analogy between the far infrared behaviour of non-abelian gauge theories and the deep ultraviolet one of asymptotically free theories is discussed.     

Renormalization group and cocycles

It is shown that some quantities appearing in the renormalization group (RG) scheme of quantum field theory can be regarded as cocycles. Simple integral representations for cocycles elucidate the arguments on some aspects of the RG equation. It is discussed how the bifurcation of the invariant charge can take place in the RG scheme.     

Renormalization group and string amplitudes

We define β-functions for open-string tachyons away from the perturbative fixed point. We show that solutions of the renormalization-group fixed-point equations generate open-string scattering amplitudes.