Rigorous Renormalization Group Research Articles

Construction of the two-dimensional sine-Gordon model for β<8π

We present a rigorous renormalization group construction of the two-dimensional massless and massive quantum sine-Gordon models in finite volume for the range 0<β<8π. We prove analyticity in the coupling constant ζ, which implies the convergence of perturbation theory. The field correlation functions and their generating functional are analyzed and shown to have the short distance asymptotics of the free field theory. In the massive case the bounds are uniform in volume and we also obtain uniform estimates on the long distance decay of correlations.

Stability of interfaces in a random environment. A rigorous renormalization group analysis of a hierarchical model

We study a hierarchical model of domain walls in aD-dimensional bond disordered Ising model at low temperatures. Using a renormalization group method inspired by the work of Bricmont and Kupiainen for the random field Ising model, we prove the existence of rigid interfaces at low enough temperatures in dimensionsD>3.

Taylor expansion of effective Hamiltonians by Monte Carlo simulations with fixed block spins

Monte Carlo simulations with fixed block spins allow the computation of Taylor expansion kernels for effective Hamiltonians. The expansion can be performed around arbitrary block spin configurations\(\bar \phi\) and does not suffer from truncation errors. Monte Carlo calculation of the Taylor kernels offers a ready possibility to check whether the effective Hamiltonian has good locality properties in the neighborhood of a given configuration\(\bar \phi\). The method is applied in a renormalization group study of the 2-dimensional critical Ising model. The results show that one has to deal with a “large field problem”, as had been expected from rigorous renormalization group studies.

Rigorous renormalisation group and disordered systems

We consider random walks in a non-symmetric random environment. We report a recent result, based on a Renormalisation Group approach, showing that for d#62;2, these walks are diffusive when the disorder is weak.

The two-dimensionalO(N) nonlinear ?-model: Renormalisation and effective actions

We establish the existence of the Wilson Renormalised trajectory of theO(N) sigma model in perturbation theory in the “effective charge.” This yields a proof of perturbative renormalisability, and is also relevant in the “small-field” analysis of the rigorous renormalisation group construction of the continuum theory.

Quantum field-theory models on fractal spacetime

Continuing the analysis of a previous paper, the present work applies rigorous renormalization group methods to the hierarchical models to establish the existence of field theories with non-Gaussian ultraviolet renormalization group fixed points in 4-e dimensions.

Rigorous renormalization group at work

We illustrate the recent progress in rigorous renormalization group on a simple example of multiscale analysis of a first order phase transition.

The U(1) Higgs model

By using rigorous renormalization group methods we construct the continuum limit of the finite-volume lattice U(1) Higgs model in two and three dimensions. The method relies on a proof of the convergence of the effective action.