Continuing the program begun by the authors in a previous paper, we develop an exact low-density expansion for the random minimum spanning tree (MST) on a finite graph and use it to develop a continuum perturbation expansion for the MST on critical percolation clusters in space dimension d . The perturbation expansion is proved to be renormalizable in d=6 dimensions. We consider the fractal dim...

QCD perturbation theory at large orders with large renormalization scales in the large β 0 limit. Abstract. We examine the QCD perturbation series at large orders, for different values of the 'large β 0 renormalization scale'. It is found that if we let this scale grow exponentially with the order, the divergent series can be turned into an expansion that converges to the Borel integral, with a...

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearise...

Power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasi-periodic solutions the issue of convergence of the series is plagued of the so-called small divisor problem. In this paper we review a method recently introduced to deal with such a problem, based on renormalisation group ideas and ...

We apply an improved Taylor expansion method, which is a variational scheme to the Ising model in two dimensions. This method enables us to evaluate the free energy and magnetization in strong coupling regions from the weak coupling expansion, even in the case of a phase transition. We determine the approximate value of the transition point using this scheme. In the presence of an external magn...

We apply a numerical basis-state method to determine the electric susceptibility of atomic hydrogen via calculations of the terms of the perturbative power series expansion up to eleventh order as well as from ab initio calculations of the field-induced polarization. The results of the perturbative calculations indicate that the applicability of the power series expansion is limited to intensit...

A precise definition of an adiabaticity parameter ν of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(τ) = ∑ l U (l)(τ) with U (l)(τ) being at least of the order ν. In particular U (0)(τ) corresponds to the adiabatic approximation and yields Berry’s adiabatic phase. It is sh...

We construct the thermodynamic limit of the critical (massless) φ model in 4 dimensions with an ultraviolet cutoff by means of a "partly renormalized" phase space expansion. This expansion requires in a natural way the introduction of effective or "running" constants, and the infrared asymptotic freedom of the model, i.e. the decay of the running coupling constant, plays a crucial role. We prov...

We present the main steps governing the theory of resonant x-ray diffraction (RXD). We focus on the derivation of the anomalous scattering amplitude from perturbation theory and starting from the low-energy expansion of the Dirac Hamiltonian. We give the main ingredients of the multipolar expansion in term of electric and magnetic transitions. We also show the expansion in terms of scattering t...

Upon combining Northrop’s picture of charged particle motion with modern liquid crystal theories, this paper provides a new description of guiding center dynamics (to lowest order). This new perspective is based on a rotation gauge field (gyrogauge) that encodes rotations around the magnetic field. In liquid crystal theory, an analogue rotation field is used to encode the rotational state of ro...