Renormalization group transformations near the critical point: Some rigorous results

Abstract

We consider renormalization group (RG) transformations for classical Ising-type lattice spin systems in the infinite-volume limit. Formally, the RG maps a Hamiltonian H into a renormalized Hamiltonian H ′ :With the help of the Dobrushin uniqueness condition and standard results on the polymer expansion, Haller and Kennedy gave a sufficient condition for the existence of the renormalized Hamiltonian in a neighborhood of the critical point. By a more complicated but reasonably straightforward application of the cluster expansion machinery, the present investigation shows that their condition would further imply a band structure on the matrix of partial derivatives of the renormalized interaction with respect to the original interaction. This in turn gives an upper bound for the RG linearization.

Explaining universality: infinite limit systems in the renormalization group method

“…At the end of this section, I will state a conjecture on the equivalence of the convergence property and the linearization* property with respect to the explanatory function of demarcating the universality classes. In the next section, I will present both heuristic and technical evidence inspired by Wilson and Kogut (1974) and Yin (2011) to motivate and prove a proposition demonstrating that, under reasonable conditions, the conjecture holds.…”

“…It is important to note here that the above specifications of the mathematical structures are not standardly discussed in the literature, in the following sense: although some authors (e.g. Yin (2011)) introduce norms on the R n b surfaces, there are very few discussions, let alone agreements, on which norms are appropriate to use. Moreover, numerous authors discuss the convergence of R n b (S c ) to S * as n → ∞ in the basin of attraction.…”

“…In this section, I will first present heuristic evidence from Wilson and Kogut (1974) suggesting the plausibility of Conjecture 1. I will then present technical evidnce from Yin (2011) to motivate a key condition for the conjecture. Furthermore, I will introduce a few reasonable auxiliary conditions about the space of Hamiltonians and the linearization*-adequate neighborhood.…”

“…In Section 4, I will respond to Batterman by pointing out that the convergence property he regards as essential in RG explanations is not as explanatorily relevant as the linearization* property, which is defined in connection to the linearization* process crucial in the RG method. Finally, in Section 5, I will present both heuristic and technical evidence inspired by Wilson and Kogut (1974) and Yin (2011) to motivate and prove a proposition demonstrating that, under reasonable conditions, linearization* properties and convergence properties provide the same explanatory functions in terms of demarcating the universality classes. Because points with linearization* properties can be finite systems and linearization* properties have additional explanatory merits, I maintain my position that in RG explanations, infinite limit systems are dispensable, if the conditions are met.…”

A cluster expansion approach to exponential random graph models

“…This transforms the probability model into a statistical mechanics model (Theorem 2.5). In Section 3 we apply cluster expansion techniques [27] and derive a convergent power series expansion (high-temperature expansion) for the limiting free energy in the case of small parameters (Theorem 3.5). Finally, Section 4 is devoted to concluding remarks.…”