469 research outputs found
A renormalisation group method. II. Approximation by local polynomials
This paper is the second in a series devoted to the development of a rigorous
renormalisation group method for lattice field theories involving boson fields,
fermion fields, or both. The method is set within a normed algebra
of functionals of the fields. In this paper, we develop a general
method---localisation---to approximate an element of by a local
polynomial in the fields. From the point of view of the renormalisation group,
the construction of the local polynomial corresponding to in
amounts to the extraction of the relevant and marginal parts of . We prove
estimates relating and its corresponding local polynomial, in terms of the
semi-norm introduced in part I of the series.Comment: 30 page
A renormalisation group method. IV. Stability analysis
This paper is the fourth in a series devoted to the development of a rigorous
renormalisation group method for lattice field theories involving boson fields,
fermion fields, or both. The third paper in the series presents a perturbative
analysis of a supersymmetric field theory which represents the continuous-time
weakly self-avoiding walk on . We now present an analysis of the
relevant interaction functional of the supersymmetric field theory, which
permits a nonperturbative analysis to be carried out in the critical dimension
. The results in this paper include: proof of stability of the
interaction, estimates which enable control of Gaussian expectations involving
both boson and fermion fields, estimates which bound the errors in the
perturbative analysis, and a crucial contraction estimate to handle irrelevant
directions in the flow of the renormalisation group. These results are
essential for the analysis of the general renormalisation group step in the
fifth paper in the series.Comment: 62 page
Expansion in high dimension for the growth constants of lattice trees and lattice animals
We compute the first three terms of the 1/d expansions for the growth
constants and one-point functions of nearest-neighbour lattice trees and
lattice (bond) animals on the integer lattice Zd, with rigorous error
estimates. The proof uses the lace expansion, together with a new expansion for
the one-point functions based on inclusion-exclusion.Comment: 38 pages, 8 figures. Added section 6 to obtain the first term in the
expansion, making the present paper more self-contained with very little
change to the structure of the original paper. Accepted for publication in
Combinatorics Probability and Computin
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