## Convergence of Migdal-Kadanoff iterations: A simple and general proof.(English)Zbl 0664.43004

Migdal-Kadanoff recursion relations for lattice gauge theories realize exact renormalization group transformations on hierarchical lattices embedded in $${\mathbb{Z}}^ d$$. The authors prove a convergence of Migdal- Kadanoff iterations using the following constructions. Let $${\mathcal G}$$ be a compact connected Lie group and $${\mathcal F}$$ be the class of real-valued functions on $${\mathcal G}$$. Defining for arbitrary $$r\in {\mathbb{N}}\setminus \{1\}$$ and $$q\in {\mathbb{N}}$$ the mapping $${\mathcal T}: {\mathcal F}\to {\mathcal F}$$ by $${\mathcal T}(g)=K(g^{*r})^ q$$ which involves a multiple convolution product of r factors, and $$K\in {\mathbb{R}}_+$$ is determined by the normalization condition at the group unit, one can demonstrate that these cover Migdal’s recursion relation for gauge models with compact connected gauge groups. Interchanging the convolution and product in the definition of $${\mathcal T}$$, we are lead to Kadanoff’s recursion relation. Estimating the rate of convergence, the authors derive lower bounds depending on the coupling strength of the initial action, for the string tension of hierarchical gauge models and for the mass gap in the hierarchical $${\mathcal O}({\mathcal N})$$ Heisenberg models in their respective critical four and two dimensions.
Reviewer: E.Kryachko

### MSC:

 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A75 Harmonic analysis on specific compact groups 81T17 Renormalization group methods applied to problems in quantum field theory 22E70 Applications of Lie groups to the sciences; explicit representations
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### References:

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