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Non-integrable Ising models in cylindrical geometry: Grassmann representation and infinite volume limit. (English) Zbl 1522.82006

Summary: In this paper, meant as a companion to [the authors, Commun. Math. Phys. 397, No. 1, 393–483 (2023; Zbl 07676189)], we consider a class of non-integrable \(2D\) Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green’s function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
05C05 Trees
15A75 Exterior algebra, Grassmann algebras

Citations:

Zbl 07676189
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References:

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