] o c n m p [Open Communications in Nonlinear Mathematical Physics |
Edited by Norbert Euler, Rafael Hernández Heredero and Da-jun Zhang. Note: This issue is not complete: deadline for submissions is May 1st, 2026.
A phenomenon of "algebraic self-similarity" on 3d cubic lattice, providing what can be called an algebraic analogue of Kadanoff--Wilson theory, is shown to possess a 4d version as well. Namely, if there is a $4\times 4$ matrix $A$ whose entries are indeterminates over the field $\mathbb F_2$, then the $2\times 2\times 2\times 2$ block made of sixteen copies of $A$ reveals the existence of four direct "block spin" summands corresponding to the same matrix $A$. Moreover, these summands can be written out in quite an elegant way. Somewhat strikingly, if the entries of $A$ are just zeros and ones -- elements of $\mathbb F_2$ -- then there are examples where two more "block spins" split out, and this time with different $A$'s.
We derive a class of discrete Painlevé equations associated with the affine Weyl group E$_7^{(1)}$. The method used is the deautonomisation of a QRT mapping belonging to the canonical form VI (according to the classification of said mappings). An equation of such a form was the first instance of a symmetric -- in QRT parlance -- discrete analogue of the Painlevé VI equation. In this paper we present an exhaustive derivation of all the discrete Painlevé equations of this class. This is made possible thanks to previous studies that established the proper lengths of singularity patterns that are compatible with integrablity, and which were already successfully applied to the study of discrete Painlevé equations associated to the affine Weyl group E$_8^{(1)}$. Given that, from the latter, one can obtain by degeneration the equations related to E$_7^{(1)}$, we decided to link the results of the present study to those of the aforementioned ones. It turns out that a bridge from E$_8^{(1)}$ to E$_7^{(1)}$ exists in almost all cases, with one exception where, while in the former case a discrete Painlevé equation does exist, in the latter we find a mapping with only periodic coefficients, devoid of secular dependence.