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We propose an algorithm base on the modulable hidden variables and steplength, which is inspired by the heuristic statistical physics and replicamethod, to study the effect of mutual correlations and the emergentWigner-Dyson distribution in a many-body system which is of the asymptotichigh-dimensional statistics regime. We consider the polaron system to illustatethe effect of IR/UV cutoff in the momentum or position space. The polaron as along-lived quasiparticle which can be found in the imcompressible state hasslow momenta and current relaxation in Fermi liquid phase. We reveal therelation between UV cutoff of polaronic momentum $\Lambda_{q}$ and its SYKbehavior. The SYK behavior of a polaron system, as well as the relation betweenscattering momentum and the related statistical behaviors has rarely beeninvestigated before. We found that the inversed momentum cutoff$\Lambda_{q}^{-1}$, which plays the role of an essential degree-of-freedom(DOF) other than the fermions, relates to the distribution and statisticalvariance of polaronic coupling term. By projecting to a 2d square lattice, weconsider this problem in position space where the DOF of polaron scatteringmomenta is replaced by another flavor (denoted as $\eta_{\Delta}$ with flavornumber of order of $O(M)$) which is determined by the site potential differece$\Delta$ as well as the site index, and we also applying the self-attentionmethod to searching for the more efficient route to exploiting the many-bodybehaviors.

Emerging polaron and Wigner-Dyson distribution in many-body disordered  system: Numerical evidence using a hidden variable based optimization  algorithm for the multi-variable PDE and ensemble-dominated weight  distribution