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We suggest how versions of Schramm's SLE can be used to describe the scaling limit of some off-critical 2D lattice models. Many open questions remain.

It is known that folding a protein chain into a cubic lattice is an NP-complete problem. We consider a seemingly easier problem: given a three-dimensional (3D) fold of a protein chain (coordinates of its C_α atoms), we want to find the closest lattice approximation of this fold. This problem has been studied under names such as "lattice approximation of a protein chain", "the protein chain fitting problem", and "building of protein lattice models". We show that this problem is NP-complete for the cubic lattice with side close to 3.8 Å and coordinate root mean square deviation.

The standard perturbation theory in QFT and lattice models leads to the asymptotic expansions. However, an appropriate regularization of the path or lattice integrals allows one to construct convergent series with an infinite radius of the convergence. In the earlier studies, this approach was applied to the purely bosonic systems. Here, using bosonization, we develop the convergent perturbation theory for a toy lattice model with interacting fermionic and bosonic fields.

Using a lattice equation of state combined with the D-dimensional Tolman–Oppenheimer–Volkoff equation and the Friedmann equations, we investigate the possibility of the formation of compact objects as well as the time evolution of the scale factor and the density profile of a self-gravitating material cluster. The numerical results show that in a ($2+1$)-dimensional space–time, the mass is independent of the central pressure. Hence, the formation of only compact objects with a finite constant mass similar to the white dwarf is possible. However, in a ($3+1$)-dimensional space–time, self-gravity leads to the formation of compact objects with a large gap of mass and the corresponding phase diagram has the same structure as the one for Neutron Star. The results also show that beyond certain critical central pressure, the star is unstable against gravitational collapse, and it may end in a black hole. Analysis of space–times of higher dimensions shows that gravity has the stronger effect in $3+1$ dimensions. Numerical solutions of the Friedmann equations show that the effect of the curvature of space–time increases with the increasing temperature, but decreases with the increasing dimensionality beyond $D=3$.