Sequential vs Net Closure of Renormalization-Group Images in the 2D Ising Model

Since the 1980s, rigorous renormalization-group (RG) theory has revealed that the image of a Gibbs measure under coarse-graining need not itself be Gibbs.
The seminal works of van Enter, Fernández, and Sokal (1993) and subsequent studies by Le Ny, Redig, and others established this non-Gibbsian phenomenon for decimation and majority-rule transforms of the low-temperature Ising model, identifying “hidden phase transitions’’ as the underlying cause of essential discontinuities in the transformed specification.
These analyses, however, treat the RG flow as a single sequence of blockings—typically dyadic—and thereby inherit the limitations of sequential convergence in a non-first-countable topology of probability measures.

In this work we reformulate the RG procedure itself as a directed inductive system of coarse-grainings, indexed by both block size and spatial tiling.
This shift from sequences to nets exposes a new structural degree of freedom: the behavior of RG images depends not only on the transformation but also on the choice of cofinal filtration through which the thermodynamic limit is taken.
We demonstrate that within the same family of majority-rule maps one can construct Gibbs-preserving and Gibbs-destroying subnets—odd scales yielding quasilocal renormalized potentials, even scales exhibiting essential discontinuities.
The resulting dichotomy shows that the RG image of a Gibbs measure is not sequentially closed in the weak topology, although it remains net-closed along properly chosen filtrations.

This observation reframes the non-Gibbs problem as a question of continuity at the level of directed systems, connecting statistical mechanics with the general topological notion that nets, rather than sequences, are the natural carriers of convergence in non-metrizable spaces.
By making this indexing structure explicit, we clarify which RG limits correspond to genuine thermodynamic phases and which arise as artefacts of an incomplete (sequential) parametrization.

#MathematicalPhysics #RenormalizationGroup #GibbsMeasures #NonGibbsian

#Nets #DirectedSystems #SequentialVsNet #TopologyOfConvergence

#ThermodynamicLimits #Quasilocality #PhaseTransitions