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Abstract:The goal of this paper is to prove singularity of three natural fields in QFT with respect to their natural base measure. The fields we consider are the following ones:
(1) The near-critical limit of the $2d$ Ising model (in the $\beta$-direction) is locally singular w.r.t the critical scaling limit of $2d$ Ising. (N.B. In the $h$-direction it is not locally singular).
(2) The $2d$ Hierarchical Sine-Gordon field is singular w.r.t the $2d$ hierarchical Gaussian Free Field for all $\beta\in[\beta_{L^2}, \beta_{BKT})$.
(3) The Hierarchical $\Phi^4_3$ field is singular w.r.t the $3d$ hierarchical GFF.
Item (1) gives the first strong indication that the energy field of critical $2d$ Ising model does not exist as a random Schwarz distribution on the plane. Item (2) has been proved to be singular for the non-hierarchical $2d$ Sine-Gordon sufficiently far from the BKT point in [GM24] while item (3) is proved to be singular for the non-hierarchical $3d$ $\Phi^4_3$ field in [BG21, OOT21, HKN24].
We believe our way to detect a singular behaviour at all scales is very much down to earth and may be applicable in all settings where one has a good enough control on the so-called effective potentials.

Submission history

From: Christophe Garban [view email]
[v1] Tue, 4 Feb 2025 18:23:35 UTC (95 KB)
[v2] Thu, 20 Nov 2025 20:41:54 UTC (96 KB)
[v3] Mon, 24 Aug 2026 09:56:24 UTC (97 KB)