Asymptotics of a planar isoperimetric problem with a white-noise volume term

Abstract

In this work we study the maximal ratio I between the white noise integrated over a set and the perimeter of the set, which can be seen as a random isoperimetric problem, and appears in the random-field Ising model (Ding-Wirth) and min-max optimal matching (Leighton-Shor). In the planar case considered here, such a ratio is scale-invariant and therefore the problem is critical. As such it requires an ultraviolet cutoff, which we impose by restricting to polygonal sets with side-length at least 1 contained in the ball B_L. Our main result establishes the leading-order asymptotics 𝔼I≈ iln^3/4L for some i∈(0,∞) and superconcentration at scale O(ln^−1/4L). This is done by relating to a simpler (1+1)-dimensional action, studied by the last two authors and C. Wagner. Such a connection is found by a classical geometric linearization of the perimeter to the Dirichlet energy, justified by Ried-Wagner large-scale regularity theory, and allows a coarse-graining and scale-by-scale iteration argument.

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BibTeX RIS

BibTeX

@misc{cheng2026asymptotics,
  title = {{Asymptotics of a planar isoperimetric problem with a white-noise volume term}},
  author = {Xiaopeng Cheng and Felix Otto and Matteo Palmieri},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08726},
  eprint = {2609.08726},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08726},
}

RIS

TY  - UNPB
TI  - Asymptotics of a planar isoperimetric problem with a white-noise volume term
AU  - Xiaopeng Cheng
AU  - Felix Otto
AU  - Matteo Palmieri
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08726
UR  - https://arxiv.org/abs/2609.08726
AB  - In this work we study the maximal ratio I between the white noise integrated over a set and the perimeter of the set, which can be seen as a random isoperimetric problem, and appears in the random-field Ising model (Ding-Wirth) and min-max optimal matching (Leighton-Shor). In the planar case considered here, such a ratio is scale-invariant and therefore the problem is critical. As such it requires an ultraviolet cutoff, which we impose by restricting to polygonal sets with side-length at least 1 contained in the ball B_L. Our main result establishes the leading-order asymptotics 𝔼I≈ iln^3/4L for some i∈(0,∞) and superconcentration at scale O(ln^−1/4L). This is done by relating to a simpler (1+1)-dimensional action, studied by the last two authors and C. Wagner. Such a connection is found by a classical geometric linearization of the perimeter to the Dirichlet energy, justified by Ried-Wagner large-scale regularity theory, and allows a coarse-graining and scale-by-scale iteration argument.
ER  -

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