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Analogue of Cardy's formula on the 60-degree parallelogram: a modular approach — arXiv:2608.29425

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.29425
License (as archived): CC0-1.0
Archived: 2026-10-17T09:10:10.416Z
SHA-256 of the archived text: 46c03b808b607d59a24afad247b8bcda7b7ad22e72897a72a0c40849630037a8
Stable URL of this page: https://metraipolis.com/documento/dc_bfeff290eaf2

How to cite

Analogue of Cardy's formula on the 60-degree parallelogram: a modular approach — arXiv:2608.29425 — arXiv (en), https://arxiv.org/abs/2608.29425 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bfeff290eaf2 (Metraipolis archive dc_bfeff290eaf2, sha256 46c03b808b607d59a24afad247b8bcda7b7ad22e72897a72a0c40849630037a8)

BibTeX

@misc{dcbfeff290eaf2,
  title = {Analogue of Cardy's formula on the 60-degree parallelogram: a modular approach — arXiv:2608.29425},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.29425},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bfeff290eaf2; sha256 46c03b808b607d59a24afad247b8bcda7b7ad22e72897a72a0c40849630037a8},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.29425. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The Cardy-Smirnov function of a four-marked planar domain encodes the conjectural scaling limit of the crossing probability for critical percolation. Cardy predicted a closed formula for this limit on the rectangle in the cases of both site and bond percolation, and Smirnov proved the formula together with conformal invariance for critical site percolation on the triangular lattice. Kleban and Zagier later showed that on the rectangle the function admits a modular interpretation: it is determined by a modular functional equation together with a mild analytic ansatz. We carry out the analogue on the $\pi/3$ parallelogram. We derive a closed conformal-map formula for the Cardy-Smirnov function as an incomplete beta integral, prove a closed modular formula realizing it as an integral of the modular form $\eta(\tau)^2\eta(3\tau)^2$, and establish a uniqueness theorem showing that a single functional equation, together with a $q$-expansion ansatz and a nondegeneracy condition, determines the function uniquely.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-01 in math.PR).
Categorie arXiv: Number Theory (math.NT) ; Mathematical Physics (math-ph); Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.29425 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T08:40:49.543Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).