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Branched polymers with loops coupled to the critical Ising model

Jan Ambjørn1,2,*, Yukimura Izawa3,4,†, and Yuki Sato5,6,‡

  • 1The Niels Bohr Institute, Copenhagen University Blegdamsvej 17, DK-2100 Copenhagen, Denmark
  • 2IMPP, Radboud University Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands
  • 3Physics Program, Graduate School of Advanced Science and Engineering, Hiroshima University Higashi-Hiroshima 739-8526, Japan
  • 4International Institute for Sustainability with Knotted Chiral Meta Matter Kagamiyama, Higashihiroshima 739-8511, Hiroshima, Japan
  • 5Department of Mechanical and System Engineering, University of Fukui 3-9-1 Bunkyo, Fukui-shi, Fukui 910-8507, Japan
  • 6Department of Physics, Nagoya University Chikusaku, Nagoya 464-8602, Japan

  • *Contact author: ambjorn@nbi.dk
  • †Contact author: izawa-yukimura@hiroshima-u.ac.jp
  • ‡Contact author: yukisato@u-fukui.ac.jp
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Phys. Rev. D 113, 066004 – Published 5 March, 2026

DOI: https://doi.org/10.1103/qs2c-v9w8

Abstract

We study the continuum limit of branched polymers (BPs) with loops coupled to Ising spins at the zero-temperature critical point. It is known that the continuum partition function can be represented by a Hermitian two-matrix model, and we propose a string field theory whose Dyson-Schwinger equation coincides with the loop equation of this continuum matrix model. By setting the matrix size to one, we analyze a convergent nonperturbative partition function expressed as a two-dimensional integral, and show that it satisfies a third-order linear differential equation. In contrast, in the absence of coupling to the critical Ising model, the continuum partition function of pure BPs with loops is known to satisfy the Airy equation. From the viewpoint of two-dimensional quantum gravity, we introduce a nonperturbative loop amplitude that serves as a solution to the Wheeler-DeWitt equation incorporating contributions from all genera. Furthermore, we demonstrate that the same Wheeler-DeWitt equation can also be derived through the stochastic quantization.

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Corrections

25 March, 2026

Correction: The citation information given in Ref. [35] was incorrect and has been fixed.

Article Text

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Phys. Rev. D 113, 066004– Published 5 March, 2026
  • Received 3 January 2026
  • Accepted 17 February 2026
  • Corrected 25 March 2026
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