The Annals of Probability

Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

Roland Bauerschmidt, Tyler Helmuth, and Andrew Swan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite-range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^{n}$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin–Wagner theorem for these sigma models. This result is of intrinsic interest for the sigma models and also implies our main theorem on the VRJP. Surprisingly, our Mermin–Wagner theorem applies even though the symmetry groups of $\mathbb{H}^{n}$ and $\mathbb{H}^{2|2}$ are nonamenable.

Article information

Ann. Probab., Volume 47, Number 5 (2019), 3375-3396.

Received: March 2018
Revised: October 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Vertex-reinforced jump process hyperbolic sigma models Mermin–Wagner theorem Dynkin isomorphism supersymmetry


Bauerschmidt, Roland; Helmuth, Tyler; Swan, Andrew. Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process. Ann. Probab. 47 (2019), no. 5, 3375--3396. doi:10.1214/19-AOP1343.

Export citation


  • [1] Angel, O., Crawford, N. and Kozma, G. (2014). Localization for linearly edge reinforced random walks. Duke Math. J. 163 889–921.
  • [2] Berezin, F. A. (1987). Introduction to Superanalysis. Mathematical Physics and Applied Mathematics 9. Reidel, Dordrecht. Translated from the Russian by J. Niederle and R. Kotecký. Translation edited by Dimitri Leĭtes.
  • [3] Brydges, D., Evans, S. N. and Imbrie, J. Z. (1992). Self-avoiding walk on a hierarchical lattice in four dimensions. Ann. Probab. 20 82–124.
  • [4] Brydges, D., Fröhlich, J. and Spencer, T. (1982). The random walk representation of classical spin systems and correlation inequalities. Comm. Math. Phys. 83 123–150.
  • [5] Brydges, D. C., Imbrie, J. Z. and Slade, G. (2009). Functional integral representations for self-avoiding walk. Probab. Surv. 6 34–61.
  • [6] Davis, B. and Volkov, S. (2002). Continuous time vertex-reinforced jump processes. Probab. Theory Related Fields 123 281–300.
  • [7] Disertori, M., Merkl, F. and Rolles, S. W. W. (2017). A supersymmetric approach to martingales related to the vertex-reinforced jump process. ALEA Lat. Am. J. Probab. Math. Stat. 14 529–555.
  • [8] Disertori, M., Sabot, C. and Tarrès, P. (2015). Transience of edge-reinforced random walk. Comm. Math. Phys. 339 121–148.
  • [9] Disertori, M. and Spencer, T. (2010). Anderson localization for a supersymmetric sigma model. Comm. Math. Phys. 300 659–671.
  • [10] Disertori, M., Spencer, T. and Zirnbauer, M. R. (2010). Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. Comm. Math. Phys. 300 435–486.
  • [11] Duncan, A., Niedermaier, M. and Seiler, E. (2005). Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models. Nuclear Phys. B 720 235–288.
  • [12] Dynkin, E. B. (1983). Markov processes as a tool in field theory. J. Funct. Anal. 50 167–187.
  • [13] Efetov, K. B. (1983). Supersymmetry and theory of disordered metals. Adv. Phys. 32 53–127.
  • [14] Fröhlich, J. and Spencer, T. (1981). On the statistical mechanics of classical Coulomb and dipole gases. J. Stat. Phys. 24 617–701.
  • [15] Le Jan, Y. (1987). Temps local et superchamp. In Séminaire de Probabilités, XXI. Lecture Notes in Math. 1247 176–190. Springer, Berlin.
  • [16] Le Jan, Y. (2011). Markov Paths, Loops and Fields. Lecture Notes in Math. 2026. Springer, Heidelberg.
  • [17] Luttinger, J. M. (1983). The asymptotic evaluation of a class of path integrals. II. J. Math. Phys. 24 2070–2073.
  • [18] Merkl, F. and Rolles, S. W. W. (2009). Edge-reinforced random walk on one-dimensional periodic graphs. Probab. Theory Related Fields 145 323–349.
  • [19] Merkl, F. and Rolles, S. W. W. (2009). Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab. 37 1679–1714.
  • [20] Mermin, N. D. (1967). Absence of ordering in certain classical systems. J. Math. Phys. 8 1061–1064.
  • [21] Mermin, N. D. and Wagner, H. (1966). Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. 17 1133–1136.
  • [22] Niedermaier, M. and Seiler, E. (2005). Non-amenability and spontaneous symmetry breaking—The hyperbolic spin-chain. Ann. Henri Poincaré 6 1025–1090.
  • [23] Sabot, C. and Tarrès, P. (2015). Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. (JEMS) 17 2353–2378.
  • [24] Sabot, C. and Zeng, X. (2019). A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs. J. Amer. Math. Soc. 32 311–349.
  • [25] Seiler, E. (2011). The strange world of non-amenable symmetries. In Mathematical Quantum Field Theory and Renormalization Theory. COE Lect. Note 30 66–77. Kyushu Univ. Fac. Math., Fukuoka.
  • [26] Shcherbina, M. and Shcherbina, T. (2018). Universality for 1d random band matrices: Sigma-model approximation. J. Stat. Phys. 172 627–664.
  • [27] Spencer, T. (2012). SUSY statistical mechanics and random band matrices. In Quantum Many Body Systems. Lecture Notes in Math. 2051 125–177. Springer, Heidelberg.
  • [28] Spencer, T. (2013). Duality, statistical mechanics, and random matrices. In Current Developments in Mathematics 2012 229–260. International Press, Somerville, MA.
  • [29] Spencer, T. and Zirnbauer, M. R. (2004). Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions. Comm. Math. Phys. 252 167–187.
  • [30] Symanzik, K. (1969). Euclidean quantum field theory. In Local Quantum Field Theory (R. Jost, ed.). Academic Press, New York.
  • [31] Sznitman, A.-S. (2012). Topics in Occupation Times and Gaussian Free Fields. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich.
  • [32] Zirnbauer, M. R. (1991). Fourier analysis on a hyperbolic supermanifold with constant curvature. Comm. Math. Phys. 141 503–522.