The Annals of Probability

Random curves on surfaces induced from the Laplacian determinant

Adrien Kassel and Richard Kenyon

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

Abstract

We define natural probability measures on finite multicurves (finite collections of pairwise disjoint simple closed curves) on curved surfaces. These measures arise as universal scaling limits of probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric, in the limit as the mesh size tends to zero. These in turn are defined from the Laplacian determinant and depend on the choice of a unitary connection on the surface.

Wilson’s algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence.

We set the framework for the study of these probability measures and their scaling limits and state some of their properties.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 932-964.

Dates
Received: October 2014
Revised: October 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947311

Digital Object Identifier
doi:10.1214/15-AOP1078

Mathematical Reviews number (MathSciNet)
MR3630290

Zentralblatt MATH identifier
1377.82037

Subjects
Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Laplacian cycle-rooted spanning forests loop-erased random walk scaling limit

Citation

Kassel, Adrien; Kenyon, Richard. Random curves on surfaces induced from the Laplacian determinant. Ann. Probab. 45 (2017), no. 2, 932--964. doi:10.1214/15-AOP1078. https://projecteuclid.org/euclid.aop/1490947311


Export citation

References

  • [1] Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419–453.
  • [2] Aizenman, M., Burchard, A., Newman, C. M. and Wilson, D. B. (1999). Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms 15 319–367.
  • [3] Aldous, D. J. (1983). On the time taken by random walks on finite groups to visit every state. Z. Wahrsch. Verw. Gebiete 62 361–374.
  • [4] Benoist, S. and Dubédat, J. (2016). An $\mathrm{SLE}_{2}$ loop measure. Ann. Inst. Henri Poincaré Probab. Stat. To appear. Available at arXiv:1405.7880.
  • [5] Biggs, N. (1997). Algebraic potential theory on graphs. Bull. Lond. Math. Soc. 29 641–682.
  • [6] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [7] Fock, V. and Goncharov, A. (2006). Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103 1–211.
  • [8] Forman, R. (1993). Determinants of Laplacians on graphs. Topology 32 35–46.
  • [9] Giné, E. and Koltchinskii, V. (2006). Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results. In High Dimensional Probability. Institute of Mathematical Statistics Lecture Notes—Monograph Series 51 238–259. IMS, Beachwood, OH.
  • [10] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2006). Determinantal processes and independence. Probab. Surv. 3 206–229.
  • [11] Itô, K. (1963). The Brownian motion and tensor fields on Riemannian manifold. In Proc. Internat. Congr. Mathematicians (Stockholm, 1962) 536–539. Inst. Mittag-Leffler, Djursholm.
  • [12] Ivashkevich, E. V., Ktitarev, D. V. and Priezzhev, V. B. (1994). Waves of topplings in an Abelian sandpile. Phys. A 209 347–360.
  • [13] Kassel, A., Kenyon, R. and Wu, W. (2015). Random two-component spanning forests. Ann. Inst. Henri Poincaré Probab. Stat. 51 1457–1464.
  • [14] Kassel, A. and Wu, W. (2015). Transfer current and pattern fields in spanning trees. Probab. Theory Related Fields 163 89–121.
  • [15] Kenyon, R. (2011). Spanning forests and the vector bundle Laplacian. Ann. Probab. 39 1983–2017.
  • [16] Kenyon, R. (2014). Conformal invariance of loops in the double-dimer model. Comm. Math. Phys. 326 477–497.
  • [17] Kenyon, R. W., Propp, J. G. and Wilson, D. B. (2000). Trees and matchings. Electron. J. Combin. 7 Research Paper 25, 34 pp. (electronic).
  • [18] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939–995.
  • [19] Lawler, G. F. and Trujillo Ferreras, J. A. (2007). Random walk loop soup. Trans. Amer. Math. Soc. 359 767–787 (electronic).
  • [20] Lyndon, R. C. and Schupp, P. E. (2001). Combinatorial Group Theory. Springer, Berlin.
  • [21] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
  • [22] Wehn, D. (1962). Probabilities on Lie groups. Proc. Natl. Acad. Sci. USA 48 791–795.
  • [23] Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996) 296–303. ACM, New York.