Spin systems from loop soups. (English) Zbl 1400.82107
Summary: We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system \(\operatorname{sgn} (\varphi)\) where \(\varphi\) is a discrete Gaussian free field.
In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by F. Camia et al. [Nucl. Phys., B 902, 483–507 (2016; Zbl 1332.82083)] and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.
In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by F. Camia et al. [Nucl. Phys., B 902, 483–507 (2016; Zbl 1332.82083)] and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.
MSC:
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 60G60 | Random fields |
| 60G18 | Self-similar stochastic processes |
| 60J65 | Brownian motion |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
Citations:
Zbl 1332.82083References:
| [1] | C. Beneš, G. F. Lawler, and F. Viklund, Scaling limit of the loop-erased random walk Green’s function, Probability Theory and Related Fields 166 (2016), no. 1, 271–319. · Zbl 1362.82026 |
| [2] | S. Benoist and C. Hongler, The scaling limit of critical Ising interfaces is CLE\(_3\), arXiv:1604.06975, 2016. |
| [3] | M. Biskup, Reflection Positivity and Phase Transitions in Lattice Spin Models, pp. 1–86, Springer Berlin Heidelberg, Berlin, Heidelberg, 2009. · Zbl 1180.82041 |
| [4] | C. Boutillier and B. de Tilière, Height representation of XOR-Ising loops via bipartite dimers, Electron. J. Probab. 19 (2014), no. 80, 33. · Zbl 1302.82014 |
| [5] | F. Camia, Scaling limits, Brownian loops, and conformal fields, Advances in disordered systems, random processes and some applications, Cambridge Univ. Press, Cambridge, 2017, pp. 205–269. |
| [6] | F. Camia, A. Gandolfi, and M. Kleban, Conformal correlation functions in the Brownian loop soup, Nuclear Physics B 902 (2016), 483–507. · Zbl 1332.82083 |
| [7] | F. Camia and M. Lis, Non-backtracking loop soups and statistical mechanics on spin networks, Annales Henri Poincaré 18 (2017), no. 2, 403–433. · Zbl 1366.82020 |
| [8] | A. Comtet, J. Desbois, and S. Ouvry, Winding of planar Brownian curves, J. Phys. A 23 (1990), no. 15, 3563–3572. · Zbl 0714.60066 · doi:10.1088/0305-4470/23/15/027 |
| [9] | J. Dubédat, SLE and the free field: Partition functions and couplings, Journal of the American Mathematical Society 22 (2009), no. 4, 995–1054. · Zbl 1204.60079 |
| [10] | J. Dubédat, Exact bosonization of the Ising model, arXiv:1112.4399, 2011. |
| [11] | H. Duminil-Copin, Random currents expansion of the Ising model, arXiv:1607.06933, 2016. |
| [12] | H. Duminil-Copin and M. Lis, On the double random current nesting field, arXiv:1712.02305, 2017. |
| [13] | E. B. Dynkin, Gaussian and non-Gaussian random fields associated with Markov processes, J. Funct. Anal. 55 (1984), no. 3, 344–376. · Zbl 0533.60061 · doi:10.1016/0022-1236(84)90004-1 |
| [14] | E. B. Dynkin, Local times and quantum fields, Seminar on stochastic processes, 1983 (Gainesville, Fla., 1983), Progr. Probab. Statist., vol. 7, Birkhäuser Boston, Boston, MA, 1984, pp. 69–83. · Zbl 0554.60058 |
| [15] | R. G. Edwards and A. D. Sokal, Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm, Phys. Rev. D 38 (1988), 2009–2012. |
| [16] | C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model, Physica 57 (1972), no. 4, 536–564. |
| [17] | B. Freivogel and M. Kleban, A Conformal Field Theory for Eternal Inflation, JHEP 0912 (2009), 019. |
| [18] | C. Garban and J. A. Trujillo Ferreras, The expected area of the filled planar Brownian loop is \(π /5\), Comm. Math. Phys. 264 (2006), no. 3, 797–810. · Zbl 1107.82023 |
| [19] | D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Communications in Partial Differential Equations 27 (2002), no. 7-8, 1283–1299. · Zbl 1034.35085 |
| [20] | R. B. Griffiths, Correlations in Ising Ferromagnets. I, Journal of Mathematical Physics 8 (1967), no. 3, 478–483. |
| [21] | R. B. Griffiths, C. A. Hurst, and S. Sherman, Concavity of Magnetization of an Ising Ferromagnet in a Positive External Field, Journal of Mathematical Physics 11 (1970), no. 3, 790–795. |
| [22] | G. Grimmett, The random-cluster model, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 333, Springer-Verlag, Berlin, 2006. · Zbl 1122.60087 |
| [23] | E. Ising, Beitrag zur Theorie des Ferromagnetismus, Z. Physik 31 (1925), 253–258. · Zbl 1439.82056 |
| [24] | D. G. Kelly and S. Sherman, General Griffiths’ Inequalities on Correlations in Ising Ferromagnets, Journal of Mathematical Physics 9 (1968), no. 3, 466–484. |
| [25] | R. Kenyon, Dominos and the Gaussian Free Field, Ann. Probab. 29 (2001), no. 3, 1128–1137. · Zbl 1034.82021 · doi:10.1214/aop/1015345599 |
| [26] | H. A. Kramers and G. H. Wannier, Statistics of the Two-Dimensional Ferromagnet. Part I, Phys. Rev. 60 (1941), 252–262. · Zbl 0027.28505 |
| [27] | G. F. Lawler, Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005. · Zbl 1074.60002 |
| [28] | G. F. Lawler and J. A. Trujillo Ferreras, Random walk loop soup, Trans. Amer. Math. Soc. 359 (2007), no. 2, 767–787 (electronic). · Zbl 1120.60037 |
| [29] | G. F. Lawler and W. Werner, The Brownian loop soup, Probab. Theory Related Fields 128 (2004), no. 4, 565–588. · Zbl 1049.60072 |
| [30] | Y. Le Jan, Markov paths, loops and fields, Lecture Notes in Mathematics, vol. 2026, Springer, Heidelberg, 2011, Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School]. |
| [31] | Y. Le Jan, Markov loops, coverings and fields, arXiv:1602.02708, 2016. |
| [32] | M. Ledoux and M. Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991, Isoperimetry and processes. · Zbl 0748.60004 |
| [33] | T. Lupu, Convergence of the two-dimensional random walk loop soup clusters to CLE, to appear in Journal of the European Mathematical Society, 2015. |
| [34] | T. Lupu, From loop clusters and random interlacements to the free field, Ann. Probab. 44 (2016), no. 3, 2117–2146. · Zbl 1348.60141 · doi:10.1214/15-AOP1019 |
| [35] | T. Lupu and W. Werner, A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field, Electron. Commun. Probab. 21 (2016), 7 pp. · Zbl 1338.60236 |
| [36] | J. Miller and S. Sheffield, Imaginary geometry I: interacting SLEs, Probability Theory and Related Fields 164 (2016), no. 3, 553–705. · Zbl 1336.60162 · doi:10.1007/s00440-016-0698-0 |
| [37] | Ş. Nacu and W. Werner, Random soups, carpets and fractal dimensions, J. Lond. Math. Soc. (2) 83 (2011), no. 3, 789–809. · Zbl 1223.28012 · doi:10.1112/jlms/jdq094 |
| [38] | S. Ouvry, Anyon model, axial anomaly and planar Brownian winding, Nuclear Phys. B Proc. Suppl. 18B (1990), 250–258 (1991), Recent advances in field theory (Annecy-le-Vieux, 1990). · Zbl 0957.82500 |
| [39] | J. Pitman and M. Yor, Asymptotic laws of planar brownian motion, Ann. Probab. 14 (1986), no. 3, 733–779. · Zbl 0607.60070 |
| [40] | Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. · Zbl 0762.30001 |
| [41] | O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees [mr1776084], Selected works of Oded Schramm. Volume 1, 2, Sel. Works Probab. Stat., Springer, New York, 2011, pp. 791–858. |
| [42] | O. Schramm and S. Sheffield, Contour lines of the two-dimensional discrete Gaussian free field, Acta Mathematica 202 (2009), no. 1, 21. · Zbl 1210.60051 |
| [43] | S. Sheffield, Exploration trees and conformal loop ensembles, Duke Math. J. 147 (2009), no. 1, 79–129. · Zbl 1170.60008 · doi:10.1215/00127094-2009-007 |
| [44] | S. Sheffield and W. Werner, Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Ann. of Math. (2) 176 (2012), no. 3, 1827–1917. · Zbl 1271.60090 · doi:10.4007/annals.2012.176.3.8 |
| [45] | F. Spitzer, Some theorems concerning \(2\)-dimensional Brownian motion, Trans. Amer. Math. Soc. 87 (1958), 187–197. · Zbl 0089.13601 |
| [46] | K. Symanzik, Euclidean Quantum Field Theory. I. Equations for a Scalar Model, Journal of Mathematical Physics 7 (1966), no. 3, 510–525. |
| [47] | T. van de Brug, F. Camia, and M. Lis, Random walk loop soups and conformal loop ensembles, Probability Theory and Related Fields 166 (2016), no. 1, 553–584. · Zbl 1357.60049 |
| [48] | W. Werner, The conformally invariant measure on self-avoiding loops, J. Amer. Math. Soc. 21 (2008), no. 1, 137–169. · Zbl 1130.60016 · doi:10.1090/S0894-0347-07-00557-7 |
| [49] | W. Werner, On the Spatial Markov Property of Soups of Unoriented and Oriented Loops, pp. 481–503, Springer International Publishing, 2016. · Zbl 1370.60192 |
| [50] | H. Weyl, Ueber die asymptotische Verteilung der Eigenwerte, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1911 (1911), 110–117. · JFM 42.0432.03 |
| [51] | M. Yor, Loi de l’indice du lacet brownien, et distribution de Hartman-Watson, Z. Wahrsch. Verw. Gebiete 53 (1980), no. 1, 71–95. · Zbl 0436.60057 · doi:10.1007/BF00531612 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
