Boundary correlations in planar LERW and UST. (English) Zbl 1441.82023
Summary: We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. We show that both probabilities, when suitably renormalized, converge in the scaling limit to conformally covariant functions which satisfy partial differential equations of second and third order, as predicted by conformal field theory. The scaling limit connectivity probabilities also provide formulas for the pure partition functions of multiple \(\text{SLE}_\kappa\) at \(\kappa =2\).
MSC:
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 05C05 | Trees |
| 60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |
References:
| [1] | Bauer, M., Bernard, D.: SLE, CFT and zig-zag probabilities. In: Proceedings of the Conference ‘Conformal Invariance and Random Spatial Processes’, Edinburgh (2003) |
| [2] | Bauer, Michel; Bernard, Denis; Kytölä, Kalle, Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales, Journal of Statistical Physics, 120, 5-6, 1125-1163 (2005) · Zbl 1094.82016 |
| [3] | Beffara, V., Peltola, E., Wu, H.: On the uniqueness of global multiple SLEs. Preprint. arXiv:1801.07699 (2018) · Zbl 1478.60225 |
| [4] | Belavin, AA; Polyakov, AM; Zamolodchikov, AB, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B, 241, 2, 333-380 (1984) · Zbl 0661.17013 |
| [5] | Belavin, AA; Polyakov, AM; Zamolodchikov, AB, Infinite conformal symmetry of critical fluctuations in two dimensions, J. Stat. Phys., 34, 5-6, 763-774 (1984) |
| [6] | Beneš, C.; Lawler, GF; Viklund, F., Scaling limit of the loop-erased random walk Green’s function, Probab. Theory Related Fields, 166, 1, 271-319 (2016) · Zbl 1362.82026 |
| [7] | Benoit, L.; Saint-Aubin, Y., Degenerate conformal field theories and explicit expressions for some null vectors, Phys. Lett., B215, 3, 517-522 (1988) · Zbl 0957.17509 |
| [8] | Chelkak, D.; Smirnov, S., Discrete complex analysis on isoradial graphs, Adv. Math., 228, 3, 1590-1630 (2011) · Zbl 1227.31011 |
| [9] | Chelkak, D.; Smirnov, S., Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., 189, 3, 515-580 (2012) · Zbl 1257.82020 |
| [10] | Courant, R.; Friedrichs, K.; Lewy, H., Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100, 1, 32-74 (1928) · JFM 54.0486.01 |
| [11] | Dubédat, J., Euler integrals for commuting SLEs, J. Stat. Phys., 123, 6, 1183-1218 (2006) · Zbl 1113.82064 |
| [12] | Dubédat, J., Excursion decompositions for SLE and Watts’ crossing formula, Probab. Theory Rel. Fields, 134, 3, 453-488 (2006) · Zbl 1112.60032 |
| [13] | Dubédat, J., Commutation relations for SLE, Commun. Pure Appl. Math., 60, 12, 1792-1847 (2007) · Zbl 1137.82009 |
| [14] | Dubédat, J., SLE and Virasoro representations: localization, Commun. Math. Phys., 336, 2, 695-760 (2015) · Zbl 1318.82007 |
| [15] | Dubédat, J., SLE and Virasoro representations: fusion, Commun. Math. Phys., 336, 2, 761-809 (2015) · Zbl 1319.81073 |
| [16] | Felder, G., BRST approach to minimal models, Nucl. Phys. B, 317, 1, 215-236 (1989) |
| [17] | Feĭgin, B.L., Fuchs, D.B.: Representations of the Virasoro algebra. In: Representation of Lie Groups and Related Topics, Volume 7 of Advanced Studies in Contemporary Mathematics, pp. 465-554. Gordon and Breach, New York (1990) · Zbl 0722.17020 |
| [18] | Felder, G.; Fröhlich, J.; Keller, G., Braid matrices and structure constants for minimal conformal models, Commun. Math. Phys., 124, 4, 647-664 (1989) · Zbl 0696.17009 |
| [19] | Flores, S.M., Peltola, E.: Monodromy invariant CFT correlation functions of first column Kac operators. In preparation (2019) |
| [20] | Flores, SM; Kleban, P., A solution space for a system of null-state partial differential equations, part I, Commun. Math. Phys., 333, 1, 389-434 (2015) · Zbl 1314.35188 |
| [21] | Flores, SM; Kleban, P., A solution space for a system of null-state partial differential equations, part II, Commun. Math. Phys., 333, 1, 435-481 (2015) · Zbl 1314.35189 |
| [22] | Flores, SM; Kleban, P., A solution space for a system of null-state partial differential equations, part III, Commun. Math. Phys., 333, 2, 597-667 (2015) · Zbl 1311.35314 |
| [23] | Fomin, S., Loop-erased walks and total positivity, Trans. Am. Math. Soc., 353, 9, 3363-3583 (2001) · Zbl 0973.15014 |
| [24] | Di Francesco, P.; Mathieu, P.; Sénéchal, D., Conformal Field Theory (1997), Berlin: Springer, Berlin · Zbl 0869.53052 |
| [25] | Gessel, I.; Viennot, G., Binomial determinants, paths, and hook length formulae, Adv. Math., 58, 3, 300-321 (1985) · Zbl 0579.05004 |
| [26] | Iohara, K.; Koga, Y., Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics (2011), Berlin: Springer, Berlin · Zbl 1222.17001 |
| [27] | Jokela, N.; Järvinen, M.; Kytölä, K., SLE boundary visits, Ann. Henri Poincaré, 17, 6, 1263-1330 (2016) · Zbl 1346.82012 |
| [28] | Kac, V., Highest weight representations of infinite dimensional Lie algebras, Proc. ICM Helsinki, 1978, 299-304 (1980) · Zbl 0425.17009 |
| [29] | Kager, W.; Nienhuis, B., A guide to stochastic Löwner evolution and its applications, J. Stat. Phys., 115, 5, 1149-1229 (2004) · Zbl 1157.82327 |
| [30] | Karlin, S.; McGregor, J., Coincidence probabilities, Pac. J. Math., 9, 4, 1141-1164 (1959) · Zbl 0092.34503 |
| [31] | Karrila, A., Kytölä, K., Peltola, E.: Conformal blocks, \(q\)-combinatorics, and quantum group symmetry. Annales de l’Institut Henri Poincaré D (2019) · Zbl 1432.81054 |
| [32] | Karrila, A.: Multiple SLE type scaling limits: from local to global. Preprint arXiv:1903.10354 (2019) |
| [33] | Kenyon, R., The asymptotic determinant of the discrete Laplacian, Acta Math., 185, 2, 239-286 (2000) · Zbl 0982.05013 |
| [34] | Kenyon, RW; Wilson, DB, Boundary partitions in trees and dimers, Trans. Am. Math. Soc., 363, 3, 1325-1364 (2011) · Zbl 1230.60009 |
| [35] | Kenyon, RW; Wilson, DB, Double-dimer pairings and skew Young diagrams, Electr. J. Comb., 18, 1, 130-142 (2011) · Zbl 1247.05025 |
| [36] | Kenyon, RW; Wilson, DB, Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs, J. Am. Math. Soc., 28, 4, 985-1030 (2015) · Zbl 1327.60033 |
| [37] | Kim, JS, Proofs of two conjectures of Kenyon and Wilson on Dyck tilings, J. Combin. Theory Ser. A, 119, 8, 1692-1710 (2012) · Zbl 1246.05039 |
| [38] | Kim, JS; Mészáros, K.; Panova, G.; Wilson, DB, Dyck tilings, increasing trees, descents, and inversions, J. Combin. Theory Ser. A, 122, C, 9-27 (2014) · Zbl 1311.05207 |
| [39] | Kozdron, M.J., Lawler, G.F.: The configurational measure on mutually avoiding SLE paths. In: Universality and Renormalization: From Stochastic Evolution to Renormalization of Quantum Fields, Fields Inst. Commun. Amer. Math. Soc., New York (2007) |
| [40] | Kytölä, K.; Peltola, E., Pure partition functions of multiple SLEs, Commun. Math. Phys., 346, 1, 237-292 (2016) · Zbl 1358.82012 |
| [41] | Kytölä, K., Peltola, E.: Conformally covariant boundary correlation functions with a quantum group. J. Eur. Math. Soc. (2019) · Zbl 1454.81112 |
| [42] | Lawler, G.F.: Conformally Invariant Processes in the Plane. American Mathematical Society, New York (2005) · Zbl 1074.60002 |
| [43] | Lawler, Gregory F., Self-Avoiding Walks, Intersections of Random Walks, 163-181 (1991), Boston, MA: Birkhäuser Boston, Boston, MA · Zbl 1228.60004 |
| [44] | Lawler, GF, The probability that planar loop-erased random walk uses a given edge, Electron. Commun. Probab., 19, 1-13 (2014) · Zbl 1303.82027 |
| [45] | Lawler, GF; Schramm, O.; Werner, W., Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab., 32, 1, 939-995 (2004) · Zbl 1126.82011 |
| [46] | Lawler, G.F., Viklund, F.: Convergence of loop-erased random walk in the natural parametrization. Preprint arXiv:1603.05203 (2016) · Zbl 1362.82026 |
| [47] | Lenells, J.; Viklund, F., Schramm’s formula and the Green’s function for multiple SLE, J. Stat. Phys., 176, 4, 873-931 (2019) · Zbl 1435.30122 |
| [48] | Lindström, B., On the vector representations of induced matroids, Bull. Lond. Math. Soc., 5, 1, 85-90 (1973) · Zbl 0262.05018 |
| [49] | Miller, J.; Sheffield, S., Imaginary geometry II: reversibility of \(\text{SLE}_\kappa (\rho_1; \rho_2)\) for \(\kappa \in (0,4)\), Ann. Probab., 44, 3, 1647-1722 (2016) · Zbl 1344.60078 |
| [50] | Panova, G.; Wilson, DB, Pfaffian formulas for spanning tree probabilities, Combin. Probab. Comput., 26, 1, 118-137 (2017) · Zbl 1376.82021 |
| [51] | Peltola, E.: Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties. Ann. Inst. H. Poincaré D (2019) · Zbl 1437.81082 |
| [52] | Peltola, E.; Wu, H., Global and local multiple SLEs for \(\kappa \le 4\) and connection probabilities for level lines of GFF, Commun. Math. Phys., 366, 2, 469-536 (2019) · Zbl 1422.60142 |
| [53] | Pemantle, R., Choosing a spanning tree for the integer lattice uniformly, Ann. Probab., 19, 4, 1559-1574 (1991) · Zbl 0758.60010 |
| [54] | Poncelet, A., Schramm’s formula for multiple loop-erased random walks, J. Stat. Mech. Theory Exp., 2018, 103106 (2018) · Zbl 1457.82158 |
| [55] | Ribault, S.: Conformal field theory on the plane. arXiv:1406.4290 (2014) |
| [56] | Rohde, S.; Schramm, O., Basic properties of SLE, Ann. Math., 161, 2, 883-924 (2005) · Zbl 1081.60069 |
| [57] | Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees, Isr. J. Math., 118, 1, 221-288 (2000) · Zbl 0968.60093 |
| [58] | Schramm, O.; Zhou, W., Boundary proximity of SLE, Probab. Theory Relat. Fields, 146, 3-4, 435-450 (2010) · Zbl 1227.60101 |
| [59] | Sheffield, S.; Wilson, DB, Schramm’s proof of Watts’ formula, Ann. Probab., 39, 5, 1844-1863 (2011) · Zbl 1238.60089 |
| [60] | Shigechi, K.; Zinn-Justin, P., Path representation of maximal parabolic Kazhdan-Lusztig polynomials, J. Pure Appl. Algebra, 216, 11, 2533-2548 (2012) · Zbl 1262.20006 |
| [61] | Wilson, D.: Generating random spanning trees more quickly than the cover time. In: Proceeding of the 28th Annual ACM Symposium on the Theory of Computing, pp. 296-303 (1996) · Zbl 0946.60070 |
| [62] | Wu, H.: Hypergeometric SLE: conformal Markov characterization and applications. Preprint arXiv:1703.02022v4 (2018) · Zbl 1473.82021 |
| [63] | Yadin, A.; Yehudayoff, A., Loop-erased random walk and Poisson kernel on planar graphs, Ann. Probab., 39, 4, 1243-1285 (2011) · Zbl 1234.60036 |
| [64] | Zhan, D., The scaling limits of planar LERW in finitely connected domains, Ann. Probab., 36, 2, 467-529 (2008) · Zbl 1153.60057 |
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