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Loop models, random matrices and planar algebras. (English) Zbl 1277.82013

A Temperley-Lieb element is a rectangle with an even number of points on the boundary and non-intersecting strings connecting the points pairwise. Each region, separated by nonintersecting strings, is colored black or white, and one point on the boundary is marked. Adjacent regions have different colors.
Let \(S_1,S_2,\dotsc, S_k\) and \(S\) be fixed Temperley-Lieb elements, and let \(t_1,t_2,\dotsc, t_k\) be real numbers. Consider the loop model \[ {\mathcal J}_t(S)= \sum_{r_i\geq 0}\prod^k_{i=1} {t^{r_i}_i\over r_i!}\,\sum \delta^{\#\mathrm{loops}}, \] where the sum is over all tangles which are non-intersecting strings on \[ S^2- \Biggl(\bigcup^{r_1}_{i=1} S_1\cup\dotsb\cup \bigcup^{r_k}_{i=1} S_k\cup S\Biggr), \] and the non-intersecting strings connect the boundary points on \(r_1\) Temperley-Lieb elements of the kind \(S_1,\dotsc, r_ k\) Temperley-Lieb elements of the kind \(S_k\) and the Temperley-Lieb element \(S\). (\(S^2\) is the standard two-dimensional sphere.) The non-intersecting strings connect the boundary points so that the black and white colors are compatible, i.e., the black and white regions match. Then, the authors prove that the loop model \({\mathcal J}_t(S)\) is equal to a matrix model provided that \[ \delta\in \Biggl\{2\cos\Biggl({\pi\over p}\Biggr)\Biggr\}_{p\geq 3}\cup [2,+\infty) \] and the numbers \(t_i\) are small. The matrix model is the limit of the integral over a Cartesian product of rectangular matrices as the size of the matrices go to infinity. The integrand is explicitly written down as perturbations of independent Gaussian measures. The subtlety of this construction can be seen in that \(\delta\) is an eigenvalue of the adjacency matrix of a certain bipartite graph. The corresponding eigenvector is used in the construction of this matrix model.
Consider the special case in which the Temperley-Lieb elements \(S_1\) and \(S_2\) are given as follows. \(S_1\) has two non-intersecting strings connecting four boundary points. \(S_1\) has one white region, and two black regions. \(S_2\) has two non-intersecting strings connecting four boundary points. \(S_2\) has one black region, and two white regions. Also, consider the Temperley-Lieb element \(B_n\) given as follows. \(B_n\) has \(n\) non-nested, non-intersecting strings connecting \(2n\) boundary points. There are \(n\) black regions, and one white region. Let \[ {\mathcal J}_t(B_n)= \sum^\infty_{r_1,r_2= 0} {t^{r_1}_1 t^{r_2}_2\over r_1!r_2!} \sum \delta^{\#\text{loops}}, \] where the sum is over all tangles which are non-intersecting strings on \[ S^2-\Biggl(\bigcup^{r_1}_{i=1} S_1\cup \bigcup^{r_2}_{i=1} S_2\cup B_n\Biggr), \] and the non-intersecting strings connect the boundary points on \(r_1\) Temperley-Lieb elements \(S_1\), \(r_2\) Temperley-Lieb elements \(S_2\) and one Temperley-Lieb element \(B_n\). The non-intersecting strings connect the points on the boundary so that the black and white colors are compatible, i.e., the black and white regions match. They explicitly evaluate the matrix model corresponding to this loop model \({\mathcal J}_t(B_n)\) using the Hubbard-Stratonovich transformation and the saddle-point technique in the large matrices limit. Consequently, the infinite series \[ C(\gamma,t)= \sum^\infty_{n=0} \gamma^n{\mathcal J}_t(B_n), \] \(\gamma\in\mathbb{C}\), is shown to be analytic in \(\gamma\) for small \(\gamma\). Moreover, they provide an explicit description of this function.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
46L53 Noncommutative probability and statistics

References:

[1] Anderson, G., Guionnet, A., Zeitouni, O.: An introduction to random matrices. Cambridge studies in advanced mathematics, Vol. 118, Cambridge: Cambridge Univ. Press, 2010 · Zbl 1184.15023
[2] Ben Arous G., Guionnet A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Th. Rel. Fields 108(4), 517–542 (1997) · Zbl 0954.60029 · doi:10.1007/s004400050119
[3] Baxter, R.J.: Exactly solved models in statistical mechanics. New York: Academic Press, 1982 · Zbl 0538.60093
[4] Bernardi O., Bousquet-Mélou M.: Counting colored planar maps: algebraicity results. J. Comb. Theory Series B 101(5), 315–377 (2011) · Zbl 1223.05123 · doi:10.1016/j.jctb.2011.02.003
[5] Borot, G., Eynard, B.: Enumeration of maps with self avoiding loops and the o(n) model on random lattices of all topologies. J. Stat. Mech. 2011, P01010 (2011)
[6] Brézin E., Itzykson C., Parisi G., Zuber J.-B.: Planar diagrams. Commun. Math. Phys. 59(1), 35–51 (1978) · Zbl 0997.81548 · doi:10.1007/BF01614153
[7] Curran, S., Jones, V.F.R., Shlyakhtenko, D.: On the symmetric enveloping algebra of planar algebra subfactors. http://arxiv.org/abs/1105.1721,2011v1 , [math.OA], 2011 · Zbl 1296.46054
[8] Di Francesco P., Golinelli O., Guitter E.: Meanders: exact asymptotics. Nucl. Phys. B 570(3), 699–712 (2000) · Zbl 0984.82024 · doi:10.1016/S0550-3213(99)00753-1
[9] Di Francesco P., Ginsparg P., Zinn-Justin J.: 2D gravity and random matrices. Phys. Rep. 254(1–2), 1–133 (1995) · doi:10.1016/0370-1573(94)00084-G
[10] Duplantier B., Kostov I.: Conformal spectra of polymers on a random surface. Phys. Rev. Lett. 61(13), 1433–1437 (1988) · doi:10.1103/PhysRevLett.61.1433
[11] Eynard B., Kristjansen C.: Exact solution of the O(n) model on a random lattice. Nucl. Phys. B 455(3), 577–618 (1995) · Zbl 0925.81129 · doi:10.1016/0550-3213(95)00469-9
[12] Guionnet, A., Jones, V.F.R., Shlyakhtenko, D.: Random matrices, free probability, planar algebras and subfactors. Quanta of Maths, Clay Mathematics Proceedings, 11, American Math. Society, Providence, pp 201–239 (2010) · Zbl 1219.46057
[13] Guionnet A., Maurel-Segala E.: Combinatorial aspects of matrix models. ALEA Lat. Am. J. Probab. Math. Stat. 1, 241–279 (2006) · Zbl 1110.15021
[14] Guionnet A., Shlyakhtenko D.: Free diffusions and matrix models with strictly convex interaction. Geom. Funct. Anal. 18(6), 1875–1916 (2009) · Zbl 1187.46056 · doi:10.1007/s00039-009-0704-0
[15] Guionnet, A.: Large random matrices: lectures on macroscopic asymptotics. Lecture Notes in Mathematics, Vol. 1957, Berlin: Springer-Verlag, 2009, Lectures from the 36th Probability Summer School held in Saint-Flour, 2006 · Zbl 1101.15024
[16] Jones V.F.R.: Index for subfactors. Invent. Math. 72(1), 1–25 (1983) · Zbl 0508.46040 · doi:10.1007/BF01389127
[17] Jones, V.F.R.: Planar algebras. http://arxiv.org/abs/math/990927v1 [math.QA, 1999]
[18] Jones, V.F.R.: The planar algebra of a bipartite graph. Knots in Hellas ’98 (Delphi), Ser. Knots Everything, Vol. 24, River Edge, NJ: World Sci. Publ., 2000, pp. 94–117
[19] Jones, V.F.R., Penneys, D.: The embedding theorem for finite depth subfactor planar algebras. http://arxiv.org/abs/1007.3173v1 [math.OA, 2010] · Zbl 1230.46055
[20] Kostov I.: The ADE face models on a fluctuating planar lattice. Nucl. Phys. B 326(3), 583–612 (1989) · doi:10.1016/0550-3213(89)90545-2
[21] Kostov I.K.: O(n) vector model on a planar random lattice: spectrum of anomalous dimensions. Mod. Phys. Lett. A 4(3), 217–226 (1989) · doi:10.1142/S0217732389000289
[22] Kostov I.: Strings with discrete target space. Nucl. Phys. B 376(3), 539–598 (1992) · doi:10.1016/0550-3213(92)90120-Z
[23] Kostov, I.: Solvable statistical models on a random lattice. Nucl. Phys. B Proc. Suppl. 45A, 13–28 (1996), Recent developments in statistical mechanics and quantum field theory (Trieste, 1995) · Zbl 0989.82507
[24] Kostov I., Staudacher M.: Multicritical phases of the O(n) model on a random lattice. Nucl. Phys. B 384(3), 459–483 (1992) · doi:10.1016/0550-3213(92)90576-W
[25] Pasquier V.: Two-dimensional critical systems labelled by Dynkin diagrams. Nuclear Phys. B 285(1), 162–172 (1987) · doi:10.1016/0550-3213(87)90332-4
[26] Temperley, H.N.V., Lieb. E.H.: Relations between the ”percolation” and ”colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ”percolation” problem. Proceedings of the Royal Society A 322, 251–280 (1971) · Zbl 0211.56703
[27] Voiculescu, D.-V., Dykema, K., Nica, A.: Free random variables. CRM monograph series, Vol. 1, Providence, RI: Amer. Math. Soc., 1992 · Zbl 0795.46049
[28] Voiculescu D.-V.: Limit laws for random matrices and free products. Invent. math 104, 201–220 (1991) · Zbl 0736.60007 · doi:10.1007/BF01245072
[29] Voiculescu D.-V.: The analogues of entropy and of Fisher’s information measure in free probability, V. Invent. Math. 132, 189–227 (1998) · Zbl 0930.46053 · doi:10.1007/s002220050222
[30] Voiculescu D.-V.: Free entropy. Bull. London Math. Soc. 34(3), 257–278 (2002) · Zbl 1036.46051 · doi:10.1112/S0024609301008992
[31] Voiculescu D.V.: Cyclomorphy. Int. Math. Research Notices No. 6, 299–332 (2002) · Zbl 1027.46089 · doi:10.1155/S1073792802105046
[32] Voiculescu, D.-V.: Symmetries arising from free probability theory. In: Frontiers in Number Theory, Physics, and Geometry I P. Cartier, B. Julia, P. Moussa, P. Vanhove, eds., Berlin-Heidelberg: Springer, 2006, pp. 231–243 · Zbl 1155.46036
[33] Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge: Cambridge Univ. Press, 1996 · Zbl 0951.30002
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