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Calculation of string correlation functions on a hierarchical tree
This article has been downloaded from IOPscience. Please scroll down to see the full text article. J. Stat. Mech. (2010) P03031 (http://iopscience.iop.org/1742-5468/2010/03/P03031) View the table of contents for this issue, or go to the journal homepage for more
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J
ournal of Statistical Mechanics: Theory and Experiment
Calculation of string correlation functions on a hierarchical tree 1
Yerevan Physic Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia 2 Institute of Physics, Academia Sonica, Nanking, Taipei 11529, Taiwan 3 National Centre for Theoretical Sciences: Physics Division, National Taiwan University, Taipei 10617, Taiwan E-mail:
[email protected] Received 26 December 2009 Accepted 4 March 2010 Published 26 March 2010 Online at stacks.iop.org/JSTAT/2010/P03031 doi:10.1088/1742-5468/2010/03/P03031
Abstract. String correlation functions in 2D gravity resemble similar objects in the statistical mechanics of directed polymer models on disordered trees. This analogy can be used to provide an approximate mapping between the two problems. Using such a mapping we derive renormalization group equations and calculate the ensuing scaling of the string correlation functions. We find it to agree with the results known for 2D gravity. The result further substantiates the similarity between the two models and shows that they share not only the free energy in the thermodynamic limit but also the gross structure of the correlation functions.
Keywords: spin glasses (theory)
c 2010 IOP Publishing Ltd and SISSA
1742-5468/10/P03031+11$30.00
J. Stat. Mech. (2010) P03031
David B Saakian1,2,3
Calculation of string correlation functions on a hierarchical tree
Contents 2
2. The multi-point correlation functions in the hierarchical tree model
4
3. Calculation of a two-point correlation function in the ultrametric space
5
4. Three-point correlation functions
5
5. Conclusion
7
Acknowledgments
8
Appendix: The generation function method 8 A.1. Two-point correlation functions . . . . . . . . . . . . . . . . . . . . . 8 A.2. Three-point correlation functions . . . . . . . . . . . . . . . . . . . . 10 References
10
1. Introduction An intriguing application of spin glasses outside their usual domain is the formal similarity noticed [1]–[3] between the partition function of the Liouville model of 2D quantum gravity and the statistical mechanics of directed polymer models on disordered trees [6, 9] which in turn is intimately related to the random energy model [4, 5, 8]. The main idea of [1]–[3] is to replace the free boson two-point correlation function in 2D which depends logarithmically on the Euclidean distance between the points on the plane with its analogue on a hierarchical tree which depends logarithmically on the hierarchical (i.e. ultrametric) distance between end points of the tree. Such a mapping between the two problems proved to be fruitful and further boosted interest in general properties of models with logarithmic correlations [10]–[15]; see also [16] for a general discussion. In [13] models have been introduced on special trees with a branching number close to 1, q → 1, and a renormalization equation has been derived, previously obtained in mathematical literature [17]. In [14], string models [18]–[24] have been connected with the models on a hierarchical tree. We took the string partition sum as a partition sum of some twodimensional field theory, and mapped this onto a similar model on a hierarchical tree. The scope of [14] was limited to the phase structure of the model. In the present work we investigate the correlation functions, which is a more involved task. Thus our goal is to compare the correlation functions of a continuous space with those in the model on a hierarchical tree to clarify the following point: how the model on the hierarchical tree resembles the original model in 2D. Again, as in the previous article, we take as a starting point the expression of string correlation functions after zero-mode integration. In the following sections we will review the string correlation functions in the conformal model approach, and then calculate their analogues in the hierarchical tree approach. doi:10.1088/1742-5468/2010/03/P03031
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J. Stat. Mech. (2010) P03031
1. Introduction
Calculation of string correlation functions on a hierarchical tree
General correlation functions in the string model. The following expression has been derived in the literature [25, 26] for describing K-point correlation functions in the string model: K αφ β φ(x ) 2 ZN ∼ Dg φ e l=1 l l exp − S(φ) − m d w gˆe , (1) 1 2 d w gˆ[φΔφ + QRφ]. S({φ(x)}) = − 8π
α2 − αQ + 2 = 0 Q > 2α.
(3)
For the spherical topology case, the zero-mode integration results in αφ s βl φ(zl ) 2 l ZN ∼ D g φ e exp[−S(φ)] d w gˆe .
(4)
Using φ(z1 )φ(z2 ) = 2 ln r(z1 , z2 )
(5)
where r(z1 , z2 ) ≡ |z1 − z2 | is the distance between two points in 2D space, one has for ZN [25, 26] N s 2 d2 wn |wn − zj |−2βn α |wn − wm |−2α (6) n=1
j=1
1≤n 2β.
(7)
We can map equation (3) onto certain models on a hierarchical tree. The last multiplier in equation (3) is the sth moment of a directed polymer partition sum. The bulk properties of the directed polymer models have been calculated for the case of a general number of branchings in [9], in the finite replica case, in [8]. The calculation of the correlation functions is a novel and non-trivial issue. Three-point string correlation functions. For N = 3, using the symmetry of the model, the following expression has been derived [25, 26]: Z3 = A3 |z1 − z2 |2(Δ3 −Δ1 −Δ2 ) |z1 − z3 |2(Δ2 −Δ1 −Δ3 ) |z3 − z2 |2(Δ1 −Δ3 −Δ2 ) doi:10.1088/1742-5468/2010/03/P03031
(8) 3
J. Stat. Mech. (2010) P03031
Here φ(w) is a field living on a closed two-dimensional surface, α, Q are certain parameters (real for d < 1), R is the surface curvature, and Dg φ is the measure. The parameters Q, α are determined by the value of d (space dimension) according to David–Distler–Kawai formulae (see review [24]–[26]): Q − l βl Q2 − 8 Q Q = (25 − d)/3; ; α= − (2) s= α 2 2 and there is an important relation:
Calculation of string correlation functions on a hierarchical tree
where one has the following scaling dimensions: Δi = βi (Q − βi )/2.
(9)
The most important issue in the latter equation is the scaling property. A3 was calculated in [25, 26] for the 2D model: s 2 d2 wn |wn |−2β1 α |1 − wn |−2β2 α |wm − wn |−2α . (10) A3 = n=1
1≤n s. Then ∞ dt tn−s−1 dn n fβ (teβx , α, v)2e2βx f (teβx , α, v)M1−2 x Z2 = (−1) n Γ(n − s) dt 0 where for the further calculations we can make the replacement M2 − 2 ≈ ev and 2 2 i∞ vα z 1 − z ln t , dz Γ(z) exp f (t, α, v) = 2π −i∞ 2 dn f (t, α, v) Γ(2u/vα2 ) −((u+vnα2 )/2vα2 ) vn2 α2 /2 e ≈e 1− √ dtn 2πvα 2 evβ /2 i∞ 2 2 dz Γ(z)ev(α z /2)−z(β+ln t) fβ (t, α, v) = 2π −i∞ dn fβ (t, α, v) Γ(2u + (β + nα)2 v/vα2 ) v(nα+β)2 /2 √ = ≈ e 1 − dtn 2πvα 2 2 × e−((2u+v(nα+β) )/8vα ) .
(A.3)
(A.4)
(A.5)
While calculating Z2 , we assume that f M2 ≈ 1. Then we should consider different possible differentiations in equation (A.1). When the differentiation is only in the term dn fβ (t, α)2 /dtn , we get ev[(β
2 /2)+((β+nα)2 /2)]
.
(A.6)
The integration via xi gives a multiplier 2 /2)(V
e((2β+nα)
−v)
.
(A.7)
Thus we get the result through equation (15). In principle we should calculate all other possibilities as well and choose the term giving the maximal contribution. While considering the paramagnetic phase, we should use equation (A.2) with M2 → eV , and look at the case when f M = 1 and 2 /2
f (t, α, v) ≈ 1 − tevα
.
(A.8)
Then we easily get the result through equation (17). doi:10.1088/1742-5468/2010/03/P03031
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J. Stat. Mech. (2010) P03031
First we integrate via yi . Let us consider the following version of the two-point correlation function in the correlated paramagnetic phase: ∞ dt tn−s−1 dn βE1 +βE2 −t i exp(αEi ) βE1 βE2 s n e e |xy . (A.2) Z2 = e e [Z(α)] = (−1) Γ(n − s) dtn 0
Calculation of string correlation functions on a hierarchical tree
A.2. Three-point correlation functions
Now we consider the three-level GREM, with branchings M1 = eV −v2 , M2 = ev2 −v1 and M3 = ev1 . v1 and v2 are defined after equation (18). Now Ei = xi + yu + hi , where x2i = V − v2 ;
yi2 = v2 − v1 ;
h2i = v1 .
(A.9)
We first average via hi : ∞ dt tn−s−1 dn n f (α, t, V − v2 )f (α, t, V − v2 )M1 −2 e(β1 +β2 )(h+y) Z3 = (−1) n β3 Γ(n − s) dt 0 × fβ1 (α, teβ1 (h+y) , v1 )fβ2 (α, teβ2 (h+y) , v1 )f (α, teβ1 (h+y) , v1 )M3 −2 .
(A.11)
We consider the differentiation of the last term. Then we get the result using equation (22). For the calculation of the paramagnetic phase we should consider f = 1. We should 2 just calculate the partition without vertex operators e(s+sα /2)V , later multiply by the exponent with Coulomb interaction of the charges eβ3 (β1 +β2 )(V −v3 )+β1 β2 (V −v1 ) . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
Chamon C C, Mudry C and Wen X G, 1996 Phys. Rev. Lett. 77 4194 Kogan I I, Mudry C and Tsvelik A M, 1996 Phys. Rev. Lett. 77 707 Castillo H E, Chamon C C, Fradkin E, Goldbart P M and Mudry C, 1997 Phys. Rev. B 56 10668 Derrida B, 1980 Phys. Rev. Lett. 45 79 Derrida B, 1981 Phys. Rev. B 24 2613 Gross D J and Mezard M, 1984 Nucl. Phys. B 240 431 Derrida B and Spohn H, 1988 J. Stat. Phys. 51 817 Cook J and Derrida B, 1991 J. Stat. Phys. 63 505 Gardner E and Derrida B, 1989 J. Phys. A: Math. Gen. 22 1975 Derrida B, Evans M R and Speer E R, 1993 Commun. Math. Phys. 15 221 Carpentier D and Le Doussal P, 2001 Phys. Rev. E 63 026110 Fyodorov Y V and Bouchaud J P, 2008 J. Phys. A: Math. Theor. 41 372001 Fyodorov Y V and Bouchaud J P, 2008 J. Phys. A: Math. Theor. 41 324009 Saakian D B, 2002 Phys. Rev. E 65 67104 Saakian D B, 2009 J. Stat. Mech. P07003 Fyodorov Y V, Le Doussal P and Rosso A, 2009 J. Stat. Mech. P10005 Fyodorov Y V, 2009 arXiv:0911.2765 Dorodnitsyn V A, The group properties and the invariant solutions of nonlinear heat transfer equation with a source, 1979 Preprint of Keldysh Inst. Appl. Math., Moscow Polyakov A M, 1981 Phys. Lett. B 103 207 Distler J and Kawai H, 1989 Nucl. Phys. B 321 504 David F, 1988 Mod. Phys. Lett. A 3 1651 Knizhnik V G, Polyakov A M and Zamolodchikov A B, 1988 Mod. Phys. Lett. A 3 819 Gupta A, Trivedi S and Wise M, 1990 Nucl. Phys. B 340 475 Dotsenko V and Fateev V, 1985 Nucl. Phys. B 251 691 Abdalla E, Abdalla M C B, Dalmazi D and Zadra A, 1994 2d-Gravity in Non-Critical Strings (Berlin: Springer) Goulian M and Li M, 1991 Phys. Rev. Lett. 66 2051 Dorn H and Otto H J, 1994 Nucl. Phys. B 429 375
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J. Stat. Mech. (2010) P03031
We consider the following expression for the three-point correlation function in the correlated paramagnetic phase: ∞ dt tn−s−1 dn β 1 E1 β 2 E2 β 3 E3 s n e e [Z(α)] = (−1) Z3 = e Γ(n − s) dtn 0 × eβ1 E1 +β2 E2 +β3 E3 e−t i exp(αEi ) |xyh . (A.10)
Calculation of string correlation functions on a hierarchical tree [27] Forrester P J and Warnaar S O, 2008 Bull. Am. Math. Soc. 45 489 [28] Fyodorov Y, 2009 private communication [29] Kozyrev S V, Methods and applications of ultrametric and p-adic analysis: from wavelet theory to biophysics, 2008 Sovrem. Probl. Mat. 12 (Steklov Math. Inst., RAS) [30] Avetisov V A, Bikulov A Kh and Kozyrev S V, 1999 J. Phys. A: Math. Gen. 32 8785 [31] Parisi G and Sourlas N, 2000 Eur. Phys. J. B 14 535 [32] Di Francesco P, Mathieu P and Senechal D, 1997 Conformal Field Theory (New York: Springer) [33] Zamolodchikov A and Zamolodchikov Al, 2001 arXiv:hep-th/0101152 [34] Saakian D B, 2000 arXiv:cond-mat/0004086
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