Elsevier

Nuclear Physics B

Volume 663, Issue 3, 28 July 2003, Pages 535-567
Nuclear Physics B

Geodesic distance in planar graphs

https://doi.org/10.1016/S0550-3213(03)00355-9Get rights and content

Abstract

We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection with decorated trees, leading to a recursion relation on the geodesic distance. The latter is solved exactly in terms of discrete soliton-like expressions, suggesting an underlying integrable structure. We extract from this solution the fractal dimensions at the various (multi)-critical points, as well as the precise scaling forms of the continuum two-point functions and the probability distributions for the geodesic distance in (multi)-critical random surfaces. The two-point functions are shown to obey differential equations involving the residues of the KdV hierarchy.

Introduction

The study of the statistical properties of random graphs is relevant for many problems in physics, such as two-dimensional quantum gravity or fluid membrane statistics. By random graphs, we here mean graphs embedded in a surface of given genus, also known as fatgraphs or maps. Indeed, these are the natural discretization of fluctuating surfaces, on which matter systems may be defined. Much is known to this day on the enumeration of maps of fixed genus either by combinatorial techniques [1] or by the use of matrix integrals [2], [3], [4]. When applied to graphs of large size, these results give rise to various scaling behaviors depending on the critical universality class at hand, described in the continuum by the coupling to 2D quantum gravity of some 2D conformal field theories (CFT) with central charges c<1. In particular, the critical behaviors are characterized by the famous KPZ scaling relations [5]. As an example, the so-called one-matrix model, which enumerates fatgraphs with vertices of arbitrary valence and of any fixed topology, displays a set of multicritical points corresponding to the CFT's with central charges c(2,2m+1), m=1,2,3,…. The first value m=1 corresponds to c=0, i.e., a model of pure 2D quantum gravity without matter, representing the universality class of generic random maps of large size. The higher order multicritical points may be reached by weighting vertices according to their valence, and by suitably fine-tuning the vertex weights.

Most results obtained so far concern global properties of the graphs which, in the continuum, can be translated into correlation functions integrated over the positions of their insertion points on the surfaces. Little is known however on more refined properties of random graphs such as the dependence of correlators on the distances between their insertion points. By distance we mean here geodesic distance along the graph, namely the length of any shortest path between say two given faces. These refined properties seem to remain beyond the reach of the standard matrix model treatment. Still, some results were obtained in Refs. [6], [7], [8], where in particular combinatorial arguments were used to derive the universal scaling two-point function for critical genus zero surfaces with two marked points at a fixed geodesic distance. However, an explicit form for the two-point function was obtained only in the pure gravity case and close to the critical point.

More recently, using a completely different combinatorial approach, the same result was recovered in Ref. [9] in the case of random tetravalent planar maps. This work relies on the existence of a bijection between planar tetravalent maps and rooted trees with vertices labelled by the geodesic distance to the root. The construction however is specific to the tetravalent case, and does not provide a closed form for the discrete solution either.

In this note, we address the general question of the enumeration of planar maps with two marked faces at a fixed geodesic distance. In the case of maps with inner vertices of arbitrary even valence, we derive explicit expressions for the generating function Gn of “two-leg diagrams”, namely, planar maps with two distinguished univalent vertices (legs) whose adjacent faces lie at a geodesic distance n of one-another, and with weights gi per inner 2i-valent vertex, i=1,2,3,…. These results are obtained via the use of a more general bijection between two-leg diagrams and decorated trees first found in Ref. [10] and extended in Ref. [11], which we adapt so as to keep track of the geodesic distance between the two legs. This leads to a simple algebraic recursion relation on n, which allows to derive a closed expression for the solution Gn. We give several explicit expressions depending on the maximum valence of the vertices of the graph. With these solutions in hand, we easily derive the continuum scaling two-point functions corresponding to the various multicritical points. This allows to recover the results of Refs. [7] and [9] in the generic critical case, and also provides the generalization to all higher order multicritical points. We also extract critical exponents such as the fractal dimension in various approaches to the multicritical points, as well as probability distributions for the (rescaled) geodesic distance in multicritical planar maps of fixed but large size.

The paper is organized as follows. In Section 2 we detail the general correspondence between two-leg diagrams and decorated trees. Section 3 is devoted to the incorporation of the geodesic distance n between the two legs in this setting and leads to a general recursion relation on n. This recursion relation is solved first in the tetravalent case (Section 4.1), and exploited to derive a critical fractal dimension dF=4 (Section 4.2) and to recover the known continuum two-point function for critical surfaces (Section 4.3). We also use our explicit solution to derive in Section 4.4 the probability distribution for rescaled geodesic distances in two-leg diagrams of fixed but large size. The general case of maps with valences up to 2(m+1) is solved in Section 5, where we discover a remarkable connection with m-soliton solutions of the KP hierarchy. This general solution is applied in Section 6 to the so-called “hard dimer” model, which displays the first higher order critical point (tricritical point). In Section 6.1 we detail the solution and display in particular the first few Gn's. This leads to the computation in Section 6.2 of the fractal dimension dF=6 and the continuum two-point function for tricritical surfaces, corresponding to a generic approach to the tricritical point. We also discuss the case of a non-generic approach along the critical line, which displays another fractal dimension dF=4 and a different two-point scaling function. We finally obtain in Section 6.3 the probability distribution for rescaled geodesic distances in tricritical surfaces of fixed but large size. All these results are extended to the case of the mth order multicritical point in Section 7, where we obtain the fractal dimension dF=2(m+1) (Section 7.1), the generic continuum multicritical two-point function (Section 7.2) and the corresponding probability distribution (Section 7.3). We gather a few concluding remarks in Section 8, namely, the relation to matrix models (Section 8.1), the extension to two-point functions of other (less relevant) operators (Section 8.2), some generalizations to other classes of planar graphs such as constellations (Section 8.3), as well as a connection to the so-called integrated super-Brownian excursion (ISE) (Section 8.4). We also gather a few technical details in Appendix A Proof of the determinant form, Appendix B Proof of the determinant form.

Section snippets

Planar maps and decorated trees

There is a general correspondence between planar maps and special classes of decorated trees [10], [11]. In this paper, we concentrate on the simpler case of maps with only vertices of even valence. In this case, there is a bijection between so-called “two-leg diagrams” and so-called rooted “blossom” trees as defined below. By two-leg diagram, we mean a planar map with two external legs (i.e., two extra univalent vertices) distinguished as in- and out-coming, whereas all inner vertices have

Geodesic distance

A nice feature of the above bijection is that it keeps track of the geodesic distance between the in- and out-coming legs in the two-leg diagrams, defined as the minimal number of edges crossed by a curve connecting the two legs (for instance, the two legs of the diagram of Fig. 1(a) are at distance 1). Indeed, this distance is nothing but the depth of the root in the corresponding blossom trees. More precisely, it was shown in Refs. [10], [11], [12] that all the edges cut in the cutting

Exact solution

Let us now show how Eq. (3.3) may be used to extract exact expressions for the Rn's. As a first simpler but instructive case, let us restrict our study in this section to the case of tetravalent maps, i.e., two-leg diagrams having only tetravalent inner vertices. Going to generating functions, this amounts to taking gk=k,2, in which case Eq. (2.1) reduces to: R=1+3gR2 with solution R=1−1−12g6g which displays the well-known critical value gc=1/12 of g for pure quadrangulations. More generally,

Exact solution in the general case

Let us now turn to the general case of maps with bounded valences, namely with arbitrary even valences 2,4,6,… up to say 2(m+1). This corresponds to keeping arbitrary vertex weights g1,g2,…,gm+1 while setting gj=0 for j>m+1. Eq. (3.3) becomes a recursion relation of order 2m, namely, expressing Rn+m in terms of Rn+j, j=m−1,m−2,…,−m. This equation is valid for all n⩾0, upon taking the initial condition

  • (i)

    R−1=R−2=⋯=Rm=0.

To completely fix the solution, the condition (i) must be supplemented by
  • (ii)

    the

The hard dimer model

As a first application of the exact solution of Section 5, let us consider the so-called hard dimer model on tetravalent planar maps [14]. In this model, we enumerate tetravalent maps whose edges may be occupied or not by a dimer, with the hardness condition that no two adjacent edges may be simultaneously occupied by dimers. Assigning as usual a weight g per vertex and an activity z per dimer, the problem may be reformulated as that of enumerating maps with both tetra- and hexavalent vertices

Fractal dimensions

In this section we consider the case of arbitrary m, in which case we may reach an mth order multicritical point. For simplicity, we set g1=0 and we introduce the following notations g2=g,gk=gk−1zk(withz2=1),V=gR,W(V)=V−k=2m+1zk2k−1k−1Vk in terms of which Eq. (2.1) reads simply W(V)=g. The mth order multicritical point is obtained by setting W′=W″=⋯=W(m)=0 which implies zk(c)=(−1)k1m+1m+1k2k−1k−16mk−1. With these, Eq. (2.1) simply reads g=W(V)=Vcm+1−(Vc−V)m+1(m+1)Vcm,Vc=m6 while gc=W(Vc)=m/(6(m

Discussion and conclusion

In this paper, we have found explicit expressions for partition functions of planar maps with arbitrary even valences and with two marked points at a specified geodesic distance. From these we have extracted various scaling functions and critical exponents corresponding to various approaches to the different critical and multicritical points. We also obtained various probability distributions for distances in maps of fixed but large number of vertices. These results rely on a purely

Acknowledgements

We thank F. David, J.-F. Delmas and G. Schaeffer for useful discussions partly while attending the semester “Geometry and statistics of random growth” held at the Institut Henri Poincaré, Paris, January–March 2003. We also thank D. Bernard, R. Conte and M. Musette for help with references on solitons.

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