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Baby Universes in 2d Quantum Gravity

arXiv:hep-th/9303149v1 26 Mar 1993NBI-HE-93-4February 1993Baby Universes in 2d Quantum GravityJan AmbjørnThe Niels Bohr InstituteBlegdamsvej 17, DK-2100 Copenhagen Ø, DenmarkSanjay JainCentre for Theoretical StudiesIndian Institute of ScienceBangalore 560012 , IndiaGudmar ThorleifssonThe Niels Bohr InstituteBlegdamsvej 17, DK-2100 Copenhagen Ø, DenmarkAbstractWe investigate the fractal structure of 2d quantum gravity, both for pure gravityand for gravity coupled to multiple gaussian fields and for gravity coupled to Isingspins. The roughness of the surfaces is described in terms of baby universes andusing numerical simulations we measure their distribution which is related to thestring susceptibility exponent γstring.1

1IntroductionThe fractal and selfsimilar structure of 2d quantum gravity is related to the entropy-(or string susceptibility-) exponent γ. This has been discussed in a recent paper [1]where the structure of so-called baby universes was analyzed.

It is convenient in thefollowing discussion to consider 2d quantum gravity with an ultraviolet cut-offandwe will consider the surfaces entering in the path integral as triangulated surfacesbuilt out of equilateral triangles [3, 4, 2]. In the case of surfaces of spherical topologya closed, non-intersecting loop along the links will separate the surface in two parts.The smallest such loop will be of length 3.

It will split the surface in two parts. If thesmallest part is different from a single triangle we call it (following the notation of[1]) a “minimum neck baby universe”, abbreviated “mimbu”.

The smallest possiblearea of a mimbu is 3 and the largest possible area will be NT/2, where NT is thenumber of triangles constituting the surface.In the case of pure 2d quantum gravity it is known that the number of distinctsurfaces of genus zero made out of NT triangles has the following asymptotic formZ(NT) ∼eµcNT Nγ−3T(1.1)where γ = −1/2. For the models which can be solved explicitly and where c < 1 wehave the following partition function:Z(µ) =XNTZ(NT)e−µNT(1.2)where Z(NT) for large NT is of the form (1.1), just with a different γ = γ(c).For c = 1 it is known that there are logarithmical corrections to (1.1), while theasymptotic form of Z(NT) is unknown for c > 1, although it can be proven that itis exponentially bounded ([2]).

If we assume (1.1) one can prove that the averagenumber of mimbu’s of area B on a closed surface of spherical topology and witharea NT (we use the notation area ≡#triangles) is given by¯nNT (B) ∼(NT −B)γ−2Bγ−2(1.3)provided NT and B are large enough.The above formalism is well suited for numerical simulations. The measurement ofthe exponent γ has always been somewhat difficult.

The first attempts used a grandcanonical updating ([6, 7, 8]), which generated directly the distribution (1.2). Thedisadvantage is that one has to fine-tune the value of µ to µc.

Later improved versionsallowed one to avoid this [9], but one still had to perform independent Monte Carlo2

simulations for a whole range of NT and γ still appeared as a subleading correctionto the determination of the critical point µc. These disadvantages disappear whenwe use (1.3).γ does not appear as a subleading correction to µc and one canuse canonical Monte Carlo simulations (the so-called link-flip algorithm [4]) whichkeeps NT fixed, and still in a single Monte Carlo simulation get a measurementof the distribution of mimbu’s all the way up to NT/2.One can therefore takea large NT and make one very long run.This allows us to avoid the problemswith long thermalization time.

In addition the actual measurement of the mimbudistribution is easy. For a given thermalized configuration one has to identify allpossible mimbu’s associated with the configuration.

This is done by picking up onelink, l0, and checking whether any links which have a vertex in common with l0 havea vertex in common which do not belong to l0. This being the case we will have aminimal neck of length 3.

For a given l0 there will always be two such, correspondingto the two triangles sharing l0. But there might be additional ones and they willdivide the surface into a mimbu and its “mother”.

By scanning over l0’s, avoidingdouble counting and repeating the process for independent configurations we canconstruct the distribution of mimbu’s.In the rest of this paper we report on the results of such numerical simulations.2Numerical simulations2.1Pure gravityThe simulations were done on lattices of size ranging from 1000 to 4000 trianglesNT and we used the standard “link flip” algorithm [4] to update the geometry. Weused of the order of 107 sweeps, where each sweep consists of NT link flips.

Afterthermalization we measured for each 10th sweep the distribution of mimbu’s, thatis we counted all areas B > 1 enclosed by boundaries of length 3. The reason forperforming the measurements so often is simply that they are not time consuming(the time it takes to make one measurement is comparable to the time it takes toperform one sweep).

The distributions are shown in fig. 1.

In order to extract γ thedistributions are fitted to equation (1.3). But as eq.

(1.3) is only asymptoticallycorrect deviations can be expected for small B. Thus a lower cut-offB0 has to beintroduced in the data to avoid the effects of this deviations.

Moreover we haveadded the simplest type of correction term which arises from the replacementBγ−2 →Bγ−21 + CB + O(1/B2)(2.1)3

in (1.3) and fitted to the formln(¯nNT ) = A + (γ −2) ln(B(1 −BNT)) + CB(2.2)for B ≥B0. A and C are some fit parameters.

Comparison of the results withand without this correction term can be seen in fig. 2 where we plot the value of γextracted with different cut-off’s B0.

We see that including the correction improvesthe results considerable.Let us assume that the values γB0 extracted from (2.2) appproach exponentiallya limiting value for large B0:γB0 = γ −c1e−c2B0. (2.3)The result of such a fit is shown in fig.

2. It is clear from fig.

2 that the assumption ofan exponential approach of γB0 to γ is not essential for the extraction of γ. We haveintroduced it at this point in order to treat all measurements consistently.

For thematter fields coupled to gravity the finite size effects will be larger and extrapolationto large B0 more important.The γ extracted in this way for different lattice sizes is:NTγ1000−0.496 ± 0.0052000−0.501 ± 0.0043000−0.504 ± 0.004which is in good agreement with the expected value of γ = −0.5. It shows that thiskind of simulations are indeed well suited to measure γ and it is thus natural to tryapply them to the case of matter couple to 2d gravity.2.2The Ising modelThe next non-trivial test of the method is to study the Ising model coupled to 2dgravity.

It has been solved analytically [5] and was found to have a 3rd-order phasetransition. The coupling to gravity is in a sense weak as it only changes the stringsusceptibility at the critical point (from γ = −1/2 to γ = −1/3).

For this reason ithas until now been considered very difficult to measure γ directly, since it required afine-tuning of both the bare cosmological constant µ and the spin coupling constantβ. On the other hand it has been verified that it is indeed possible to extract theother known critical exponents [10] since for these exponents it is possible to use thecanonical ensemble in the simulations.4

The Ising spins are placed in the center of the triangles and they interact withthe spins on neighbouring triangles. This corresponds to placing them on verticesin the dual graph.

In that case the critical point has been found explicitly and isβc = 0.7733... [11]. The (canonical) partition function of the model isZNT (β) =XT∼NTX{σi}eβ P σiσj(2.4)where the summation is over all triangulations with NT triangles.In the simulations we used a Swendsen-Wang cluster algorithm [12] to updatethe Ising spins and lattices sizes NT = 1000 and 2000.

We made runs for severalvalues of the coupling in the interval 0.6 ≤β ≤0.95 and then fitted the distributionsto eq. (2.2).

In this way we could extract values γB0(β) and by assuming a relationlike (2.3):γB0(β) = γ(β) −c1(β)e−c2(β)B0(2.5)we have extracted the values for γ(β) shown in fig. 3.

Examples of γB0(β) andthe exponential fit (2.5) for different values of β are shown in fig. 4.

We observe amarked increase in the dependence on B0 when β approaches βc.We get, as expected, the pure gravity value of −0.5 for couplings far below andabove βc. In the vicinity of the phase transition we see on the other hand a clearpeak and the peak values agree well with the exact value γ = −1/3.

We concludethat the method for extracting γ works well in this case too, although it should beclear that the amount of numerical work needed is much larger in this case than inthe case of pure gravity.2.3Gaussian fieldsThe gaussian fields xµ, µ = 1, . .

. , D are placed on the sites i of the triangulationT.

They can be viewed as representing an immersion i →xµi of our abstract tri-angulation T into RD, i.e. a model for non-critical strings and they also representa coupling of matter with central charge c = D to gravity.The multiple gaus-sian models do not interact directly with each other but only through their mutualinteraction with the geometry.

The (canonical) partition function is given byZNT =XT∼NTZYi∈T\{i0}dDxi e−P(xµi −xµj )2(2.6)where the summation is over all triangulations T with NT triangles. One site is keptfixed in order to eliminate the translation mode.

No coupling constant appears inthe action as it can be absorbed in a redefinition of the gaussian fields.5

Again we have performed simulations with up to 107 sweeps for lattice sizesranging from 1000 to 4000 triangles. We have used from one to five gaussian fieldsand a standard Metropolis algorithm to update them.In fig.5 we show howthe distributions of baby universes change with increased c (normalized with thedistribution for pure gravity).Fitting these distributions to the functional form(2.2) and extracting γ as above yields the results shown in fig.

6. The results arecompatible with earlier estimates [9].It is seen that γ is too small for c = 1 where it is known that γ = 0.

But in thecase c = 1 we know that the asymptotic form (1.1) is not correct. It should bemultiplied with logarithmic corrections.

If we include these we get for c = 1 that(1.3) is replaced by [1]¯nNT (B) ∼[(NT −B)B]γ−2 [ln(NT −B) ln B]α . (2.7)In this formula we have left γ and α as variables.

Model calculations give α = −2,but it is not known whether this power is universal and the model has not beensolved analytically in the case of one gaussian field.If we fit to (2.7) in the way described above (including also the 1/B correction)we extract for c = 1 the following values of γ and α for different lattice sizes:NTγα1000−0.22 ± 0.05−0.5 ± 0.42000−0.14 ± 0.07−1.0 ± 0.44000−0.09 ± 0.08−1.2 ± 0.4Both γ and α moves towards the expected values 0 and -2 as a function of NT, butthe finite size effects are clearly larger here than for pure gravity.In fig. 6 the results of a fit to (2.7) for c > 1 is included.

It is seen that γextracted in this way exceeds the theoretical upper bound γ = 1/2 ([6]). In additionthe power α decreases from -1.2 for c = 1 to -5 for c = 5.

We conclude that eitherlogaritmic corrections are not the right ones to include for c > 1 or finite size effectsare so large that they make the fits unreliable.What is clear from the analysis is that γ increases with c for c in the range 0−5.According to (1.3) this means that the number of baby universes of a given size willincrease, i.e. the fractal structure will be more pronounced with increasing c. Wehave illustrated this in fig.

7, which shows two “typical” surfaces corresponding toc = 0 and c = 5. It should be emphasized that the pictures are only intended tovisualize the internal structure, i.e.

the connectivity of the surfaces11The surfaces are constructed in the following way: Given the connectivity matrix of the trian-6

3DiscussionWe have verified that the technique of extracting the entropy exponent γ directlyfrom the distribution of baby universes is superior to the methods used until nowfrom a practical point of view. A single (although long) Monte Carlo run for a fixedvalue of NT is sufficient for extracting γ and we get the correct results for c < 1.On the other hand the situation in the case c > 1 has not really improved muchcompared to the earlier measurements [9].

We get in fact similar results, and thisshows that the method also works in the case c > 1 and the ambiguity in extractingγ for c > 1 is that we do not know the correct functional form to be used in thefits. It is clear that it would be most interesting if we could reverse the procedureand use the data to obtain knowledge about the corrections to (1.1) for c > 1.

Ourdata are not yet good enough to do this in a convincing way, but the problem isclearly not due to the baby universe technique introduced in this paper, but due tothe inefficiency of the flip algorithm used to update the triangulations.References[1] S. Jain and S.D.Mathur, Phys.Lett. 286B (1992) 239.

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168B (1986) 273;177B (1986) 89.gulation we choose arbitrary coordinates for the vertices in R3. Next we introduce an attractionbetween neighbouring vertices in order to keep the surface together and extrinsic curvature tosmooth out the surface during a Monte Carlo simulation.

When the surface reach a configurationwithout self-intersection we put a pressure in the interior and a Coulomb repulsion between distantvertices and in this way we blow up the surface as a balloon.7

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Figure captionsFig.1 The distribution of baby universes in the case of pure gravity.Fig.2 Fitted values of γB0 for different cutoff’s B0 for pure gravity. Values are shownfor fits with and without the correction term included.

The curve shows a fitusing (2.3) resulting in γ = −0.496 ± 0.005 (errors are 95% confidence limitsfor a χ2-test).Fig.3 Fitted values of γ vs the coupling β in the case of one Ising model coupled togravity. Results are shown for two lattice sizes, NT = 1000 and 2000.Fig.4 Fitted values of γB0(β) for various β as a function of the cut-offB0.

The curvesrepresent fits to (2.5).Fig.5 The distributions of baby universes for up to five Gaussian fields coupled to2d gravity. The values are normalized with the distribution for pure gravity.Fig.6 Fitted values of γ vs central charge in the case of multble Gaussian fields.Results are shown for fits without (circles) and with (squares) a logarithmiccorrection term included.Fig.7 3d illustration of the fractal structure of the surfaces for c = 0 (fig.

7a) andc = 5 (fig. 7b).

NT = 200 is used.9

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