Matrix models and graph colouring

Physxcs Letters B 306 (1993) 245-251 North-Holland

P H YSIC S k ETT ERS B

Matrix models and graph colouring Giovanni M. Cicuta 1 Dtparttmento dt Ftszca, Umversltgzdl Ban, and INFN, Seztone dt Barl, VzaAmendola 173, I-70126 Ban, Italy Luca Molinari 2 and Emilio Montaldi Dtparttmento dt Ftstca, Untversttd dl Mdano, and INFN, Seztone dt Mdano, Vza Celorta 16, 1-20133 Mtlan, Italy Received 7 January 1993 Edttor: R Gatto

We study an edge-colounngproblem on random planar graphs which is one of the simplestvertex models that may be analyzed by standard methods of large N matrix models. The mam features of the saddle point solution and its crmcal behavlour are described. At the critical value of the couphng ger the eigenvaluedensity u (~.) xs found to vamsh at the border of the support as 12-al 2/3.

1. I n t r o d u c t i o n Vertex models on r a n d o m planar graphs are an obvious step in investigating the statistical mechanics of matter coupled with two-dimensional q u a n t u m gravity. Although m a n y of them are easily formulated as multimatrix models in the large N limit, it is so far generally impossible to solve them [ 1 ]. In this letter we analyse a simple vertex model on planar r a n d o m graphs that corresponds to an interesting colouring problem ~1. We consider the partition function

Z(g)=f~A~B~,Cexp(-½Tr(A2+B2+C2)-~NTr(ABC+BAC)),

(1)

where A, B and C are N × N hermitean matrices. I n the large N limit Z ( g ) is the generating functional of the planar graphs of valence three, such that the three edges incident in each vertex are coloured with the three distinct colours A, B and C. The same weight is gaven to proper colourings of the edges A, B, C being oriented clockwise or anti-clockwise or anti-clockwise. This edge colouring problem is known to be closely related to the face colouring of the same class of planar cubic graphs [ 3,4 ] ~:2 A gaussian integration over the matrix C yields

Z(g)=f@A~Bexp(-½Tr(A2+Bz)+g2Tr(A2B2+ABAB))~

.

(2)

A n integration over angular variables is then performed, to obtain 1 E-mall address: [email protected] @BARI.INFNIT. 2 E-malladdress: [email protected]. ~l An early attempt to evaluate the number of proper colourlngsby quantum field methods as gavenin ref. [2 ]. ~2 See the Talt theorem m ref. [ 5 ].

ElsevierSciencePubhshers B.V.

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Z(g)=

PHYSICSLETTERSB

d2zI-[ ()[,-)[s)Zexp -½ ~ )[~ --

l<)

2,Bexp

t~l

3 June 1993

-½TrBZ+~Tr(A~B2+AdBAaB) ,

(3)

where Aa is the diagonal form of the matrix A, with elgenvalues )[,. Again, a gausslan integration over the matrix B can be done, and one obtains

Z(g,= f. i_i=dA, i_i()[_)[j)2exp(_½ ~,=1)[,)~(1 g2 U

N

2

l
( ) i t ..~)[j ) 2 )\-1/2 |

N

(4) "

In eqs. (2) - (4) irrelevant factors have been neglected. Rather than the partition function ( 1 ), we may consider the following one:

Z(g)=fgA~B~Cexp(-½Tr(A2+B2+C2)-~NTF(ABC) ),

(5)

that corresponds to an edge colourlng problem where only one definite orientation of the colourings has been kept. We then have / ( g ) = f f i d2, l-I ()[~-)[s)2exp -½ ~ )[3 t=l

t
1-- ~)[~2s

(6)

k=l

Both eqs. (4) and (6) are rather retractable by the orthogonal polynomial method, because of the presence of "nonlocal" terms in the effective action. However, they may be analyzed by the saddle point method in the large N limit. In this way, one obtains a singular integral equation for the elgenvalue density. In this letter we restrict ourselves to the evaluation of the large N limit of the simpler model described by eqs. ( 5 ) and (6).

2. T h e planar limit

In the large N limit, the contribution of the planar graphs to the partition function (6) is obtained from the saddle point of the effective action Vef~(2), after introducing the continuous distribution of eigenvalues 2(x) 1

N=X,

--

2z = 2 ( x ) ,

l lf

~,=

=

o dx,

Z ( g ) = ] ~2(x) exp(-NzVef~[)[] ) , 1

(7b)

1 1

geff[)[] =½ f dx)[(x) 2 - f f 0

(7a)

dxdy{logl)[(x)-)[(y)l-llog[l-g2)[(x))[(Y)]}.

(7c)

0 0

In terms of the eigenvalue density u (2) = dx/d2, the saddle point equation and the normalization condition are

)[__g2 ff d/t 1_/tu(g) g 2 ) [ ~ = 2 ~ f d # u(/t), ~ 1= 3f dy u(/z),

(Sa,b)

where N denotes the principal part prescription. We are interested in investigating a solution u()[) vanishing outside a finite interval. Therefore the product )[/z is bounded and for g small enough the denominator 1 _g2)[/~ is positive definite. Eq. (8a) is consistent with the ansatz that u (2) is an even functaon of)[ with support - a ~<)[~
(ga)2<

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U(~)=

~

PHYSICS LETTERS B

3 June 1993

(9)

dl~41_g4a2#2 1_g22]~].

1--g 4 --a

By writing (10) eq. (9) becomes, for 0~< t~< 1 and s e t t i n g A = 1

(ag)4< 1, 1

A~ x~l-x) dP(x) l+fdXKA(t,x)q)(x) q)(t)=l- ~ dxN[ 1-Ax 1-Ax~t 0

(lla)

0

and the normalization condition (8b) is now 1

(11b) 0

The main advantage of eq. ( 11 a) over eq. (9) is to exhibit the dependence of the unknown function q~(t) on a single parameter A. The normalization equation ( 11 b) may next be used to determine a 2. For any 0 < A < 1, Ka (t, x) is the kernel of a compact operator on L z (0, 1 ). The square of its norm ilgll 2 = ~ 0

0

dxdtX(1-x) 1 (A-2)log(1-A)-2A 1-Ax (1-Axt) 2 = 4~2A

(12)

is a strictly increasing function of A, for 0
q)A(t) =

1 -- ( ~ A ) -- (~6A)Z(8t+3) --

(~A)3(80t2+32t+ 19) +

1 4 (i~A) (896t 3 + 3 6 8 t 2 + 2 3 2 t + 158) + O ( A 5) .

(13) F r o m this and eq. ( 1 l b ) we get a 2 = 4 [ 1 + (~6A) +

6(~A)2+48(~A)3+447(~A)4+O(AS)]

=411 + g 4 + 8 g 8 + 89912+ 1157g16+O(g2°)

] .

(14) To compute the free energy in a simple way, it is useful to allow a parameter loz other than ½ multiplying the quadratic term in the partition function (5). We then have a function Z(o6 g) = oz - (3/2)~2Z(1, g/o~3/2), after a simple rescaling of matrix variables. The planar free energy

1 E=-

Z(c~, g)

N~limo~N-5 log Z(oz, 0)

(15) 247

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cA(t)

05

i

,

i

,

i

,

i

,

°o

i

,

i

,

I

,

l

,

I

05

i

I

1

t

Fig 1. Oa(t) is plotted forA= 1-2-", n= 1, 4, 7, 17 (from top to bottom). The rapid fall of q~A(t) close to t= 1 and for A-~I, suggests a zero of q~c~(t) an t= 1.

is a function of the adimensional variable g/oL 3/2 only, implying the simple differential equation 0E 2 0E g~g + 5a~a =0.

(16)

However, using (15) and the rescaling (7a): 0E

1

0 a -- N~m 2-~hm

3

(17)

(Tr(A2+B2+C2)} - ~ .

Taking ot = 1 and integrating eq (16) we finally get a formula for the free energy in terms o f the second m o m e n t of the eigenvalue density: -E(g)=

i(

dg' o g'

-1+

i

d222u(2)

--a

)!( =

dg' g'

-It

dXx/~I--X)~A(X) 0

)

= ¼g4+ ~gS + ~11 ~ 1 2 ÷ ~_~ 267 ~ 1 6 ÷ ~ 6049 _ ~~ 2 0 ÷ O ~ l ~24 ) .

(18)

The free energy is a power series in g4, although the perturbative expansion of eq. (5) has nonvanishing terms at each order o f g 2. To understand this fact, we write Z(g)/Z(O)=exp[ -F(g, N) ] and evaluate the first few Feynman diagrams:

1 g2 + ~.. 1 ( 6 N 2 + 18)g 4+ ~-.t 120 ~{ 1 6 + ~59--g6 ) + O ( g S N 2 ) 2v~% --N2E(g). - F ( g , N ) = I + ~.t

(19)

This indicates that terms of order g4n+2 are depressed by a factor N - 2 with respect to terms of order g4n. This unusual feature is due to the clockwise ordering of the model eq. (5) and does not occur In the model eq. ( 1 ).

3. The critical eigenvalue density In the planar limit of matrix models, the perturbatlve expansion of the free energy usually has a finite radius 248

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of convergence g < g¢r, even though for finite N t h e radius might be zero [ 7 ]. The same finite radius characterizes the perturbative series for the size a of the support. In the saddle point analysis, the value g=g¢~ is also found to correspond to a zero of the eigenvalue density, at the boundary of its support, of degree higher than the generic value ½. The saddle point eq. (8a) can be transformed into the Fredholm equation (9) provided that the only singular integral is the one on the RHS of the equation; this requires A < 1. However, we shall consider the critical function O c t ( t ) = lim CA(t).

(20)

Eq. (9) implies that Cot(t) vanishes at t = 1, so that the density u(2) has in 2 = _+a a zero of degree higher than ½. In fig. 2 we show the decreasing values of Ca (t) in t = 1, as A approaches its critical value. To investigate the behaviour o f the critical solution as t--, 1 - , we plotted in fig. 3 some points near t = 1 of the function CA(t), for a value of A very close to 1. The linear behaviour in the logarithmic plot, if one neglects the possibility of logarithmic factors, suggests the power law decay Ccr (t) ~ ( 1 -- t) ~. The following simple argument allows to evaluate 7= ~. Let us assume that the solution of eq. ( 11 a) for A = 1 has the form Ccr (t) = ( 1 - t) 7f(t), with 0 < f ( 1 ) < ~ . Using the fact that C = ( 1 ) = 0, eq. ( 1 la) becomes 1

1

f(t)= ~

1

(1--x)~-lx3/Zf(x) = 1 dx 1 - t x ~

( l - - t ) 1-' o

(l--y) y,_af(1--y) dy ( 1 - t y ) '+3/2 \l-ty}

(21)

o

after a change of variable. Taking now t = 1 we get the condition -log CA(t)

--log@a(1)

18 2

,

,

,

~

,

,

,

~

,

,

,

,

,

~

,

,

,

,

,

~

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,

+++ ÷ 4+ 4+ + + + + ÷

13

1

4-

++ ++

m

o -12

m B , i ml

-6

0.8

.

,

,

I

.

,

,

.

,

.

.

,

.

r

9

0

to# (i -

.

A)

Fig 2. log q~a(t= 1 ) is plotted versus log( 1 - A ). For values of A close to 1 the figure shows the vamshlng of @,~( 1 ) approximately described by ~bA(1 ) = C( 1 - A ) " The asymptotic straight lane corresponds to the choice p = -~ and log C = 0.25

,

.

.

i

I

i

I

,

.

I

.

.

,

,

12

--log (1 - At) Fig 3. F o r A = 1 - 2 -~7, the plot shows q)A(t) for 0 . 9 9 7 5 < t < 1, suggesting the approximate power law OA(t),,~K(1-At) q, q=0.158.

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12 2

i

45

I

i

[]

i

t

I

[]

,

i

,

I

,

I

i

I

,

I

i

n Inl n~ mlli! m E []

m

m

39

I [ I I ' I I I I I T [ I I ' I I I I I ' I ' I -10

- 5

0

log (1 - a)

Fig. 4. T h e h a l f size a o f the s u p p o r t o f t h e e l g e n v a l u e d e n s i t y is a m o n o t o m c i n c r e a s i n g f u n c t i o n o f A T h e figure s h o w s t h e satu r a t a o n o f a 2 as A--* 1 -

1 f(1)= ~-~F(7)F(1-y)f(1),

(22)

which implies ~,= I- This value is actually in close agreement with the numerical result of fig. 3. It follows that

bl()~)~ [~.2--a2[1/2+1/6

2-~ +_a.

(23)

Once the solution q)A (t) is found, the width a of the support of the elgenvalue distribution can be numerically computed through eq. ( 1 lb). In fig. 4 we give a 2 for a sequence of values A. It shows that a z grows monotonically from 4 (free model with semicircle distribution) and stabilizes at the critical value a o2~ 4.504. Although the n u m b e r gcr 4 = act-4 IS quite small, the perturbative expansion (14) diverges at this value, since a 2 (A) in eq. ( 1 l b ) is not analytic in A (and hence in g). This behaviour, occurring together with the appearance of a higher order zero of u (2) at the edge ,~= _+a, was observed in the simpler one-matrix models [7 ].

Acknowledgement

One of us (G.M.C.) thanks A. D ' A d d a for useful discussions on vertex models.

Note added

In either model described by eq. ( 1 ) or eq. (5) the loops formed by sequences of bonds with colourings ABAB ... are self-avoiding loop coverings of planar trivalent graphs. This interpretation, together with the edge critical behaviour (23), show some similarity with the results for O (n) models on random graphs, extensively studied by Kostov and others [ 8 ]. The partition function in Kostov's paper is written as a sum over non intersecting loops on cubic planar graphs, which become dense in the low temperature phase, leaving no lattice site empty. In this phase, the 250

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etgenvalue d e n s i t y b e h a v e s as u (2) ~ (a - 2 ) 1- 1/h, where 2 cos (re/h) = n . It is possible that at the critical p o i n t , K o s t o v ' s m o d e l for n = 1 a n d ours, describe the s a m e " c o n d e n s a t i o n o f loops", since they b o t h exhibit the edge e x p o n e n t ]. M o d e l ( 1 ), which has the s a m e edge critical b e h a v i o u r ( 2 3 ) as m o d e l ( 5 ) , will be discussed i n a f o r t h c o m i n g paper. We t h a n k the referee for h a v i n g suggested to use this interesting remark.

References [ 1 ] See, e.g, P. Gmsparg, Tneste lectures ( 1991 ), report LA-UR-91-4101. [2] N. Nakanlshi, Commun. Math. Phys. 32 (1973) 167. [3] See, e.g, B. Bellobas, Graph theory (Springer, Berlan, 1979), exercise (27), p. 101. [4] G.M. Cleuta, Matrix models m statistical mechamcs and quantum field theory, talk, in: Proc. NATO Workshop (Newton Institute, Cambridge, September 1992), ed. H Osborn, to appear. [5] S. Flonm and R.J Wilson, Edge-colourmgs of graphs (Pitman, London, 1977) p. 26. [ 6 ] N.I. Muskhehshvdi, Singular integral equations ( Noordhoff, Gronmgen, 1953 ). [ 7 ] E Brezm, C. Itzykson, G. Pans1 and J B Zuber, Commun. Math Phys 59 (1978) 35. [8] I. Kostov, Mod. Phys. Lett A 4 (1989) 217; M. Gaudm and I Kostov, Phys. Lett. B 220 (1989) 200, B Duplantler and I. Kostov, Nucl. Phys B 340 (1990) 491; B. Eynard and J. Zlnn-Justln, Nucl. Phys. B 386 (1992) 558.

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