ASYMPTOTICS OF 10j SYMBOLS JOHN C. BAEZ, J. DANIEL CHRISTENSEN, AND GREG EGAN Abstract. The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a `degenerate spin network', where the rotation group SO (4) is replaced by the Euclidean group of isometries of R3. We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulas in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1=16 times the Riemannian ones. 1. Introduction In the Ponzano-Regge model of 3-dimensional Riemannian quantum gravity [1], an am- plitude is associated with each tetrahedron in a triangulation of spacetime. The amplitude depends on the tetrahedron's six edge lengths, which are assumed to be quantized, taking values proportional to 2j + 1 where j is a half-integer spin. One can compute this ampli- tude either by evaluating an SU (2) spin network shaped like a tetrahedron, or by doing an integral. Approximating this integral by the stationary phase method, Ponzano and Regge argued that when all six spins are rescaled by the same factor ~, the ~ ! 1 asymptotics of the amplitude are given by a simple function of the volume of the tetrahedron and the Regge calculus version of its Einstein action. Nobody has yet succeeded in making their argument rigorous, but Roberts [2, 3] recently proved their asymptotic formula by a different method. This result lays the foundation for a careful study of the relation between the Ponzano-Regge model and classical general relativity in 3 dimensions. Our concern here is whether a similar result holds for the Barrett-Crane model of 4- dimensional Riemannian quantum gravity [4]. In this model an amplitude is associated with each 4-simplex in a triangulation of spacetime. This amplitude, known as a 10j symbol, is a function of the areas of the 10 triangular faces of the 4-simplex. Each triangle area is proportional to 2j + 1, where j is a spin labelling the triangle. The amplitude can be computed by evaluating an SU (2) x SU (2) spin network whose edges correspond to the ______________ Date: October 20, 2002. 1 2 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, AND GREG EGAN triangles of the 4-simplex: vvoHHH~)) j23vvvv~~))Hj12HHH vvv ~~ )) HHH ovv)______j13________Ho_~~HHv)) ))HHj24~~~H )j25vvv~~)) )) HH~~HH vvvv)~~) j34)))~~jHHHvjvv)) j15~~ ) ~~ 35HHH14vvv~~)) ))~~vvvv HHHH~~)) o_____j45____o_~~v There is also an integral formula for the 10j symbol [7]. The problem is to understand the asymptotics of the 10j symbol as all 10 spins are rescaled by a factor ~ and ~ ! 1. Barrett and Williams [8] applied a stationary phase approximation to the integral for the 10j symbol, focusing attention on stationary phase points corresponding to nondegenerate 4-simplices with the specified face areas. They showed that each such 4-simplex contributes to the 10j symbol in a manner that depends on its Regge action. They pointed out the existence of contributions from degenerate 4-simplices, but did not analyse them. In Section 2 of this paper we begin by applying Barrett and Williams' estimate of the 10j symbols to the case where all 10 spins are equal. We find that the contribution of their stationary phase points to the 10j symbols is of order ~-9=2 . However, our numerical calculations show that the 10j symbols are much larger, of order ~-2 . This means we must look elsewhere to explain the asymptotics of the 10j symbol. In Section 3 we analyse the contribution of `degenerate 4-simplices' to the integral for the 10j symbols. They do not correspond to stationary phase points in the integral for the 10j symbols; instead, the integrand has a strong maximum at these points. We argue that the contribution of a small neighborhood of these points is asymptotically proportional to ~-2 . We give a formula expressing the constant of proportionality as an integral over the space of degenerate 4-simplices. We also reduce this to an explicit integral in 5 variables. In Section 4, we numerically compare these results to the 10j symbols as calculated using the algorithm developed by Christensen and Egan [9]. Our formula for the contribution of degenerate 4-simplices closely matches the actual asymptotics of the 10j symbols. Thus, even though our argument that these asymptotics are dominated by degenerate 4-simplices is not rigorous, we feel confident that the resulting formula is correct. In Section 5 we discuss a new sort of spin network, associated to the representation theory of the Euclidean group, which arises naturally in our analysis of the contribution of degenerate 4-simplices. Generalizing our results on the Riemannian 10j symbols, we conjecture formulas for the asymptotics of a large class of Riemannian spin networks in terms of these new `degenerate spin networks'. We verify this conjecture in a number of simple cases. In Section 6 we formulate a similar conjecture for Lorentzian spin networks. Taken with the previous one this conjecture implies that the ~ ! 1 asymptotics of a Lorentzian spin network in this class are the same, up to a constant, as those of the corresponding Riemannian spin network. For example, as ~ ! 1, the Lorentzian 10j symbol should be asymptotic to 1=16 times the corresponding Riemannian 10j symbol! We conclude by presenting some numerical evidence that this is the case. 2. Stationary Phase Points The 10j symbols can be defined using a Riemannian spin network _ also known as a `balanced' spin network [4] _ whose underlying graph is the complete graph on five vertices. ASYMPTOTICS OF 10j SYMBOLS 3 The ten edges of the graph are labelled with half-integer spins jkl = 0, 1_2, 1, . .,.where k and l refer to the vertices connected by each edge. In this approach, an edge labelled by the spin j corresponds to the representation j j of Spin(4) = SU (2) x SU (2), and the 10j symbols are computed using the representation theory of this group. However, to analyse the asymptotics of the 10j symbol, it is easier to use the integral formula due to Barrett [7]. This is: voHHHv~)) j23vvvv~~))HHj12HH vvv ~~ )) HHH ovv)______j13________Ho_~~HHv)) ))HHj24~~~H )j25vvv~~)) P 2j Z Y R dh1 dh5 (1) )) HH~~HH vvvv)~~) = (-1) k 0 half of the xz-plane. This corresponds to integrating over all possible values for ,23, inserting a factor equal to the volume of S(a23) with our chosen measure, which is a23, and also integrating over all possible azimuthal coordinates for ,34 and inserting a factor of 2ss. In what follows, we will choose coordinates so that the 12 x 12 matrix J(,) whose deter- minant we require is block diagonal, with two 3 x 3 blocks and one 6 x 6 block. It turns out to be convenient to parameterise ,34, not by its angle from the z-axis, but by the length s1 := |,23 - ,34|. Since ,13+,23 = ,34+,35, these four vectors can be positioned to form the sides of a (possibly non-planar) quadrilateral. Then s1 is the length of one of the diagonals. With the vectors ,23 and ,34 fixed and the lengths of ,13 and ,35 specified, the only remaining freedom this quadrilateral has, if it is to remain closed, is the `hinge angle', ff1, between the two triangles that meet along the diagonal. The vectors ,13 and ,35 have 4 degrees of freedom in all, and ASYMPTOTICS OF 10j SYMBOLS 11 specifying ff1 removes one of them, leaving 3 which break the quadrilateral. With our chosen measure on N , the product of the measure for the coordinates we are integrating over, and the Jacobian determinant for the 3 that break the quadrilateral, is: ___1_____. (4ss)3a23 We can treat a second 4-tuple of vectors more or less identically. If the vector ,12 has an azimuthal angle of OE, and we define s2 := |,23 - ,12|, then the quadrilateral formed by ,23, ,12, ,25 and ,24 can be assigned a `hinge angle' of ff2. This specifies all the degrees of freedom that allow this quadrilateral to remain closed. Once again, the product of the measure for the coordinates we are integrating over, and the Jacobian determinant for the 3 that break the quadrilateral, is: ___1_____. (4ss)3a23 So far, we have specified 5 degrees of freedom for N = SO (3): s1, s2, ff1, ff2 and OE. No further continuous degrees of freedom remain. The three vectors we have yet to specify must form triangles that complete two quadrilaterals with vectors that have already been parameterised, and because these two triangles have a vector in common, there is no `hinge' freedom left. Specifically, the three vectors ,14, ,15 and ,45 must form a tetrahedron by fitting over a triangular base that has, as two of its sides, the vectors: v := ,35 + ,25 w := -,24 - ,34. Given their fixed lengths, this determines ,14, ,15 and ,45 completely, apart from the freedom to locate the apex of the tetrahedron on either side of the plane spanned by v and w. This freedom can be accounted for with a factor of 2 in the integral. The final contribution to the Jacobian comes from a 6 x 6 block involving all the coordinates of ,14, ,15 and ,45, and with our chosen measure this is: __________________1___________________, (4ss)3 6V (a14, a15, a45, |w|, |v - w|, |v|) where V is the volume of the tetrahedron as a function of its edge lengths. Combining these results, we can rewrite (25) as: fifio)HH fi Z Z Z Z Z )vvv_H_~~H))fifi ~-2 ds1 ds2 dff1 dff2 dOE (27) fififio))o~~~~HHHHvvvv))fi~ ________4 _______________________________. o__o~~vfideg 96ss a23 s1 s2 ff1 ff2 OEV (a14, a15, a45, |w|, |v - w|, |v|) The integrals over the si are taken over intervals determined by the four sides of the quadri- laterals for which they are the diagonal lengths, and the angular variables range from 0 to 2ss, with the proviso that any part of the domain where the tetrahedron is not geometri- cally possible must be excluded. In numerical calculations, this can be dealt with by setting the integrand to zero wherever Cayley's determinant formula for the squared volume of the tetrahedron yields a negative value. We note that the integrand here is unbounded, and we have not proved that (27)converges, but our numerical calculations suggest that it does. 12 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, AND GREG EGAN 4. Numerical Data To test our hypothesis that the asymptotics of 10j symbols are dominated by the contri- bution of degenerate 4-simplices, we used the algorithm described in [9] to calculate values for several sets of 10j symbols. The figure below shows log-log plots for the absolute values of the Riemannian 10j symbols as a function of ~, where ~ is the parameter by which the areas akl were multiplied. The legend shows the base spins jkl; multiplying akl by ~ was achieved by replacing the jkl with: Jkl = ~jkl+ ~_-_1_2. ASYMPTOTICS OF 10j SYMBOLS 13 The lines on the plot show the ~-2 asymptotic behaviour described by equation (27); the integrals were evaluated numerically with Lepage's VEGAS algorithm [17 ], and found to have values of 0.680, 0.341, 0.209, 0.110, 0.0841 and 0.0446 respectively. In summary, our numerical data supports the following conjecture. Let us say that the spins jkl are `admissible' if for each vertex in the 10j symbol, the spins labelling the four incident edges satisfy the tetrahedron inequalities and sum to an integer. Then: Conjecture 1. If the ten spins jkl are admissible, the ~ ! 1 asymptotics of the Riemannian 10j symbols are given by: fifioHv) fi Z fifiovHHv)))o_~~~HHHvv)))fifi~ 16~-2 Y KD (|y - y |) dy2_ . .d.y5_. fi o__o~~~~~HHvvv)fifi (R3)4k = f^(,)g^(,) d,, ,2S(a) 16 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, AND GREG EGAN we form a Hilbert space Ha consisting of all solutions f with < 1. This Hilbert space becomes a representation of the Euclidean group where each group element g acts via (gf)(x) = f(g-1 x). In fact, this representation is unitary and irreducible. The functions exp (i, . x) form a `basis', in the sense that any element of Ha can be expressed as in equation (28) for a unique square-integrable function f^on the sphere. We define a `degenerate spin network' to be a directed graph with each edge e labelled by a representation of this form, or equivalently, a positive number ae. An intertwiner between tensor products of these representations can be defined at each vertex by taking the product of functions from the representations labelling the incoming edges, multiplying it by the product of complex conjugates of functions from representations labelling the outgoing edges, and integrating the result over R3. Given this, the standard way to evaluate such a spin network [18 ] would be to take a `trace': that is, integrate over a basis label ,e 2 S(ae) for each edge of the graph. The result would be: Z Y " Z #Y dx __v_ Q R3 S(a )exp (i(xs(e)- xt(e)) . ,e) d,e 2ss2. v2V e2E e v2V Here E denotes the set of edges of the graph, V denotes its set of vertices, and the vertices s(e) and t(e) are the source and target of the edge e. However, just as in the Lorentzian case [12 ], this gives a divergent integral, because the integrand is invariant when we simultaneously translate all the vectors xv 2 R3 by the same amount. More generally, if the underlying graph of our spin network consists of several connected components, and we translate the vectors xv where v lies in any one component, the integrand does not change. To keep things simple, let us consider only spin networks whose underlying graph is connected. In this case we can sometimes obtain a well-defined integral by `gauge-fixing' one of the vectors rather than integrating over it: that is, setting xv1 = x 2 R3 for some vertex v1 2 V . If we let V 0= V - {v1}, this gives the following formula for evaluating a degenerate spin network with edges labelled by the numbers ae: Z Y "Z #Y (29) ID (a) = Q exp (i(xs(e)- xt(e)) . ,e) d,e dxv_2. R3 e2E S(ae) v2V 02ss v2V 0 As in the Lorentzian case [19 ], one can show that if this integral converges, the result does not depend on our choice of the special vertex v1 or the point x 2 R3. Assuming the integral does converge, we can use the Kirillov trace formula (23) to reexpress it as: Z Y Y (30) ID (a) = Q KDae(|xs(e)- xt(e)|) dxv_2. R3 e2E v2V 02ss v2V 0 When the evaluation of a degenerate spin network converges, it always obeys a very simple scaling law as we multiply all the edge labels ae by the same constant ~. Using the scaling property of the degenerate kernel noted in equation (20), we find that (31) ID (~a) = ~|E|-3(|V |-1)ID (a), where |E| is the number of edges in the underlying graph and |V | is the number of vertices. We have already argued that the asymptotics of the Riemannian 10j symbols are governed by the corresponding degenerate spin network. We can generalize this argument as follows. Fix a connected graph. If we label each edge e by a positive integer ae _ or equivalently a spin je with ae = 2je + 1 _ we obtain a Riemannian spin network, whose evaluation we ASYMPTOTICS OF 10j SYMBOLS 17 define by: Z Y Y (32) IR (a) = Q KRae(d(xs(e), xt(e))) dxv_2. S3 e2E v2V 02ss v2V 0 Here d(x, y) is the distance between points x, y in the unit 3-sphere in R4 as measured by the induced Riemannian metric. This formula is equivalent to the standard integral formula [20 ], except that we have omitted the usual signs in order to simplify the relationshipQto degenerate spin networks. Fixing a small open ball U around some point (x, . .,.x) 2 v2V 0S3 we define the `degenerate contribution' to this integral to be: Z Y Y (33) IRdeg(a) = 2|V |-1 KRae(d(xs(e), xt(e))) dxv_2. U e2E v2V 02ss Just as we included a factor of 16 in equation (17), here we include a factor of 2|V |-1 to take into account the contribution of anti-parallel degenerate points; as before there is no cancellation between these if the spins labelling edges incident to each vertex sum to an integer, as they must for the spin network to have a nonzero value. Using the same nonrigorous argument as in Section 3, we see that as ~ ! 1, Z Y Y IRdeg(~a)~ 2|V |-1 KD~ae(|xs(e)- xt(e)|) dxv_2 U e2E v2V 02ss Z Y Y (34) = 2|V |-1~|E|-3(|V |-1) KDae(|ys(e)- yt(e)|) dyv_2 ~U e2E v2V 02ss ~ 2|V |-1~|E|-3(|V |-1)ID (a) = 2|V |-1ID (~a), Q where U now denotes an open ball around the origin of v2V 0R3, and we made the change of variables ye = ~xe. In short, this argument suggests that the asymptotics of the degenerate contribution to the value of a Riemannian spin network are proportional to those of the corresponding degenerate spin network: IRdeg(~a) ~ 2|V |-1ID (~a), and we know the latter are very simple: ID (~a) = ~|E|-3(|V |-1)ID (a). This is particularly interesting when we also have IR (~a) ~ IRdeg(~a), because then we can compute the asymptotics of a Riemannian spin network by evaluating a degenerate spin network: IR (~a) ~ 2|V |-1~|E|-3(|V |-1)ID (a). When can we expect this to occur? Clearly we should at least demand that the degenerate contribution outweigh the contribution of stationary phase points. A simple power-counting argument as in Section 2 suggests that the contribution of stationary phase points is of order: 8 3_ > 2 (35) IRstat(~a) = > O(~- 1_2) |V | = 2 : O(~) |V | = 1, 18 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, AND GREG EGAN where the graphs with one or two vertices are different because there is less need for `gauge- fixing'. Comparing these asymptotics to those of the degenerate contribution, we can for- mulate the following: Conjecture 2. Given a connected graph with more than two vertices and |E| > 3_2|V |, or two vertices and |E| > 2, or one vertex and |E| > 1, as ~ ! 1 we have IR (~a) ~ 2|V |-1~|E|-3(|V |-1)ID (a) as long as the integral defining ID (a) converges and the spins je labelling the edges incident to each vertex are admissible. Here we say the spins labelling the edges incident to some vertex are `admissible' if they sum to an integer and each is less than or equal to the sum of the rest. We do not yet have general criteria for when the integrals associated to Euclidean spin networks converge, and as we shall see, the relevant theorems are bound to be a bit different than in the Lorentzian case [19 ]. The simplest test of this conjecture is the `theta network', with two vertices joined by three edges, labelled by positive integers a, b, and c. When the corresponding spins are admissible, the Riemannian theta network evaluates to: 0 a 1 R __________________________________________________________* *_________________________ B ___________________________________________________________* *_______________________bC (36) B@o_____________________________________________oC= 1. __________________________________________________________* *____________________A _c__________________________________________ The degenerate theta network can also be explicitly evaluated; assuming without loss of generality that a b c: 0 a 1 D 8 _________________________________________________________________* *__________________>0c > a + b B __________________________________________________________________* *________________bC< (37) B@o_____________________________________________oC= 1_ c = a + b _________________________________________________________________* *_____________A>:41_ _c__________________________________________ 2 c < a + b. Since the Riemannian network's spins are admissible, the third inequality must hold for the corresponding areas a, b, c. Thus in this case the conjecture gives an exact formula for the Riemannian spin network. The next simplest case is the `4j symbol': the spin network with two vertices joined by four edges, labelled by positive integers a, b, c and d. Without loss of generality let us assume a b c d. As noted in a previous paper in this series [6], the Riemannian 4j symbol counts the dimension of a space of SU (2) intertwiners. Using this it follows that: 0 a___________________1R * * @ ________________________________________________________________________ * * @ B __________________________________________________________________________b____* *_____________________________________________@ (38) B@o_______________________________________________________________________________* *_____________________________________________@ __________________________________________________________________cA * * @ ______________________________________________ * * @ d The corresponding degenerate spin network evaluates to: 0 a___________________1D * * @ ________________________________________________________________________ * * @ B __________________________________________________________________________b____* *_____________________________________________@ (39) B@o_______________________________________________________________________________* *_____________________________________________@ __________________________________________________________________cA * * @ ______________________________________________ * * @ d * * @ so the conjecture is again exact. ASYMPTOTICS OF 10j SYMBOLS 19 An interesting check on our hypotheses is the tetrahedral spin network. This has four vertices and |E| = 3_2|V |, so the hypotheses of Conjecture 2 do not apply: we expect the stationary phase contribution to the Riemannian tetrahedral network to be comparable to the degenerate contribution. The degenerate tetrahedral network evaluates to: 0 o1 1 D flfl11| BB flflfl11d||C B aflflfl|1c11CC 1 (40) BBB flflfol|eqq111qfMMMMCC= ____________________, B@oflfl__________11qqqqqMMMMCC24ssV (a, b, c, d, e, f) b oA where V (a, b, c, d, e, f) is defined as the volume of the tetrahedron dual to the tetrahedral network. Each triangle in this dual tetrahedron corresponds to a vertex of the tetrahedral network, and the three sides of the triangle have lengths equal to the labels on the three edges incident to the network vertex: flflo111| flflb11|| fflflflfl1e11| (41) V (a, b, c, d, e, f) = the volume of flflqo|qMM111M. flflcqqqqqaMM11MMM ofl_____d_______oq On the other hand, the Riemannian tetrahedral network evaluates to the square of the SU (2) tetrahedral network, the basic building-block of the Ponzano-Regge model. Thanks to the calculation of Ponzano and Regge [1], later made rigorous by Roberts [2], this means that: 0 o1 1 R flfl11| BB flflfl11d||C BB aflflfl|1c11CC cos2(S + ss_) BB flflfol|eqq111qfMMMMCC~ ______________4_______a_b_c_d_e_f_ (42) B@oflfl__________11qqqqqMMMMCC12ssV ( 2, 2, 2, 2, 2, 2) b oA 1 + cos2(S + ss_4) = ______________________. f 24ssV ( a_2, b_2, c_2, d_2, e_2, __2) Here we are dealing with a dual tetrahedron whose edge lengths are 1_2akl, where akl ranges over a, b, c, d, e, f as 1 k < l 4. This tetrahedron has Regge action X S = 1_2akl`kl, 1 k 0} with its induced Riemannian metric. We define the Lorentzian kernel KL by: (44) KLa(OE) := sinaOE_sinhOE. We warn the reader that this convention differs from that of most previous papers [5, 12, 19], which include a factor of a in the denominator. Including that factor would divide any Lorentzian spin network by the product of its edge labels, so for example, it would divide the asymptotics of the Lorentzian 10j symbols as defined here by a factor of ~10. The same line of argument by which we arrived at our conjecture concerning asymptotics of Riemannian spin networks applies to Lorentzian ones. The most important difference is that no factor of 2|V |-1appears, since there are no `antipodal points' in hyperbolic space. We have not investigated criteria for the existence of stationary phase points, but where they are present their exponents will be the same as in the Riemannian case, leading us to make: Conjecture 3. Given a connected graph with more than two vertices and |E| > 3_2|V |, or two vertices and |E| > 2, or one vertex and |E| > 1, as ~ ! 1 we have IL (~a) ~ ~|E|-3(|V |-1)ID (a) as long as the integral defining ID (a) converges and the positive numbers ae labelling edges incident to each vertex are admissible. Here we say the positive numbers labelling the edges incident to some vertex are `admissible' if each is strictly less than the sum of the rest. Again the simplest test of this conjecture is the Lorentzian theta network. Translating their result into our notation, a calculation of Barrett and Crane [12 ] shows that for any a, b, c > 0, 0 a 1 L ___________________________________________________________________________________ BB __________________________________________________________________________________b_CC1_ @ o__________________________________________________________________________________oA= 4 [* *f(-a + b + c) + f(a - b + c) + f(a + b - c) -@ __________________________________________________________________________ c_________ where ss f(k) = tanh( __2k). ASYMPTOTICS OF 10j SYMBOLS 21 As the conjecture predicts, the asymptotics of this match those of the degenerate theta network, which are given by: 0 a 1 D ___________________________________________________________________________________ BB__________________________________________________________________________________b_CC 1_ @ o__________________________________________________________________________________oA= 4 [sign(* *-a + b + c) + sign(a - b + c) + sign(a + b - @ __________________________________________________________________________ c_________ It is worth noting that while the integral for the Lorentzian theta network converges even when we take the absolute value of the integrand, this fails for the degenerate theta network. This makes it more challenging to find criteria for convergence of degenerate spin networks, since we cannot simply mimic the theory that applies in the Lorentzian case [19 ]. Barrett and Crane also worked out the Lorentzian 4j symbols, obtaining: 0 a___________________1L ________________________________________________________________________ BB __________________________________________________________________________b_____________________* *_____________________________________________@ @ o________________________________________________________________________________________________* *_____________________________________________@ ____________________________________________________________________________c d -g(a + b - c - d) - g(a - b + c - d) - g(a - b - c + d) - g(a + b + c + d)], where g(k) = k_2coth( ss_2k). From equation (39) one can show: 0 a___________________1D _________________________________________________________* *_______________ BB __________________________________________________________* *________________b____________________________@ @ o__________________________________________________________* *_____________________________________________@ ________________________________________________________* *____________________c d 1_[h(-a + b + c + d) + h(a - b + c + d) + h(a + b - c + d) + h(a + b + c - d) 4 -h( a + b - c - d) - h(a - b + c - d) - h(a - b - c + d) - h(a + b + c + d)], where h(k) = k_2sign(k). Thus the conjecture is also confirmed in this case. Next, consider the tetrahedral spin network. As in the Riemannian case, Conjecture 3 does not apply, but we can still predict the asymptotic behaviour of the contribution of the fully degenerate point: 0 o1 1 L flfl11| BB flflfl11~||C BB ~flflfl| 1~11CC 1 BB flflflo|~qq111q~MMMMCC~ _____________________+ SP B@oflfl__________11qqqqqMMMMCC24ssV (~, ~, ~, ~, ~, ~) ~ oA p __ = __2_4ss~-3 + SP, 22 JOHN C. BAEZ, J. DANIEL CHRISTENSEN, AND GREG EGAN where `SP ' represents the contribution from stationary phase points, if any. Below is a log-log plot comparing this prediction to numerical data. The horizontal axis in this graph represents ~, while the vertical axis represents the value of the tetrahedral spin network. The most interesting test of Conjecture 3 is the 10j symbol. If the conjecture is true, the Lorentzian 10j symbol should be asymptotic to the degenerate 10j symbol, and therefore asymptotic to 1=16 times the Riemannian 10j symbol. Since the Riemannian 10j symbol is positive [5], this in turn would imply that the Lorentzian and degenerate 10j symbols are positive in the ~ ! 1 limit. It is difficult to compute the Lorentzian 10j symbol, but we have numerically checked the conjecture in the special case where all the edges are labelled by the same number ~. In this case the conjecture says that: 0 )H 1 L ~vvovHH~Hv~~)) BB ovv))~____HHo_~~HHv))C Z Y dx dx B@ ~))~~~~HHH~~~Hvvvv~)))CC:= KL~(OEkl) __2_2. ._.5_2 ))~~~HHH~~~H~vvvv))A (H3)4 k