3 and that ¯ Bφ ( x ) is well approximated by □ φ ( x ) when the characteristic length scale, λ , on which φ ( x ) varies is large compared to l . ¯ B is therefore effectively sampling the value of the field only in a neighbourhood of x of size of order l and the mean, at least, of B is about as local as it can possibly be, given the discreteness. To see roughly how this can happen, notice that the in- tegrand in (4) is negligible for ξ > α 4 where α is such that e − α 4 << 1. The significant part of the integration range therefore lies between the past light cone of x and the hyperboloid ξ = α 4 and comprises a part within a neigh- bourhood of x of size αl – whence the local contribution – and the rest which stretches off far down the light cone. It is this second part of the range which threatens to in- troduce nonlocality but because it can be coordinatized by ξ itself and some coordinates η a on the hyperboloid the integration over it will be proportional to If φ is nearly constant over length scale αl , the ξ integra- tion is close to zero and the contribution is suppressed. The fluctuations in Bφ ( x ), however, are a different matter: if the physical IR cutoff L is fixed and the dis- creteness scale sent to zero, i.e. , the number of causet ele- ments N grows, simulations show the fluctuations around the mean grow rather than die away and Bφ ( x ) will not be approximately equal to the continuum □ φ ( x ). To dampen the fluctuations we follow [18] and introduce an intermediate length scale l$_{k}$ ≥ l and smear out the ex- pressions above over this new scale, with the expectation that when l$_{k}$ >> l the inhering averaging will suppress the fluctuations via the law of large numbers. Thus we seek a discrete operator, B$_{k}$ , whose mean is given by (4) but with l replaced by l$_{k}$ : where now ξ := l − 4 k V ( x, y ). Working back, one can show that the discrete operator, B$_{k}$ , with this mean is where ϵ = ( l/l$_{k}$ ) 4 and B$_{k}$ reduces to B when ϵ = 1. B$_{k}$ effectively samples φ over elements in 4 broad bands with a characteristic depth l$_{k}$ , the bands’ contributions being weighted with the same set of alternating sign coefficients as in B . Since (7) is just (4) with l replaced by l$_{k}$ , the mean of B$_{k}$φ ( x ) is close to □ φ ( x ) when the characteristic scale over which φ varies is large compared to l$_{k}$ . Now, however, numerical simulations show that the fluctuations are tamed. Points were sprinkled into a fixed causal interval in M 4 between the origin and t = 1 on the t axis, at varying density ρ = N V , where volume V = π $_{24}$. For each N , 100 sprinklings were done and for each sprinkling, B$_{k}$φ was calculated at the topmost point of the interval for φ = 1 and l$_{k}$ = 0 . 16. For N = 5000, the mean was µ = 9 , 35 and the standard deviation s.d = 134 . 8. For N = 10000, µ = − 4 . 00 and s.d. = 102 . 6 and for N = 20000, µ = 1 . 12 and s.d. = 58 . 8. These results indicate that the fluctuations do die away, as anticipated, as N increases and are consistent with the dependence N ( − 1 / $^{2)}$. Further results will appear elsewhere. The operators B and B$_{k}$ derived in both 2D (in [18]) and 4D are defined in terms of the order relation on C alone and so can be applied to a scalar field on any causet. If, therefore, ( M , g ) is a (2D or 4D) curved spacetime and φ is a scalar field on M , we can compute B$_{k}$φ ( x ) on a sprinkling into M and calculate its mean. Let V$_{2}$ and V$_{4}$ be the volumes of the intervals in 2D and 4D respectively, ξ$_{2}$ := V$_{2}$ ( x, y ) l − 2 k and ξ$_{4}$ := V$_{4}$ ( x, y ) l − 4 k . Then, in the presence of curvature, and in 2D and 4D respectively. These expressions can be evaluated using Riemann normal coordinates and in both cases we find The limit is a good approximation to the mean when the field φ varies slowly over length scales l$_{k}$ and the radius of curvature r >> l$_{k}$ . If the damping of fluctuations found in simulations in flat space are indicative of what happens in curved space then, for a fixed large enough IR cutoff, L , the nonlocality length scale l$_{k}$ can be chosen such that l << l$_{k}$ << L and the value of B$_{k}$φ for a single sprinkling will be close to the mean. If B$_{k}$ is applied to the constant field φ = − 2, we therefore obtain an expression that is close to the scalar curvature of the approximating spacetime. ∫ d $^{3}$η ∫ α 4 0 dξe − $^{ξ}$(1 − 9 ξ + 8 ξ 2 − 4 3 ξ $^{3}$) φ ( ξ, η $^{a}$) . (6) ¯ B (2) k φ ( x ) = 2 l 2 k [ − φ ( x ) + 2 l 2 k ∫$_{y}$ ∈ J − ( x ) d 2 $_{y}$√ − g e − ξ 2 (1 − 2 ξ$_{2}$ + 1 2 ξ 2 $_{2}$) φ ( y ) ] (10) ¯ B (4) k φ ( x ) = 4 √$_{6}$$_{l}$ 2 k [ − φ ( x ) + 1 l 4 k ∫$_{y}$ ∈ J − ( x ) d 4 $_{y}$√ − g e − ξ 4 (1 − 9 ξ$_{4}$ + 8 ξ 2 4 − 4 3 ξ 3 $_{4}$) φ ( y ) ] , (11) ¯ B$_{k}$φ ( x ) = 4 √$_{6}$$_{l}$ 2 k [ − φ ( x ) + 1 l 4 k ∫$_{y}$ ∈ J − ( x ) d $^{4}$y φ ( y ) e − $^{ξ}$(1 − 9 ξ + 8 ξ 2 − 4 3 ξ $^{3}$) ] , (7) B$_{k}$φ ( x ) = 4 √$_{6}$$_{l}$ 2 $_{k}$[ − φ ( x ) + ϵ ∑ y ≺ x f ( n ( x, y ) , ϵ ) φ ( y ) ] , (8) lim l $_{k}$→ 0 ¯ B ( i ) k φ ( x ) = ( □ − 1 2 R ( x ) φ ( x ) . (12) f ( n, ϵ ) = (1 − ϵ ) n [ 1 − 9 ϵn 1 − ϵ + 8 ϵ $^{2}$n ! ( n − 2)!(1 − ϵ ) 2 − 4 ϵ $^{3}$n ! 3( n − 3)!(1 − ϵ ) 3 ] . (9)