L.B. Richmond and N.C. Wormald, Almost all maps are asymmetric, J. Combin. Theory B 63 (1995) 1--7.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... generalisation of the work of Farahat and Higman [2] and Macdonald to arbitrary structure constants of the class algebra of the symmetric group ring [4] and, more recently, the combinatorial investigation into free probability theory [26] Since almost all maps have only the trivial automorphism [22], asymptotic results for maps with a large number of edges can be obtained from a study of rooted maps. If l ff;fi is the number of maps with n edges, face partition ff 2n and vertex partition fi 2n; then the genus series for maps in locally orientable surfaces is defined to be M(x;y; z) ....

L.B.Richmond and N.C.Wormald, Almost all maps are asymmetric, J. Combinatorial Theory B. 63 (1995), 1--7.

....characterized by its underlying graph together with a cyclical ordering of edges around each vertex. Following Tutte [48, 49] we consider rooted maps, that is, maps with an oriented edge called the root this simpli es the analysis without essentially a ecting statistical properties (see [42] and Section 5) In order to represent maps on the plane, a point of the sphere must be placed at in nity; by convention we always choose it so that the root runs along the in nite face counterclockwise. Figure 1 illustrates this convention. From now on, unless explicitly mentioned, all maps are ....

....the number of distinct rootings of an unrooted map with n edges is equal to 2n unless the map has a symmetry. But the probability that a random unrooted map has a symmetry is exponentially small in all families of Table 2, a fact that follows from the elegant analysis of Richmond and Wormald in [42]. The proof is then easily completed by following [42] 5.3. Random sampling algorithms. Random sampling algorithms for various families of maps have been described by Schae er in [43, 44] Here, we show that all classes of maps described in Section 5.1 are amenable to ecient random generation and ....

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Richmond, L. B., and Wormald, N. C. Almost all maps are asymmetric. Journal of Combinatorial Theory. Series B 63, 1 (1995), 1-7. 48 C. BANDERIER, P. FLAJOLET, G. SCHAEFFER, AND M. SORIA

....characterized by its underlying graph together with a cyclical ordering of edges around each vertex. Following Tutte [48, 49] we consider rooted maps, that is, maps with an oriented edge called the root this simplifies the analysis without essentially affecting statistical properties (see [42] and Section 5) In order to represent maps on the plane, a point of the sphere must be placed at infinity; by convention we always choose it so that the root runs along the infinite face counterclockwise. Figure 1 illustrates this convention. From now on, unless explicitly mentioned, all maps are ....

....the number of distinct rootings of an unrooted map with n edges is equal to 2n unless the map has a symmetry. But the probability that a random unrooted map has a symmetry is exponentially small in all families of Table 2, a fact that follows from the elegant analysis of Richmond and Wormald in [42]. The proof is then easily completed by following [42] 5.3. Random sampling algorithms. Random sampling algorithms for various families of maps have been described by Schaeffer in [43, 44] Here, we show that all classes of maps described in Section 5.1 are amenable to efficient random generation ....

[Article contains additional citation context not shown here]

Richmond, L. B., and Wormald, N. C. Almost all maps are asymmetric. Journal of Combinatorial Theory. Series B 63, 1 (1995), 1--7.

....characterized by its underlying graph together with a cyclical ordering of edges around each vertex. Following Tutte [43, 44] we consider rooted maps, that is, maps with an oriented edge called the root this simpli es the analysis without essentially a ecting statistical properties (see [38] and Section 5) In order to represent maps on the plane, a point of the sphere must be placed at in nity; by convention we always choose it so that the root runs along the in nite face counterclockwise. Figure 1 illustrate this convention. From now on, unless explicitly mentioned, all maps are ....

....the number of distinct rootings of an unrooted map with n edges is equal to 2n unless the map has a symmetry. But the probability that a random unrooted map has a symmetry is exponentially small in all families of Table 2, a fact that follows from the elegant analysis of Richmond and Wormald in [38]. The proof is then easily completed by following [38] 5.3. Random sampling algorithms. Random sampling algorithms for various families of maps have been described by Schae er in [39, 40] Here, we show that all classes of maps described in Section 5.1 are amenable to ecient random generation and ....

[Article contains additional citation context not shown here]

Richmond, L. B., and Wormald, N. C. Almost all maps are asymmetric. Journal of Combinatorial Theory. Series B 63, 1 (1995), 1-7.

....too far away to interact or share the same anchor. In both cases the corresponding elements in B will satisfy the same atomic formulas. 6. Conclusion Richness properties have been proved for many families of rooted maps by Bender et al. 4, 3, 5, 6, 7] Thanks to the work of Richmond and Wormald [8] these results hold as well for unrooted maps. Using the previously described machinery it implies 0 1 laws for the following families of maps on a surface of given genus: all maps, smooth maps, 2 connected maps, 3 connected maps, triangular maps, 2 connected triangular maps and 3 connected ....

Richmond (L. B.) and Wormald (N. C.). -- Almost all maps are asymmetric. Journal of Combinatorial Theory. Series B, vol. 63, n 1, 1995, pp. 1--7.

....cells from A (the same but with nontrivial automorphism group) Earlier Tutte remarked that it is very intuitive that almost all triangulations have no nontrivial automorphism. Many rigorous results appeared afterwards, see [18, 21, 22] Proof for the case of disk triangulations see in [23]. 3.2 Metrics and Curvature The metric structure is defined once it is defined for each closed cell so that on the edges the lengths are compatible. There are two basic approaches for defining the metric structure: Dynamical Triangulations when all edges have length one and Quantum Regge ....

....mechanism. Tutte [13, 14, 15] has begun to study the asymptotics for C(N;m) and developed a beautiful and efficient quadratic method. Afterwards many authors contributed by developing the method itself and obtaining asymptotics for various classes A (see review [19] and more recent papers [23]) The main idea of Tutte are the following recurrent equations for C(N;m) N = 0; 1; m = 2; 3; C(N;m) C(N Gamma 1; m 1) X N1 N2=N Gamma1;m 1 m2=m 1 C(N 1 ; m 1 )C(N 2 ; m 2 ) m 3; N 1 C(0; 2) 1; C(0; m) 0; m 2 These equations are easily derived as follows from the ....

L. Richmond, N, Wormald. Almost all maps are asymmetric. J. Comb. Theory, B63, 1995, 1-7.

....In many cases, it is known that the number of n edged such maps is asymptotic to AB n =n 1 5=4 (unrooted case) or 4AB n =n 5=4 (rooted case) 7) where is the Euler characteristic of the surface, B depends on the type of map, and A depends on both the type of map and the surface. See [22] for a proof of (7) for the unrooted case and for further references. Zvonkin [28, p. 290] remarks that it is sometimes necessary in physics to consider maps which are not connected. In that case, each component is embedded in a separate surface. Since generating functions are often not available, ....

L.B. Richmond and N.C. Wormald, Almost all maps are asymmetric, J. Combin. Theory, Ser. B 63 (1995) 1--7.

....the simplest classes of maps. Two results in this direction for the sphere are [9] and [16] Problem 7. Develop some general tools or results that can be used to conclude that almost all maps in various classes in an arbitrary surface are asymmetric. I misjudged this one. Richmond and Wormald [17] found an elegant result based on the fact that a typical map has many copies of any given submap. 4 The previous problem is probably quite difficult, so perhaps one should (temporarily ) settle for a (still probably difficult) special case such as Problem 8. Prove that almost all maps in ....

....of any given submap. 4 The previous problem is probably quite difficult, so perhaps one should (temporarily ) settle for a (still probably difficult) special case such as Problem 8. Prove that almost all maps in some particular class in an arbitrary surface are asymmetric. In addition to [17], see the results [6] for some enumeration by vertices and faces. It is not clear what the easiest class would be. Connectivity might help establish asymmetry, so something like 3 connected maps might be easier to deal with than all maps. The asymptotic study of various classes of maps in ....

L.B. Richmond and N.C. Wormald, Almost all maps are asymmetric, J. Combin. Theory Ser. B 63 (1995) 1--7.

....A map is a planar graph given together with an embedding in the plane considered up to continuous deformations. Following Tutte, we consider rooted maps, that is, maps with an oriented edge called the root this simpli es this analysis without essentially a ecting statistical properties (see [16] and Section 4) Generically, we take M and C to be two classes of maps, with Mn , Cn the subsets of elements of size n (typically elements with n 1 edges) Here, C is always a subset of M that satis es additional properties (e.g. higher connectivity) The elements of M are then called the basic ....

....= k) that a map of size n has a core of size k has a local limit law of the map Airy type with centering constant 0 and scale parameter c. The technique of [10] relates the size of the core to the size of the largest component in random maps. Also, since maps have almost surely no symmetries [16], the analysis extends to unrooted maps. As a consequence: Theorem 6. i) Consider any scheme of Table 1 with parameters 0 and c. Let X n be the size of the largest component of in a random map of size n with uniform distribution. Then Pr X n = b 0 n xn 2=3 c = cA(cx) n ....

Richmond, L. B., and Wormald, N. C. Almost all maps are asymmetric. J. Combin. Theory Ser. B 63, 1 (1995), 1-7.

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L.B. Richmond and N.C. Wormald, Almost all maps are asymmetric, J. Combin. Theory B 63 (1995) 1--7.

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L.B. Richmond and N.C. Wormald, Almost all maps are asymmetric, J. Combin. Theory B 63 (1995) 1-7.

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L.B. Richmond and N.C. Wormald, Almost all maps are asymmetric, J. Combin. Theory B 63 (1995) 1-7.

....has less than cn vertex disjoint copies of G 0 is O(e n ) 3 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 2.02 2.04 0 0.2 0.4 0.6 0.8 1 Figure 2: The plot of 0 (t) for 0 t 1: It is interesting to note that almost all graphs or maps have no symmetries. See [10] for graphs; see [7] and [1] for maps. The situation is di erent for 2 connected planar graphs: Theorem 3 There is a constant C 1 such that almost all 2 connected planar graphs G, labeled or unlabeled, have a symmetry group of order at least C v(G) where v(G) is the number of vertices of G. As can be seen ....

L.B. Richmond and N.C. Wormald, Almost all maps are asymmetric, J. Combin. Theory Ser. B 63 (1995) 1-7.

....(e.g. 2 connected) on S. Let Mn (S) be those with n edges and M n;k (S) those with n edges and k vertices. It was shown in [3,4,5,6] that almost all maps in Mn (S) contain many copies of any reasonable planar submap. Various applications were given in those papers and Richmond and Wormald [13] used the result to show that almost all of the maps in Mn (S) are asymmetric. In this paper, we extend these ideas to M n;k (S) Since the Euler relation v Gamma e f = S) relates the number of vertices, edges and faces, n and k can equally well stand for any two of these three numbers. 2 ....

....is all 3 connected maps; 3 (d) P is any planar map with face degrees in D and M(S) is all maps with face degrees in D. Positive density of a submap implies a 1 law for that submap. See [3] for a definition. Perhaps more importantly, if we have submaps with positive density, it follows from [13] that, for every closed interval I ae (a; A) there is a constant c so that, for all sufficiently large n and each k 2 nI, the fraction of maps in M n;k (S) which have symmetries is less than c Gamman for all suffciently large n. The results in Theorem 1 are probably not best possible since ....

L. B. Richmond and N. C. Wormald, Almost all maps are asymmetric, J. Combin. Theory, Ser. B , to appear.

....for (iv) follows from [5] and [7] Large representativity for (v) follows from [5] and [20] Large representativity for (vi) follows from [5] and [21] Large representativity for (vii) follows from [5] and [22] The results in these references are all stated for rooted maps. However, by [29] they are valid as stated here. Remark. For maps on the plane, large representativity is trivial and so only richness needs to be verified for a 0 1 law to hold. Richness has been proved for many classes of planar maps. See [3] and [4] for a discussion. The proofs of richness require that the ....

L. Bruce Richmond and Nicholas C. Wormald. Almost all maps are asymmetric. J. Combin. Theory Ser. B, 63:1--7, 1995.

.... iy)j (2 ) 1=2 jyj x 1=2 exp( 2)jyj) Related functions occur in the theory of random walks as discussed by Hughes [10] Corollary 2 contrasts with the result in [9] where it was shown that almost all maps in a certain family have a unique maximum component , for various families. In [12] it was shown that almost all maps are asymmetric. Similarly, almost all triangulations of a polygon are asymmetric. This is much simpler to prove: the numbers with given symmetries can be derived, and each is exponentially smaller than the total number of maps. Since the number of symmetries is ....

L.B. Richmond and N.C. Wormald, Almost all maps are asymmetric, J. Combin. Theory, Ser. B 63 (1995), 1-7. 23

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