Introduction The Poissonian City Variance and efficiency Flows Conclusion References. The Poissonian City. Wilfrid Kendall.

Transcription

1 The Poissonian City Wilfrid Kendall Mathematics of Phase Transitions Past, Present, Future 13 November 2009

2 A problem in frustrated optimization Consider N cities x (N) = {x 1,..., x N } in square side N.

3 A problem in frustrated optimization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess road network G = G(x (N) ) connecting cities by:

4 A problem in frustrated optimization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess road network G = G(x (N) ) connecting cities by: network total road length len(g)

5 A problem in frustrated optimization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess road network G = G(x (N) ) connecting cities by: network total road length len(g) (minimized by Steiner minimum tree ST(x (N) ));

6 A problem in frustrated optimization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess road network G = G(x (N) ) connecting cities by: network total road length len(g) (minimized by Steiner minimum tree ST(x (N) )); versus average network distance between two random cities, average(g) = 1 N(N 1) dist G (x i, x j ), i j

7 A problem in frustrated optimization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess road network G = G(x (N) ) connecting cities by: network total road length len(g) (minimized by Steiner minimum tree ST(x (N) )); versus average network distance between two random cities, average(g) = 1 N(N 1) dist G (x i, x j ), (minimized by laying tarmac for complete graph). i j

8 Questions Aldous and Kendall (2008) provide answers for the First Question Consider a configuration x (N) of N cities in [0, N] 2 as above, and a well-chosen connecting network G = G(x (N) ). How does large-n trade-off between len(g) and average(g) behave? (And how clever do we have to be to get a good trade-off?)

9 Questions Aldous and Kendall (2008) provide answers for the First Question Consider a configuration x (N) of N cities in [0, N] 2 as above, and a well-chosen connecting network G = G(x (N) ). How does large-n trade-off between len(g) and average(g) behave? (And how clever do we have to be to get a good trade-off?) Note: len(st(x (N) )) is no more than O(N) (Steele 1997, 2.2);

10 Questions Aldous and Kendall (2008) provide answers for the First Question Consider a configuration x (N) of N cities in [0, N] 2 as above, and a well-chosen connecting network G = G(x (N) ). How does large-n trade-off between len(g) and average(g) behave? (And how clever do we have to be to get a good trade-off?) Note: len(st(x (N) )) is no more than O(N) (Steele 1997, 2.2); Average Euclidean distance between two randomly chosen cities is at most 2N;

11 Questions Aldous and Kendall (2008) provide answers for the First Question Consider a configuration x (N) of N cities in [0, N] 2 as above, and a well-chosen connecting network G = G(x (N) ). How does large-n trade-off between len(g) and average(g) behave? (And how clever do we have to be to get a good trade-off?) Note: len(st(x (N) )) is no more than O(N) (Steele 1997, 2.2); Average Euclidean distance between two randomly chosen cities is at most 2N; Perhaps increasing total network length by const N α might achieve average network distance no more than order N β longer than average Euclidean distance?

12 Further Questions Today I focus on answers to yet further questions including: Question about fluctuations Given a good compromise between average(g) and len(g), how might the variance behave?

13 Further Questions Today I focus on answers to yet further questions including: Question about fluctuations Given a good compromise between average(g) and len(g), how might the variance behave? Question about true geodesics The upper bound is obtained by controlling non-geodesic paths. How might true geodesics behave?

14 Further Questions Today I focus on answers to yet further questions including: Question about fluctuations Given a good compromise between average(g) and len(g), how might the variance behave? Question about true geodesics The upper bound is obtained by controlling non-geodesic paths. How might true geodesics behave? Question about flows Consider a network which exhibits good trade-offs. What can be said about flows in this network?

15 Answer to first question (I) Augment Steiner tree by a low-intensity invariant Poisson line process Π.

16 Answer to first question (I) Augment Steiner tree by a low-intensity invariant Poisson line process Π. Unit intensity is 1 2 d r d θ: we will use this and scale.

17 Answer to first question (I) Augment Steiner tree by a low-intensity invariant Poisson line process Π. Unit intensity is 1 2 d r d θ: we will use this and scale. Pick two cities x and y at distance n = N units apart.

18 Answer to first question (I) Augment Steiner tree by a low-intensity invariant Poisson line process Π. Unit intensity is 1 2 d r d θ: we will use this and scale. Pick two cities x and y at distance n = N units apart. Remove lines separating the two cities;

19 Answer to first question (I) Augment Steiner tree by a low-intensity invariant Poisson line process Π. Unit intensity is 1 2 d r d θ: we will use this and scale. Pick two cities x and y at distance n = N units apart. Remove lines separating the two cities; focus on cell C x,y containing the two cities.

20 Answer to first question (II) Upper-bound network distance between two cities by

21 Answer to first question (II) Upper-bound network distance between two cities by mean semi-perimeter of cell, 1 [ ] 2 E len C x,y = n + 4 ( log n + γ + 5 )

22 Answer to first question (II) Upper-bound network distance between two cities by mean semi-perimeter of cell, 1 [ ] 2 E len C x,y = n + 4 ( log n + γ + 5 ) Aldous and Kendall (2008) apply this to resolve our First Question, and use other methods from stochastic geometry to show that the resolution is nearly optimal.

23 Links to random metric spaces The study of the metric space generated by the line process forms a chapter in the theory of random metric spaces:

24 Links to random metric spaces The study of the metric space generated by the line process forms a chapter in the theory of random metric spaces: Vershik (2004) builds random metric spaces out of random distance matrices (compare MDS in statistics); almost all such metric spaces are isometric to Urysohn s celebrated universal metric space.

25 Links to random metric spaces The study of the metric space generated by the line process forms a chapter in the theory of random metric spaces: Vershik (2004) builds random metric spaces out of random distance matrices (compare MDS in statistics); almost all such metric spaces are isometric to Urysohn s celebrated universal metric space. But these spaces are definitely not finite-dimensional!

26 Links to random metric spaces The study of the metric space generated by the line process forms a chapter in the theory of random metric spaces: Vershik (2004) builds random metric spaces out of random distance matrices (compare MDS in statistics); almost all such metric spaces are isometric to Urysohn s celebrated universal metric space. But these spaces are definitely not finite-dimensional! The Brownian map has been introduced as the limit of random quadrangulations of the 2-sphere (for example, Le Gall 2009).

27 Links to random metric spaces The study of the metric space generated by the line process forms a chapter in the theory of random metric spaces: Vershik (2004) builds random metric spaces out of random distance matrices (compare MDS in statistics); almost all such metric spaces are isometric to Urysohn s celebrated universal metric space. But these spaces are definitely not finite-dimensional! The Brownian map has been introduced as the limit of random quadrangulations of the 2-sphere (for example, Le Gall 2009). But these spaces are definitely not flat!

28 Links to random metric spaces The study of the metric space generated by the line process forms a chapter in the theory of random metric spaces: Vershik (2004) builds random metric spaces out of random distance matrices (compare MDS in statistics); almost all such metric spaces are isometric to Urysohn s celebrated universal metric space. But these spaces are definitely not finite-dimensional! The Brownian map has been introduced as the limit of random quadrangulations of the 2-sphere (for example, Le Gall 2009). But these spaces are definitely not flat! Baccelli, Tchoumatchenko, and Zuyev (2000) link to geometric spanners; they exhibit 4 π -spanner paths in Poisson-Delaunay triangulations.

29 Links to random metric spaces The study of the metric space generated by the line process forms a chapter in the theory of random metric spaces: Vershik (2004) builds random metric spaces out of random distance matrices (compare MDS in statistics); almost all such metric spaces are isometric to Urysohn s celebrated universal metric space. But these spaces are definitely not finite-dimensional! The Brownian map has been introduced as the limit of random quadrangulations of the 2-sphere (for example, Le Gall 2009). But these spaces are definitely not flat! Baccelli, Tchoumatchenko, and Zuyev (2000) link to geometric spanners; they exhibit 4 π -spanner paths in Poisson-Delaunay triangulations. Famous conjecture (late 1940 s) by D. G. Kendall: large cells in Poisson line process tessellation are nearly circular.

30 Links to random metric spaces The study of the metric space generated by the line process forms a chapter in the theory of random metric spaces: Vershik (2004) builds random metric spaces out of random distance matrices (compare MDS in statistics); almost all such metric spaces are isometric to Urysohn s celebrated universal metric space. But these spaces are definitely not finite-dimensional! The Brownian map has been introduced as the limit of random quadrangulations of the 2-sphere (for example, Le Gall 2009). But these spaces are definitely not flat! Baccelli, Tchoumatchenko, and Zuyev (2000) link to geometric spanners; they exhibit 4 π -spanner paths in Poisson-Delaunay triangulations. Famous conjecture (late 1940 s) by D. G. Kendall: large cells in Poisson line process tessellation are nearly circular. Now known to be true (Miles, Kovalenko).

31 Introducing the Poissonian city Consider an idealized network in a disk of radius n:

32 Introducing the Poissonian city Consider an idealized network in a disk of radius n: long-range connections use Poisson line process;

33 Introducing the Poissonian city Consider an idealized network in a disk of radius n: long-range connections use Poisson line process; connect points x and y thus:

34 Introducing the Poissonian city Consider an idealized network in a disk of radius n: long-range connections use Poisson line process; connect points x and y thus: proceed from x in opposite direction to y,

35 Introducing the Poissonian city Consider an idealized network in a disk of radius n: long-range connections use Poisson line process; connect points x and y thus: proceed from x in opposite direction to y, till one hits the line process,

36 Introducing the Poissonian city Consider an idealized network in a disk of radius n: long-range connections use Poisson line process; connect points x and y thus: proceed from x in opposite direction to y, till one hits the line process, proceed clockwise or anti-clockwise round cell formed by lines not separating x from y,

37 Introducing the Poissonian city Consider an idealized network in a disk of radius n: long-range connections use Poisson line process; connect points x and y thus: proceed from x in opposite direction to y, till one hits the line process, proceed clockwise or anti-clockwise round cell formed by lines not separating x from y, till y occludes x,

38 Introducing the Poissonian city Consider an idealized network in a disk of radius n: long-range connections use Poisson line process; connect points x and y thus: proceed from x in opposite direction to y, till one hits the line process, proceed clockwise or anti-clockwise round cell formed by lines not separating x from y, till y occludes x, then proceed directly to y.

39 Introducing the Poissonian city Consider an idealized network in a disk of radius n: long-range connections use Poisson line process; connect points x and y thus: proceed from x in opposite direction to y, till one hits the line process, proceed clockwise or anti-clockwise round cell formed by lines not separating x from y, till y occludes x, then proceed directly to y. Thus we can connect all pairs of points in the disk.

40 Density of point of maximal y-coordinate Density / intensity Using u = 2 n x ( 1, 1) and v = y n > 0, 1 ( 4 (sin β + sin γ sin(β + γ)) exp ( v 3 ( ) 1 u 2 2 exp ) 1 2 (η n) d x d y ) v2 1 u 2 d u d v

41 Density of point of maximal y-coordinate Density / intensity Using u = 2 n x ( 1, 1) and v = y n > 0, 1 ( 4 (sin β + sin γ sin(β + γ)) exp ( v 3 ( ) 1 u 2 2 exp ) 1 2 (η n) d x d y ) v2 1 u 2 ASYMPTOTICALLY d u d v Location of maximum is uniformly distributed;

42 Density of point of maximal y-coordinate Density / intensity Using u = 2 n x ( 1, 1) and v = y n > 0, 1 ( 4 (sin β + sin γ sin(β + γ)) exp ( v 3 ( ) 1 u 2 2 exp ) 1 2 (η n) d x d y ) v2 1 u 2 ASYMPTOTICALLY d u d v Location of maximum is uniformly distributed; Conditional height of maximum is length of Gaussian 4-vector.

43 Variance and growth process (I) Consider cell boundary as path with x-coordinate X τ, parametrized by τ (excess over geodesic distance).

44 Variance and growth process (I) Consider cell boundary as path with x-coordinate X τ, parametrized by τ (excess over geodesic distance). Let Θ be angle with positive x-axis;

45 Variance and growth process (I) Consider cell boundary as path with x-coordinate X τ, parametrized by τ (excess over geodesic distance). Let Θ be angle with positive x-axis; Θ jumps at Poisson point process intensity 1 2 ;

46 Variance and growth process (I) Consider cell boundary as path with x-coordinate X τ, parametrized by τ (excess over geodesic distance). Let Θ be angle with positive x-axis; Θ jumps at Poisson point process intensity 1 2 ; A jump Θ = Θ Θ obeys P [Θ Θ θ Θ ] = 1 cos θ 1 cos Θ.

47 Variance and growth process (II) Define σ (n) by X σ (n) = n.

48 Variance and growth process (II) Define σ (n) by X σ (n) = n. Then M τ = 1 2 X τ 3 4 τ is almost a martingale, and X σ (n) is almost a self-similar Markov process.

49 Variance and growth process (II) Define σ (n) by X σ (n) = n. Then M τ = 1 2 X τ 3 4τ is almost a martingale, and X σ (n) is almost a self-similar Markov process. Hence (tautologically!) σ (n) = 2 ( 3 log n + log Θ log X ) σ (n) 2M σ (n) n

50 Variance and growth process (II) Define σ (n) by X σ (n) = n. Then M τ = 1 2 X τ 3 4τ is almost a martingale, and X σ (n) is almost a self-similar Markov process. Hence σ (n) = 2 ( 3 log n + O(1) + log X ) σ (n) 2M σ (n) n Dispose of log Θ2 0 2 contribution to second moment by integration;

51 Variance and growth process (II) Define σ (n) by X σ (n) = n. Then M τ = 1 2 X τ 3 4τ is almost a martingale, and X σ (n) is almost a self-similar Markov process. Hence σ (n) = 2 3 log n + ( O(1) 2M σ (n) ) Dispose of log X σ (n) n contribution to second moment using Lamperti transformation and work of Bertoin and Yor (2005);

52 Variance and growth process (II) Define σ (n) by X σ (n) = n. Then M τ = 1 2 X τ 3 4τ is almost a martingale, and X σ (n) is almost a self-similar Markov process. Hence σ (n) = 2 3 log n + ( O(1) 2M σ (n) ) M σ (n) derives from a uniformly integrable L 2 martingale;

53 Variance and growth process (II) Define σ (n) by X σ (n) = n. Then M τ = 1 2 X τ 3 4τ is almost a martingale, and X σ (n) is almost a self-similar Markov process. Hence and thus σ (n) = 2 3 log n + ( O(1) 2M σ (n) ) E [σ (n)] = 2 log n + O(1) 3 Var [σ (n)] = 20 ( ) 27 log n + O log n.

54 Variance and growth process (II) Define σ (n) by X σ (n) = n. Then M τ = 1 2 X τ 3 4τ is almost a martingale, and X σ (n) is almost a self-similar Markov process. Hence and thus σ (n) = 2 3 log n + ( O(1) 2M σ (n) ) E [σ (n)] = 2 log n + O(1) 3 Var [σ (n)] = 20 ( ) 27 log n + O log n We deduce that perimeter length fluctuations are O ( log n )..

55 What about true geodesics? The above fluctuation theory shows that true geodesics typically have smaller excess than our paths;

56 What about true geodesics? The above fluctuation theory shows that true geodesics typically have smaller excess than our paths; however

57 What about true geodesics? The above fluctuation theory shows that true geodesics typically have smaller excess than our paths; however the number of top-to-bottom crossings of a true geodesic is stochastically bounded in any region [na, nb] (0, n), so all but a stochastically bounded number of short-cuts must be within O(n/ log n) of start or end, affecting coefficient of log(n) but no more;

58 What about true geodesics? The above fluctuation theory shows that true geodesics typically have smaller excess than our paths; however the number of top-to-bottom crossings of a true geodesic is stochastically bounded in any region [na, nb] (0, n), so all but a stochastically bounded number of short-cuts must be within O(n/ log n) of start or end, affecting coefficient of log(n) but no more; indeed

59 What about true geodesics? The above fluctuation theory shows that true geodesics typically have smaller excess than our paths; however the number of top-to-bottom crossings of a true geodesic is stochastically bounded in any region [na, nb] (0, n), so all but a stochastically bounded number of short-cuts must be within O(n/ log n) of start or end, affecting coefficient of log(n) but no more; indeed we can prove that any path of length n built using the Poisson lines must have mean excess at least ( log 4 5 ) log n + o(log n). 4

60 How much better can a true geodesic be? Methods:

61 How much better can a true geodesic be? Methods: [ n ] E (sec θ x 1) d x n 1 [ ] E θx 2 d x

62 How much better can a true geodesic be? Methods: [ n ] E (sec θ x 1) d x n 1 [ ] E θx 2 d x P [ θ x u] ( E [ exp( ul + x )]) 2

63 How much better can a true geodesic be? Methods: [ n ] E (sec θ x 1) d x n 1 [ ] E θx 2 d x P [ θ x u] ( E [ exp( ul + x )]) 2 2 p p e p2 /2 p e s2 /2 d s 4 p p

64 How much better can a true geodesic be? Methods: [ n ] E (sec θ x 1) d x n 1 [ ] E θx 2 d x P [ θ x u] ( E [ exp( ul + x )]) 2 2 p p e p2 /2 p e s2 /2 d s 4 p p Birnbaum (1942) Sampford (1953)

65 Flow in a model network (I) Consider an idealized network in a disk of radius n, conditioned to have a line passing through the origin: Consider the region in 4-space determined by pairs of points for which such a connection passes through o.

66 Flow in a model network (I) Consider an idealized network in a disk of radius n, conditioned to have a line passing through the origin: Consider the region in 4-space determined by pairs of points for which such a connection passes through o. What is the limiting distribution of the 4-volume of this region as n?

67 Flow in a model network (II) Mean value of V n is asymptotically 2n 3.

68 Flow in a model network (II) Mean value of V n is asymptotically 2n 3. Dominant contribution comes from opposing pairs of points nearly aligned with conditioned line l (angle of order about 1 n ).

69 Flow in a model network (II) Mean value of V n is asymptotically 2n 3. Dominant contribution comes from opposing pairs of points nearly aligned with conditioned line l (angle of order about 1 n ). Very heavy variance calculations show, second moment of V n /n 3 is uniformly bounded ("no congestion").

70 Flow in a model network (II) Mean value of V n is asymptotically 2n 3. Dominant contribution comes from opposing pairs of points nearly aligned with conditioned line l (angle of order about 1 n ). Very heavy variance calculations show, second moment of V n /n 3 is uniformly bounded ("no congestion"). In fact a limiting construction, involving a network based on an improper anisotropic line process, generates a non-trivial limiting distribution for V n /n 3.

71 Improper Anisotropic Line Process (I) Limiting process: scale x-axis by n and y-axis by n. The resulting improper line process: has an improper rose of directions (there is a singularity at vertical directions);

72 Improper Anisotropic Line Process (I) Limiting process: scale x-axis by n and y-axis by n. The resulting improper line process: has an improper rose of directions (there is a singularity at vertical directions); possesses a special affine symmetry group;

73 Improper Anisotropic Line Process (I) Limiting process: scale x-axis by n and y-axis by n. The resulting improper line process: has an improper rose of directions (there is a singularity at vertical directions); possesses a special affine symmetry group; can be easily shown to yield non-degenerate flow volume.

74 Improper Anisotropic Line Process (II)

75 Improper Anisotropic Line Process (II)

76 Improper Anisotropic Line Process (II)

77 Improper Anisotropic Line Process (II)

78 Improper Anisotropic Line Process (II)

79 Improper Anisotropic Line Process (II) Open questions:

80 Improper Anisotropic Line Process (II) Open questions: analytical characterization of flow volume distribution?

81 Improper Anisotropic Line Process (II) Open questions: analytical characterization of flow volume distribution? good simulation methodology?

82 What I don t know yet Computing User Equilibrium for flows.

83 What I don t know yet Computing User Equilibrium for flows. We can in principle calculate asymptotic mean flow at any particular point in the disk, and hence compute User / Wardrop / Nash Equilibria so as to minimize objective including distance travelled and traffic encountered if

84 What I don t know yet Computing User Equilibrium for flows. We can in principle calculate asymptotic mean flow at any particular point in the disk, and hence compute User / Wardrop / Nash Equilibria so as to minimize objective including distance travelled and traffic encountered if users can choose which of two routes (left or right) to travel;

85 What I don t know yet Computing User Equilibrium for flows. We can in principle calculate asymptotic mean flow at any particular point in the disk, and hence compute User / Wardrop / Nash Equilibria so as to minimize objective including distance travelled and traffic encountered if users can choose which of two routes (left or right) to travel; or users can choose to use only some of the available lines (thinning the Poisson line process).

86 What I don t know yet Computing User Equilibrium for flows. We can in principle calculate asymptotic mean flow at any particular point in the disk, and hence compute User / Wardrop / Nash Equilibria so as to minimize objective including distance travelled and traffic encountered if users can choose which of two routes (left or right) to travel; or users can choose to use only some of the available lines (thinning the Poisson line process). Moving away from lines.

87 What I don t know yet Computing User Equilibrium for flows. We can in principle calculate asymptotic mean flow at any particular point in the disk, and hence compute User / Wardrop / Nash Equilibria so as to minimize objective including distance travelled and traffic encountered if users can choose which of two routes (left or right) to travel; or users can choose to use only some of the available lines (thinning the Poisson line process). Moving away from lines. results will not change qualitatively if lines replaced by long segments. How long?

88 What I don t know yet Computing User Equilibrium for flows. We can in principle calculate asymptotic mean flow at any particular point in the disk, and hence compute User / Wardrop / Nash Equilibria so as to minimize objective including distance travelled and traffic encountered if users can choose which of two routes (left or right) to travel; or users can choose to use only some of the available lines (thinning the Poisson line process). Moving away from lines. results will not change qualitatively if lines replaced by long segments. How long? results will not change qualitatively if line segments replaced by curves of low curvature. How low?

89 Conclusion Aldous and Kendall (2008) showed that

90 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost;

91 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently.

92 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. (and Poisson line processes are not computationally hard!)

93 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. (and Poisson line processes are not computationally hard!) Random variation of network distance is controlled.

94 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. (and Poisson line processes are not computationally hard!) Random variation of network distance is controlled. Near-geodesics are pretty good.

95 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. (and Poisson line processes are not computationally hard!) Random variation of network distance is controlled. Near-geodesics are pretty good. Traffic flow in the network scales well.

96 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. (and Poisson line processes are not computationally hard!) Random variation of network distance is controlled. Near-geodesics are pretty good. Traffic flow in the network scales well. User equilibrium? Line segments or curves?

97 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. (and Poisson line processes are not computationally hard!) Random variation of network distance is controlled. Near-geodesics are pretty good. Traffic flow in the network scales well. User equilibrium? Line segments or curves? Same problem in 3-space or higher dimensions?

98 Conclusion Aldous and Kendall (2008) showed that the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. (and Poisson line processes are not computationally hard!) Random variation of network distance is controlled. Near-geodesics are pretty good. Traffic flow in the network scales well. User equilibrium? Line segments or curves? Same problem in 3-space or higher dimensions? QUESTIONS?

99 Bibliography This is a rich hypertext bibliography. Aldous, D. J. and W. S. Kendall (2008, March). Short-length routes in low-cost networks via Poisson line patterns. Advances in Applied Probability 40(1), 1 21,, and Baccelli, F., K. Tchoumatchenko, and S. Zuyev (2000). Markov paths on the Poisson-Delaunay graph with applications to routing in mobile networks. Advances in Applied Probability 32(1), Bertoin, J. and M. Yor (2005). Exponential functionals of Lévy processes. Probability Surveys 2, (electronic),. Birnbaum, Z. W. (1942). An inequality for Mill s ratio. Annals of Mathematical Statistics 13, Le Gall, J.-F. (2009). Geodesics in large planar maps and in the Brownian map. Acta Mathematica to appear.

100 Sampford, M. R. (1953). Some inequalities on Mill s ratio and related functions. Annals of Mathematical Statistics 24, Steele, J. M. (1997). Probability theory and combinatorial optimization, Volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Stoyan, D., W. S. Kendall, and J. Mecke (1995). Stochastic geometry and its applications (Second ed.). Chichester: John Wiley & Sons. (First edition in 1987 joint with Akademie Verlag, Berlin). Vershik, A. M. (2004). Random and universal metric spaces. In Dynamics and randomness II, Volume 10 of Nonlinear Phenom. Complex Systems, pp Dordrecht: Kluwer Acad. Publ.