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Introduction Useful references Chronology Introduction Useful references Chronology

Syllabus of lectures

Part1 geometry: a brief introduction: Poisson Processes and The basic building blocks of . Spatial Transportation Networks Part2 Poisson line processes and Network Efficiency: Curious results arising from careful investigation of measures of effectiveness of networks. Wilfrid S. Kendall Part3 Traffic flow in a Poissonian city: [email protected] Using Poisson line processes to model traffic intensity in

Department of , University of Warwick a random city. Part4 Scale-invariant random spatial networks (SIRSN): 21st-22nd June 2018 Novel random metric spaces arising from the methods used to analyze effectiveness of networks. Support: Turing, EPSRC K031066, K013939, INI “Random Geometry”. Part5 Random flights on the SIRSN. How can we move about on a SIRSN modelled on Poisson line processes?

1 2 Introduction Useful references Chronology Introduction Useful references Chronology

Useful references for stochastic geometry A short chronology of stochastic geometry (The study of random pattern) Kingman (1993): concise and elegant 1. Georges-Louis Leclerc, Comte de Buffon introduction to Poisson processes; (7 September, 1707 – 16 April, 1788); Stoyan, WSK, and Mecke (1995): now in an 2. Władysław Hugo Dionizy Steinhaus rd extended 3 edition (Chiu, Stoyan, WSK, and (14 January, 1887 – 25 February, 1972); Mecke, 2013); 3. Luís Antoni Santaló Sors Schneider and Weil (2008): magisterial (9 October, 1911 – 22 November, 2001); introduction to the links between stochastic and 4. Rollo Davidson integral geometry; (8 October, 1944 – 29 July, 1970); WSK and Molchanov (2010): collection of essays 5. D.G. Kendall and K. Krickeberg coined the phrase on developments in stochastic geometry. “stochastic geometry” when preparing an Oberwolfach workshop, 1969 (also Frisch and Hammersley, 1963); Last and Penrose (2017): more on Poisson processes. 6. Mathematical morphology (G. Matheron and J. Serra); 7. arising from Molchanov (1996b): Boolean models, a way to (J. Kerstan, K. Matthes, . . . ); build random sets from Poisson point processes. 8. Combinatorial geometry (R.V. Ambartzumian).

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Lecture I Poisson point processes Humans have evolved to recognize patterns . . . Poisson Point Processes and Friends

A quiz about random pattern

Definitions of Poisson point processes

Palm theory for Poisson point processes

Examples of Poisson point processes

Marked Poisson point processes

The Boolean model

. . . we are less good at recognizing complete . 5 6 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model

Puzzle Constructive definition of Poisson 6 Using 10 years of accumulated evolutionary programming . . . Definition: unit intensity Poisson point pattern in a unit square . . . which of these patterns fail to be completely random? 1. Draw a Poisson(1) N; 2. Conditional on N n, plant n independent random = points uniformly in [0, 1]2.

This can be viewed as a binomial point process of N points, with total number of points N being Poisson-randomized. Replace unit square [0, 1]2 by region R, and Poisson(1) by Poisson((R)), to produce a unit intensity Poisson point pattern in a more general planar region R. If R is of infinite area, break it up into disjoint sub-regions of finite area and work independently in each sub-region.

Replace Poisson(area(R)) by Poisson(λ area(R)) to produce × a Poisson point pattern of intensity λ in general region R.

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Axiomatic definition of General point processes Definition: Poisson point (alternative abstract definition) A (stationary) point pattern is Poisson (“truly random”) if: 1. We can think of a point process Φ as a taking non-negative integer values (attention focussed 1. number of points in any prescribed region has a Poisson on Φ(A) the number of points in a Borel set A); distribution with mean defined by intensity measure ν (NB: ν is diffuse). EG: ν(region) λ (area of region); 2. or (equivalently) as a random closed set which is locally = × finite (attention focussed on whether the set hits 2. numbers of points in disjoint regions are independent. compact test sets K, so whether Φ(K) > 0. Basic construction: each very small region independently has 3. The variety of general point processes is huge. a very small chance of holding a point. Nevertheless, the distribution of a point process is Remark: Powerful theoretical result determined by its of void Properties1 and2 are both automatically implied if the void P [Φ(K) 0] for all compact K. = of a measurable region failing to hold any points 4. Indeed this holds more generally for random closed sets at all is exp( ν(region)) (consequence of result of Choquet). − Ξ: the distribution of Ξ is determined by the system of k ν(A) Computations: P [N(A) k] (ν(A)) e− , E [N(A)] ν(A). void probabilities P [Ξ K ] for all compact K. = = k! = ∩ = ∅ Much more theory: Kingman (1993), Chiu et al. (2013).

9 10 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model

Palm distributions (I) Palm distributions (II) Consider a general point process Φ on R2 with (σ -finite) intensity measure ν. How to condition on specified x Φ? ∈ 1. Intensity measure Definition: Palm distribution of a general point process ν(B) E [# x B Φ ] E x Φ I [x B] = { ∈ ∩ } = ∈ ∈ = If Φ is a general point process on R2 with σ -finite intensity E x B Φ 1 ; P  ∈ ∩ measure ν, then its Palm distribution P at x R2 satisfies 2. LetP be some (measurable) set of possible point x ∈ Y patterns φ (EG: K φ : φ(K) for given compact K). E h(x, Φ) h(x, φ)Px( d φ)ν( d x) ,   2 Y = Y = { = ∅} = ZR Z Campbell measure (B ) E x B Φ I [Φ ] ; xXΦ C × Y = ∈ ∩ ∈ Y  ∈  3. Radon-Nikodym theorem: (B P) P ( )ν( d x) ; C × Y = B x Y for all bounded measurable h(x, φ), for x R2 and φ a loc- ∈ 4. Use theory of regular conditional probabilities:R ally finite point pattern on R2. P ( ) can be made into a probability measure; x · 5. We interpret P ( ) as the Palm distribution at x. x · 6. The reduced Palm distribution is obtained by deleting the point x from the Palm distribution. 11 12 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model

Palm distribution of a Poisson process Application: When is a point process Poisson? Theorem: Slivnyak Is the randomness of a pattern “unstructured”? The Palm distribution of a planar Poisson point process Φ at Use statistical tests for x R2 is the superposition δ P( d φ), namely Φ with added ∈ x ∗ of point at x. point counts; Proof. Deciding whether a pattern Consider an arbitrary (compact) set K R2 and a bounded is completely random is ⊂ Borel set B R2. relatively easy (answer is ⊂ usually no!). Deciding what Let VK be the set of patterns placing no points in K. sort of pattern can be hard. Consider (B, VK) E x B I [Φ K ] . C = ∈ ∩ = ∅ But (B K,VK) 0, whileP (B K,VK) B K P(VK)ν( d x), Compare Ripley (1977) C ∩ = C \ = \ when P is the original Poisson distributionR (independence of (a) First-contact distribution Poisson process in disjoint regions). H(r ) of distance from fixed This agrees with the results obtained by replacing location to nearest point, ( d x, d ξ) by ν( d x) (δ P( d ξ)). (b) Nearest-neighbour C × x ∗ Choquet’s capacitability theorem implies measures agree. locations distribution . in Western Australia. D(r ) 13 14 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model

Example: Alpha Mephistopheles and the bus company Centres of traces of α-particles on a detector in a 60 60 × Classic bus paradox: mean between Poisson buses is square (unit of 2µ m). twice the mean of interval between bus arrivals. This follows immediately from Slivnyak’s theorem! For renewal point processes the picture can be stranger. 1. The aspirant devil Mephistopheles is forced to take a job with Hades Regional Bus Company. 2. He is responsible for operation of the No12 bus service between city centre and university campus. 3. Contract requires him to deliver mean time between bus arrivals of 10 minutes. 4. How bad can he make the service while still (statistically speaking) fulfilling the contract? In mean-value terms, Mephistopheles can make matters Locations of centres of traces (Stoyan et al., 1995, Figure 2.5ff). infinitely bad . . .

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Marked point processes Palm distribution for marks (sketch) Many constructions and applications use the notion that Independently marked Poisson point process Ψ, typical points of the pattern carry “marks”. marked point [x, m], independent mark distribution Uniform 0 1 . 1. In applications the points of a pattern are often [ , ] accompanied by extra measurements (diameters of We want to impose (Palm) conditioning by requiring a trees, yields of ore deposits, . . . ). point to lie at x, but without conditioning the corresponding mark m. 2.A marked point process on R2 can be viewed as a point Distribution of Ψ can be obtained by process on R2 M, where M is mark . × 1. Constructing un-marked point process Φ from Ψ; 3. Under stationarity we can factorize intensity measure as 2. Ordering points of Φ (eg: lexicographical); ν(A B) λ Leb(A) (B), where is the mark 3. Assigning marks from an i.i.d. sequence. × = × M M distribution. Campbell equation for marked process: E x B Φ I [Ψ ] B Px( )ν( d x). 4. If additionally marks are independent then factorization ∈ ∩ ∈ Y = Y carries through to distribution of process. IntegrateP out the marks R and apply Slivnyak to E x Φ h(x, Ψ) h(x, ψ)Px( d ψ)ν( d x). 5. An independently marked Poisson process is a Poisson ∈ = 2 RecognizehP Palmi distributionRR as superposition of (a) Ψ, process on R M. e e × and (b) point x with independent random mark m. 17 18 Quiz Definitions Palm Examples Marks Boolean model Quiz Definitions Palm Examples Marks Boolean model

Palm distribution for marked Poisson point process The Boolean model

Suppose we need a model for a Theorem: Slivnyak for independent marks random set rather than a The Palm distribution of a planar marked Poisson point pro- random point pattern. cess Ψ at x R2 (independently marked with identically Simplest approach that might ∈ distributed marks of distribution ) is the superposition work: plant a random set on top M δ ; P( d ψ), namely Ψ with added point at x furnished of each point in a Poisson [x M] ∗ with independent random mark M of distribution . process and take the union. M Theory: Hall (1988), Molchanov (1996b).

MS lesions (white ) in human brain.

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Boolean model notation A result which bridges to the next lecture Germs defined by points of a Poisson process; Grains are (independent Suppose we have a high-intensity Boolean model using random compact) sets disks for grains. What is the approximate statistical placed on each grain behaviour of the vacant regions? (control size to ensure At high intensity, vacancies are small and rare. In the vacant regions exist); limit, boundaries are locally straight: the vacancy will the Boolean model is the look approximately like a cell formed by dividing up union of all grains. space by many random lines (Hall, 1988). Boolean models can also be Similar results under much weaker conditions, and in used to define interacting higher dimensions too (Hall, 1988; Molchanov, 1996a).

point processes (Baddeley Simulated Boolean model matched to But what do we mean by a random collection of lines? and van Lieshout, 1995; image of potassium deposit (grains WSK, van Lieshout, and are disks of random radius). Baddeley, 1999; WSK, 1997). 21 22 History Lines Idealization Hierarchy Complement Further questions History Lines Idealization Hierarchy Complement Further questions

Lecture II Georges-Louis Leclerc, Comte de Buffon (7 September, 1707 – 16 April, 1788) Poisson Line Processes French encyclopædist, corresponded with Daniel and Network Efficiency Bernoulli; Published “Essai d’Arithmétique Morale” (Comte de Buffon, 1829) History (web translation by Hey, Poisson line processes and Network Efficiency Neugebauer, and Pasca, 2010); A Mathematical Idealization Measured and valued ; Striking mixture of empiricism A Transportation Hierarchy and rationalism; A Complementary Result “Calculate π: drop needle Some further questions randomly on ruled , count mean proportion of hits.”

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A statistical puzzle Estimating π with Buffon

Since de Buffon, several people have published (recreational!) attempts to estimate π this way. Stigler (1991) gives a list, eg citing Lazzarini (1901). Suppose you have to design such an experiment. Unit-length needle, unit-ruled floor. Say you must choose the number of trials n. The number of successes X is Binomial(n, 2/π). Measure success by 1/ 2/π X/n . | − | Choose n [300, 400]p, say. ∈ Which choice would you make? Log of mean of 1/ 2/π X/n | − | for X with distributionp Binomial(n, 2/π).

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Continued fraction for π Dualizing Buffon, with Steinhaus

Hugo Steinhaus (1887-1972) observed something which we can view as the dual of the Buffon phenomenon; 1 π 3 1 = + 7 1 + 15 1 + 1 Replace needle by convex polygon, or convex curve, + 1 292 1 + 1 + ... perimeter 2`, diameter less than 1. Randomize lines not 1 355 polygon! Probability of hitting ruled line is 2`/π (see, 3 . ≈ + 7 1 = 113 eg, Gnedenko, 1998, Ch. 2.4, Example 5). + 15 1 + There is a notion of invariant measure on the space of So consider 2 X lines in the plane; ... π − 355 Invariant measure of set of lines hitting a regularizable curve is proportional to the length of the curve (Steinhaus, 1930). Mean number of random lines hitting curve?

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Random lines Illustration of invariant line measure How to build random patterns of lines? The key to understanding invariant line measure: 1. We could use the theory of Random Closed Sets: use a good parametrization of (undirected) lines in the plane. characterize a random closed set Ξ by void-probabilities P [Ξ A] P [Ξ A ]. 6⇑ = ∩ = ∅ 2. More constructively, we could try the Boolean model: take union of (possibly random) sets placed on Poisson points. 3. However, the Boolean model does not adapt well to random lines, since random lines are not localized. Parametrize the line ` by fixing a reference point o and a 4. Simple solution: parametrize lines ` by signed reference line through the point. Define: perpendicular distance r and angle θ: 4.1 each line ` corresponds to a point (r , θ) in (a) (signed) perpendicular distance r of ` from o; representation space; (b) angle θ (0, π] between ` and reference line. ∈ 4.2 view Poisson line process as Poisson point process in Invariant line measure is then given by representation space, µ being invariant line measure. 1 5. The representation space is a cylinder with twist: in fact d r d θ , an infinite Möbius strip, or punctured projective plane. 2 6. Calculations: often reduce to probabilities that there are (Normalization by 1 : no lines of particular forms. 2 unit measure for set of lines hitting unit segment). 29 30 History Lines Idealization Hierarchy Complement Further questions History Lines Idealization Hierarchy Complement Further questions

Poisson Line Process Tools for line processes

Buffon The length of a curve equals the mean number of hits by 1 Definition: Poisson line process a unit-intensity Poisson line process (intensity 2 d r d θ); A (stationary) Poisson (undirected) line process of intensity Slivnyak Suppose we condition a Poisson process on placing a λ > 0 is obtained from a Poisson point process on R [0, π) “point” z at a specified location. The conditioned process 1 × with intensity 2 λ dr dθ as follows: for each point (r , θ) in the is again a Poisson process with added z; point process, a line is generated at signed distance r from Angles Generate a planar line process from a unit- the origin and making an angle θ with the x-axis. intensity Poisson point process on a refer- (Stationary) Poisson directed line processes can be obtained ence line `, by constructing lines through the points p whose angles θ [0, π) to ` from undirected Poisson line processes by assigning a ∈1 direction to each line. A natural parameter space would then are independent with density 2 sin θ. The result is a unit-intensity Poisson line pro- be R [0, 2π). × cess. Intensity measure in these coordin- 1 ates: 2 sin θ d p d θ.

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To 3-space and beyond A Useful Exercise for planar lines:

Compute probability that point A is separated from line The notion of line processes in higher dimensions is segment BC by a unit intensity Poisson line process. important in generalizations. Let T be triangle ABC. Let [T ] be the set of lines intersecting . Let Poisson line processes in Rd: T each line ` [T ] score 1 for each ∈ + Parametrize by $ “direction” of (undirected) line (point intersection with AB or AC, 1 for d 1 − on hemisphere: more properly RP − ), intersection with BC. and x location on perpendicular hyperplane. Integrating score against invariant line measure µ, count contributions from intersections with AB, AC and BC: Invariant measure now c d x ν ( d $). d × d µ([AB]) AB , µ([AC]) AC , µ([BC]) BC . = | | = | | − = −| | Coordinate x is “twisted” by $: measure theory doesn’t Lines intersecting AB and BC, or AC and BC, score zero. see this. Lines intersecting AB and AC score 2. (Ignore Variant parametrization replaces x by p, intersection of measure-zero cases of vertex intersections.) Deduce invariant line measure of lines intersecting AB ` with reference hyperplane. 1 and AC is 2 ( AB AC BC ). Invariant measure now cd sin θ d p νd 1( d $). | | + | | − | | × − Accordingly required probability is 1 1 exp ( AB AC BC ) . − − 2 | | + | | − | |   33 34 History Lines Idealization Hierarchy Complement Further questions History Lines Idealization Hierarchy Complement Further questions

Some challenging facts about line processes Transportation Network Questions 1. The Steinhaus method of determining length by the line measure of intercepts can be viewed as lying at the heart of modern stereology: take a random slice (line or plane), count geometrical features along the slice, deduce estimates of 3d geometric structure. The resulting geometric calculations can be complicated! 2. D.G. Kendall conjecture. Consider a plane tessellated by an invariant Poisson line process. Consider the cell containing the origin. Suppose that we condition on this cell having large area. What is its limiting shape? Answer: Circular (Miles, Kovalenko, Schneider and others . . . ). 3. Rollo Davidson’s conjecture. Consider an invariant line process with finite mean-square counts on compact sets, Mastiles Lane, Yorkshire Dales and with no parallel lines. Must it be a ? (Poisson process with randomized intensity measure.) Answer: No. Kallenberg and Kingman describe a beautiful construction based on a random lattice. 35 36 History Lines Idealization Hierarchy Complement Further questions History Lines Idealization Hierarchy Complement Further questions

An ancient optimization problem Modern variants British Railway British Railway network network UK Motorways: A Roman before Beeching after Beeching Emperor’s dilemma:

PRO: Roads are needed to move legions quickly around the country; CON: Roads are expensive to build and maintain; Pro optimo quod faciendum est?

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A Mathematical Idealization Main Question (a problem in frustrated optimization) Aldous and WSK (2008) provide answers for the (N) Consider N cities x x1, . . . , x in square side √N. Question = { N } Assess road network G G(x(N)) connecting cities by: (N) √ 2 = Consider a configuration x of N cities in [0, N] as above, network total road length len(G) and a well-chosen connecting network G G(x(N)). How (N) = (minimized by Steiner minimum tree ST(x )); does large-N trade-off between len(G) and (G) be- versus have? average network distance between two random cities, 1 (And how clever do we have to be to get a good trade-off?) average(G) distG(xi, xj), = N(N 1) − XXi≠j Note: (minimized by laying tarmac for complete graph). len(ST(x(N))) O(N) (eg, Steele, 1997, §2.2); ≤ Perhaps the average ratio would be a good measure of Average Euclidean distance between two randomly performance? chosen cities is at most √2N; 1 dist (x , x ) Perhaps increasing total network length by const Nα G i j × N(N 1) x x might achieve average network distance no more than − XXi≠j k i − jk order Nβ longer than average Euclidean distance? 39 40 History Lines Idealization Hierarchy Complement Further questions History Lines Idealization Hierarchy Complement Further questions

Answer to main question (I) Answer to main question (II)

Upper-bound “network distance” between two cities by Augment Steiner tree by a low-intensity invariant Poisson mean semi-perimeter of cell, line process Π1. 1 1 E len ∂ x,y Unit intensity is 2 d r d θ: we will use this and scale. 2 C h i Pick two cities x and y at distance n √N units apart. = Remove lines separating the two cities; Aldous and WSK (2008) answer Main Question using this,

focus on cell x,y containing the two cities. and use other methods from stochastic geometry to C show that the resolution is nearly optimal.

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The key construction Some stochastic geometry (I)

We have

E len ∂ 2 x y E # Π∗ ∂ . Cx,y − | − | = 2 ∩ Cx,y h i h  i This is the total intensity of the intersection point process Π∗ Π∗ thinned by removing z Π∗ Π∗ when 1 ∩ 2 ∈ 1 ∩ 2 z is separated from both x and y by Π1∗. (Remember, line process renormalized to unit intensity.) We can appeal to a variant of Slivnyak’s theorem: the Compute mean length of ∂ by use of independent retention probability for z is the probability that no line Cx,y unit-intensity invariant Poisson line process Π2, and of Π1∗ hits both of segments xz and yz, namely determine the mean number of hits. 1 1 exp x z y z x y exp (η n) , It is convenient to form Π2∗ by deleting from Π2 those − 2 | − | + | − | − | − | = − 2 − lines separating x from y. (Mean number of hits     where η x z y z . contributed by these lines: 2 x y 2n.) = | − | + | − | | − | =

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Some stochastic geometry (II) Mean perimeter length as a double integral

Theorem: Mean perimeter length

The intensity of the completely unthinned intersection π process Π1 Π2 is . E len ∂ 2 x y ∩ 2 Cx,y − | − | = h i The intensity of Π1∗ Π2∗ is obtained by careful 1 1 ∩ (α sin α) exp 2 (η n) d z computation of the probability that the intersection lines 2 R2 − − − ZZ   of a point of Π1 Π2 do not hit xy, using the “Angle” ∩ construction from given above. The resulting intensity is: Note that α α(z) and η η(z) both depend on z. = = π α sin α α sin α − − , 2 × π = 2 Fixed α: locus of z is circle. where α is the exterior angle of the triangle ∆xyz at z. Fixed η: locus of z is ellipse.

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Asymptotics A Transportation Hierarchy

Theorem: Asymptotic cell perimeter Careful asymptotics for n show that → ∞ 1 E len ∂ 2 Cx,y = h 1 i 1 n 4 (α sin α) exp 2 (η n) d z + R2 − − − ≈ ZZ   4 5 Use a hierarchy of: n log n γ 1. a (sparse) Poisson line process; + 3  + + 3 2. a rectangular grid at a moderately large length scale; where γ 0.57721 ... is the Euler-Mascheroni constant. = 3. the Steiner minimum tree ST(x(N))); Thus a unit-intensity invariant Poisson line process is within 4. a few boxes from a grid at a small length scale, to avoid log of providing connections which are as efficient as O( n) potential “hot-spots” where cities are close (boxes are Euclidean connections. connected to the cities).

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Answering the main question (example)

Theorem: Main efficiency result For any configuration (N) in square side √ and for any se- x N 1000 simulations quence wN there are connecting networks GN such that: at n 1000000: = → ∞ average 21.22, len(G ) len(ST(x(N))) o(N) s.e. 0.23, N = + asymptotic 21.413. 1 average(GN ) xi xj o(wN log N) = N(N 1) k − k + Vertical exaggeration: − XXi≠j √n The sequence w can tend to infinity arbitrarily slowly. { N }

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A Complementary Result Main ideas of proof

Theorem: Complementary Efficiency Result Given a configuration of N cities in [0, √N]2 which is L N = o( log N)-equidistributed: random choice XN of city can be coupledp to uniformly random point YN so that

XN YN E min 1, | − | - 0 ;   LN  → Use tension between two facts: (a) efficient connection of a random pair of cities forces a then any connecting network GN with length bounded above by a multiple of N connects the cities with average connection path which is almost parallel to the Euclidean path, and length exceeding average Euclidean connection length by at (b) the coupling means such a random pair is almost an 2 least Ω( log N) . independent uniform draw from [0, √N] p (equidistribution), so a random perpendicular to the Euclidean path is almost a uniformly random line. 51 52 History Lines Idealization Hierarchy Complement Further questions History Lines Idealization Hierarchy Complement Further questions

Further Questions Location of maximal lateral deviation (I)

Question Fluctuations: Given a good compromise between average(G) and len(G), how might the variance behave? Question True geodesics: The upper bound is obtained by controlling non-geodesic paths. How might true geodesics behave? Question Flows: Consider a network which exhibits good trade-offs. What can be said about flows of traffic in this network? Question Scale-invariance: Can we generate scale-invariant random structures resembling suitable compromise networks? Red dots indicate maximum lateral deviation. (NB: vertical exaggeration!)

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Location of maximal lateral deviation (II) Variance and growth process (I)

Remark: Density / intensity 2 Using u x ( 1, 1) and v y > 0, = n ∈ − = √n

1 1 (sin β sin γ sin(β γ)) exp (η n) d x d y 4 + − + − 2 −   v3 v2 exp d d 2 2 u v ≈ 1 u2 −1 u ! − −  ASYMPTOTICALLY Consider cell boundary as path with x-coordinate Xτ , Location of maximum is parametrized by τ (excess over geodesic distance). Let Θ be angle with positive x-axis; uniformly distributed; 1 Θ jumps at Poisson point process intensity ; Conditional height of 2 A jump ∆Θ Θ Θ obeys maximum is length of − = − − 1 cos θ Gaussian 4-vector. P [Θ Θ θ Θ ] − . − − ≤ | − = 1 cos Θ − − 55 56 History Lines Idealization Hierarchy Complement Further questions History Lines Idealization Hierarchy Complement Further questions

Variance and growth process (II) Theorem: Near-geodesic fluctuations Sketch of argument: Let σ (n) be the geodesic distance between the half-x-axis 2 2 t Θ0 ( , 0] and x n. Then 1. Xt X0 2Θ0− 0 2 d s; −∞ = ≈ + Θs 2. log (Θ0/Θ ) ξ R a non-decreasing Lévy process; t ≈ t 3 2 2 3. Mt ξt 4 t is an L martingale. E [σ (n)] log n O(1) = − = 3 + 4. Define a by σ (n) by X n. So σ (n) = 20 2 σ (n) 2 2 Var [σ (n)] log n O log n . n Xσ (n) X0 2Θ0− 0 Θ0/Θs d s. = 27 + = ≈ + q  5. Now exp(2ξσ (n)) is (approximately)R a self-similar process, about which much information is available; Hence perimeter length fluctuations are O log n . 6. Analyze the tautology Check: Earlier today, we calculated mean p  2 4 σ (n) log n 2M log exp(2ξ )/n . semi-perimeter-length as n log n .... Each ≈ 3 − σ (n) + σ (n) + 3 + semi-perimeter excess contains contributions from two of 7. Control log exp(2ξσ (n))/n using work of Bertoin and these paths (growth from left, growth from right), so this Yor (2005);  2 approach is in agreement with the 8. Deduce σ (n) 3 (log n O(1) 2Mσ (n)); ≈ + − 2 stochastic geometry calculations. now Mσ (n)) can be shown to be L . 57 58 History Lines Idealization Hierarchy Complement Further questions History Lines Idealization Hierarchy Complement Further questions

What about true geodesics? How much better can a true geodesic be? The above can be used to show true geodesics typically have smaller excess than our paths, Key tools: nevertheless Measure excess with the number of “top-to-bottom” crossings of a true n n geodesic is stochastically bounded in any region 1 2 E (sec θx 1) d x 2 E θx d x 1 − ≥ 1 [na, nb] (0, n), so all but a stochastically bounded Z  Z h i ⊆ number of “short-cuts” must be within O(n/ log n) of start or end, affecting coefficient of log(n) but no more; indeed 2 P [ θx u] E exp( uL+) Theorem: Lower bound on mean excess | | ≥ ≥ − x Approximate by integrals involving  Gaussian kernels . . . Any path of length n built using the Poisson lines must have mean excess at least 2 p2/2 ∞ s2/2 4 e e− d s 5 p2 4 p ≤ Zp ≤ p2 8 3p log 4 log n o(log n) . + + + +  − 4 + q q Birnbaum (1942) Sampford (1953)

59 60 City Calculations Impropriety Empirical comparisons Conclusion City Calculations Impropriety Simulation Empirical comparisons Conclusion

Lecture III Traffic in a random city

Traffic Flow in a Poissonian City

Traffic in the City

Calculations for traffic distribution

Improper line processes

Effective simulation of limiting distribution

Empirical comparisons

Conclusion

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The Poissonian City (I) Real city plan Design an idealized city! Complete the 2 routes using off-road sections.

63 64 City Calculations Impropriety Simulation Empirical comparisons Conclusion City Calculations Impropriety Simulation Empirical comparisons Conclusion

Palm distributions The Poissonian City (II) Aldous idea: Every point-pair (x, y) contributes Recall Slivynak’s theorem: if we condition on a Poisson infinitesimal dx dy split between two near-geodesics. process placing a point at the origin . . . Compute 4-volume of random polytope! . . . then the reduced Palm distribution, the residual point pattern (minus the conditioned point), is again Poisson with the same intensity. Best understood in the same way as we understand abstract conditional expectation: plugging in the Palm distribution in certain formulae gives the correct answer. (Reductionist point of view: just “integration by parts”.) Similarly for Poisson line processes: If we condition on a line through the centre, then the remainder of the line process is again Poisson. So what could we then say about traffic at the centre?

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More sophisticated approaches (I) More sophisticated approaches (II): Poisson locations scattered along Π

Options: We consider a Poissonian city formed Up to constant factors (readily computable using by unit intensity Poisson line process Π. Slivnyak’s theorem) we get essentially the same mean Independently, source / destination value behaviour if instead traffic is generated points form a Poisson point process uniformly between pairs of points lying on the lines only; whose intensity is Lebesgue linear measure along . between points of a point pattern that is Poisson along Π the lines (using Lebesgue length measure); Let 2f (Π) indicate whether x1 lies on near-geodesic between the points of intersections of lines; x0,x1,x2 x1 x2, with x0 ball(z0, ), x1 ball(z1, ρ1). → ∈ ∈ the process used to generate traffic along fibres or at Our task is to compute intersections can be random if it is of finite mean and does not depend on location. E  fx0,x1,x2 (Π) xX0 Ψ xX1 Ψ xX2 Ψ  ∈ ∈ ∈  67 68 City Calculations Impropriety Simulation Empirical comparisons Conclusion City Calculations Impropriety Simulation Empirical comparisons Conclusion

More sophisticated approaches (III): Results for more sophisticated approaches (I) Palm theory for lines of Π We consider a Poissonian city formed We consider a Poissonian city formed by unit intensity Poisson line process Π. by unit intensity Poisson line process Π. Independently, source / destination Independently, source / destination points form a Poisson point process points form a Poisson point process whose intensity is Lebesgue linear whose intensity is Lebesgue linear measure along Π. measure along Π.

Theorem: Mean traffic between Poisson points Palm theory for `2, change coordinates, cutoff for x2: If source / destination points are scattered by a (condi- π 3 tionally independent) Poisson process along Π then mean E fx0,x1,x2 (Π ` ) traffic through a “typical” point is given by integrating 2 ball(z0,ε) ball(z1,ρ1) ball(z2,ρ2) ∪ { ∗} x0   Z Z Z 3   π E fx0,x1,x2 (Π ` ) against x1 and x2, conditional on d x2 d x1 dx0 asymptotically as n 2 ∪ { ∗} → ∞ x 0 belonging to Π via a randomly oriented line ` . ∗

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Results for more sophisticated approaches (II) Illustration of traffic routes between intersection points

Consider again a Poissonian city formed by unit intensity Poisson line process Π. Source / destination points are formed by the line intersections of Π.

Theorem: Mean traffic between intersections If source / destination points are formed by the intersections of the lines of Π then mean traffic through a “typical” point x0 3 π is given by integrating 2 E fx0,x1,x2 (Π ` ) against 4 ∪ { ∗} x1 and x2, conditional on x0 belonging to Π via a randomly oriented line ` . ∗

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Measure congestion? Questions:

1. What is the mean traffic flow through the centre? 2. Is there a limiting scaled distribution? 3. Is the limiting distribution accessible to computation?

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Mean traffic computation Where does most of the traffic come from? Mean traffic flow through centre can be obtained by direct arguments from stochastic geometry: Theorem: Mean traffic flow through centre For ρ √r 2 s2 2r s cos θ, the traffic flow T through the = + − n centre of a Poissonian city built on a disk of radius n has mean value

π n n 1 E [Tn] exp 2 (r s ρ) r d r s d s θ d θ . = 0 0 0 − + − Z Z Z   Asymptotically for large city radius (n ), → ∞ E [T ] 2n3 O n2√n . n = +  

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Existence of limiting distribution? Improper line processes to the rescue (I)

Consider stretching the city. Vertically, rescale by 1/√n. Horizontally, rescale Is there a limiting by 1/n. 3 distribution for Tn/n as (Affine n ? → ∞ transformation.) New coordinates y − and y . + New (improper) intensity (as n ) 1 → ∞ is dy dy , 4 − +

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Improper line processes to the rescue (II) Computing mean traffic (I) Infinitely many nearly vertical lines running near any point. Change coordinates to heights z and z of − + intercepts on x u and x v. 1 = − = z 2 ((1 u)y (1 u)y ); − = 1 + − + − + z ((1 v)y (1 v)y ). + = 2 − − + + + Compute dz dz using formalism of differential forms: − ∧ + 1 u 1 v 1 u 1 v dz dz + + dy dy − − dy dy −∧ + = 2 2 −∧ ++ 2 2 +∧ − u v + dy dy , = 2 − ∧ + hence 1 1 dy dy dz dz . 4 − + = 2(u v) − + Centre sees no traffic when separated off by lines. + Coupling argument: limiting distribution non-degenerate. 79 80 City Calculations Impropriety Simulation Empirical comparisons Conclusion City Calculations Impropriety Simulation Empirical comparisons Conclusion

Computing mean traffic (II) Computing mean traffic (III)

Measure of lines separating AB from O: We now compute the mean 4-volume of (A, B) above ( u, v) − Area of a rectangle such that AB is separated from O: h h /(2(u v)); − + + 2 ∞ ∞ 1 u v plus area of a triangle exp d d 1 h h h h (u/v)h h /(2(u v)); Z0 Z0 −4(u v) rv + + ru − ! − + 2 + × + + + plus area of another triangle 2(u v) . 1 = + 2 (v/u)h h /(2(u v)). − × − + Now we can integrate out u > 0 and v > 0. Suppose ( u, v) − is contained in the region ( u0, v0) ( 1, 1): total mean h h u h h v h h − ⊆ − − + + + − − traffic through the origin is given by 2(u v) + v 4(u v) + u 4(u v) + + + 2 1 u v 1 u0 v0 h h . 2 2(u v) du dv u0v0(u0 v0). = 4(u v) v + + u − 2 × × 0 0 + = + + r r  Z Z

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Mean traffic flow in regular region Mean traffic flow over the disk

Theorem: Mean traffic flow Theorem: Mean traffic flow (case of disk) Consider a general convex body C, and traffic derived from Consider traffic derived from a Poissonian city based on a a Poissonian city based on a Poisson line process of intensity Poisson line process of intensity n in a unit disk. Consider a n. Consider a point x in the region, and Palm condition on point x in the region, and Palm condition on a line ` passing a line ` passing through this point with further conditioning through this point without further conditioning on the direc- on the direction of the line. Suppose that the two intervals tion of the line. The traffic flow Tn through x under this con- (`(θ) C) x have w (θ) and w (θ). The traffic ditioning has mean value asymptotic to ∩ \{ } − + flow Tn through x under this conditioning has mean value π 2 2 2 2 4 2 asymptotic to E [Tn] (1 r ) 1 r sin θ dθ (1 r )E(r ) . ∼ π − Z0 q − = π − E 3 [Tn] w (θ)w (θ)(w (θ) w (θ))n . where r [0, 1] is the distance of x from the disk centre. ∼ − + − + + ∈

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There is a limiting distribution, but how to access it? Seminal curves WSK (2014): Can’t find analytic expression for the limiting distribution 3 limn Tn/n . →∞ L We now can  recognize Tn as 4-dimensional volume of random polytope (use 1 Focus on upper half-plane (quadrants Q ) and subset of 4 d y d y ). ± − + source/destinations in Q Q not separated from o. Can we produce − × + Identify two sub-families of improper line process Π : achievable ∞ approximating Π , ` Π : slope(`) , ` intercepts x-axis . ∞ ± = { ∈ ∞ = ± ∓ } simulations? Define seminal curves Γ as the concave lower envelopes ± of the unions of lines in Π , . 85 86 ∞ ± City Calculations Impropriety Simulation Empirical comparisons Conclusion City Calculations Impropriety Simulation Empirical comparisons Conclusion

A series expression (I) Recursive representation of Γ ± WSK (2014): Theorem: Representing the seminal curves Consider of slope-change of Γ Γ in reversed time: = + 1 S0 > S1 > S2 > . . . > 0. = Successive tangents `0, `1, `2,..., ` : slopes Γ (s) for S n 0 n ≥ s > Sn 1. + Let Y Γ (S ) S Γ (S ) be intercept of ` on y axis, n = n − n 0 n n 1 1 4 2 En 1 , Sn 1 = Sn + Yn + + Yn Use regions in figure to bound 4-volume . . . Γ 0(Sn 1) Γ 0(Sn) Un 1 . + = + Sn 1 + 1 1 + p Γ ( s) d s Γ (s) d s for En Exponential(1), Un Uniform[0, 1], independent of Z0 − − ! × Z0 + ! ∼ ∼ each other and Γ (S0) Γ (1), Γ 0(S0) Γ 0(1). ∞ ∞ = = (Γ (1), Γ 0(1) joint distribution is computable.) Leb2(C+) Leb2(∆+) Leb2(C−) Leb2(∆−) + n n + n n n 0 n 0 X= X= 87 88 City Calculations Impropriety Simulation Empirical comparisons Conclusion City Calculations Impropriety Simulation Empirical comparisons Conclusion

Simulation Error Estimation Empirical comparisons

Theorem: L1 simulation error The L1 error of the 4-volume approximation We can now compare traffic distribution to traffic on the 1 1 UK railway system before Beeching (1963) (approximate I[o ∂ (x,y)] d x d y Γ ( s) d s Γ (s) d s UK by ellipse . . . ). Q Q ∈ C ≈ 0 − − !× 0 + ! Z + Z − Z Z N N density computable using elliptic integrals, or Leb2(C+) Leb2(∆+) Leb2(C−) Leb2(∆−) numerically. + n n + n n n 0 n 0 X= X= Remarkably, mean traffic distribution appears to have an is bounded above by unexpected approximate affine invariance!

20 N 20 N 3− 6− . 7 × + 27 ×

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UK railway traffic and Poissonian cities Conclusion

Traffic at the centre of the Poissonian city: There is a limiting distribution under scaling. There is a theoretical basis for developing an effective simulation algorithm. Pre-Beeching rail traffic is How to build an actual implementation? more highly Analytical representation? concentrated than Perfect simulation? Poissonian city traffic. Links to Gamma distributions? “Grid”-based cities. Replacing lines by fibre fragments?

Data from Beeching (1963); Is there a scale-invariant approach? annotation by Gameros Leal (2017).

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Lecture IV Google holidays for cats Scale-Invariant Spatial Transportation Networks

Googling for cats

Scale-invariant Random Spatial Networks

About Π-paths

Properties of Π-geodesics Pre-SIRSNs and SIRSNs

Random Metric Spaces

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SIRSN axioms Formal definition What would a scale-invariant (random) network look like? Definition: (Strong) SIRSN (Aldous, 2014; Aldous and Ganesan, 2013) A SIRSN is a random mechanism for generating routes Input: set of nodes x1,..., x ; 2 n Route(x, y) between locations x, y R , such that Output: ∈ 1. Given x, y the route Route(x, y) is almost surely random network N(x1, . . . , x ) connecting nodes. n uniquely defined; (1) Scale-invariance: for each Euclidean similarity λ, 2. Let be the union of routes between various (N(λx1, . . . , λxn)) (λN(x1, . . . , xn)). N(x1, . . . , xn) L = L locations . For each Euclidean similarity , (2) If D1 is length of fastest route between two points at unit xi λ ; distance apart then E [D1] < . (N(λx1, . . . , λxn)) (λN(x1, . . . , xn)) ∞ L = L (3) Weak SIRSN property: the network connecting points of 3. D(x, y) E len Route(x, y) < . = ∞ (independent) unit intensity Poisson point process has 4. (Strong form of SIRSN property) re-use of routes means finite average length per unit area. that the “fibre process” (Strong) SIRSN property: the network connecting points x≠y Φ(Route(x, y) (ball(x, 1) ball(y, 1)) of of dense Poisson point process has finite average length ∈ \ ∪ Slong-range routes has finite mean length per unit area per unit area of “long-range” part of network for a “dense Poisson process” Φ. (more than distance 1 from source or destination). 95 96 Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics

Model for these axioms: Aldous’ dyadic SIRSN Aldous’ dyadic SIRSN SIRSN by randomization Aldous (2014) “binary hierarchy” model: 1. Consider integer lattice Z2. Verticals at x (2k 1)2h, = + horizontals at y (2k 1)2h, have height h. 1 = + 2. Fix γ ( 2 , 1). A path along the lattice travels at speed h ∈ γ− when on a segment of height h. 3. Mark segments randomly to help choose between non-unique geodesics. r 2 4. Extend to dyadic lattice r 2− Z . 5. Force stationarity by takingS weak limit of random translations. 6. Force isotropy by taking random rotation. 7. Force scaling symmetry by taking scaling by random factor c (1, 2). ∈

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The Poisson line process SIRSN candidate Impropriety and marks Need to take care when defining random point or line Proposed by Aldous (2014). processes which are dense! 1. Fill plane with dense speed-marked Poisson line process. There are strange measure-theoretic issues: (Aldous and (Parametrize each line by signed perpendicular distance Barlow, 1981; WSK, 2000). from origin r , angle θ [0, π), and positive speed v: Such processes are often described as “improper”: the ∈1 use intensity measure (γ 1)v γ d v d r d θ.) 2 − − intensity measure ν is not locally finite. 2.“ Π-paths”: Lipschitz paths integrating resulting Typical way to resolve the problem: add marks, for measurable orientation field, (almost always) obeying example lines with speed-limits: speed-limit. We want to define a Poisson line process that is “dense everywhere”. 3. Connect pairs of points by fastest routes (“ -geodesics”) Π Introduce a notion of speed v, a random speed-limit travelling entirely on line pattern at maximum speed. mark for each line `. 4. For γ > 2, almost surely, all pairs are connected. Almost View the process as a point process in an extended surely, fastest connection for a specified pair is unique. representation space, typical point (v, r , θ). For intensity measure use f (v) d v d r d θ, locally finite 5. Strong SIRSN axioms hold(WSK, 2017; Kahn, 2016). on v-r -θ space even if it does not project down to local finite measure on r -θ space.

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About Π-paths (I) When infinity is infinitely far away Theorem: An a priori upper bound for path space Seek shortest-time paths (“temporal geodesics” or Suppose γ d 2. Fix T < , consider Π-path ξ begun in Π-geodesics) built using line process Π. ≥ ≥ ∞ unit ball. There is C < depending on realization of Π such Require γ > 1, or fast lines will go everywhere. ∞ that almost surely ξ(t) C for all t T . Introduce maximum speed limit, upper-semi-continuous | | ≤ ≤ V : Rd [0, ). → ∞ Proof. A Π-path is locally Lipschitz, integrates measurable Set V (r ) to be speed of fastest line within r of origin. orientation field determined by Π, obeys speed limit. Map line (`, v) , ( r d 1, v (γ 1)) to a point in [0, )2: this | | − − − ∞ If γ > d then: maps Π to a planar Poisson process! (γ 1) d 1 there is an a priori random bound on distance travelled S (V (r ))− − changes at r − P0 < P1 < P2 < . . .. by Π-path in fixed time; = | | = space of paths up to time is closed, weakly closed in Pn Pn 1 1/(Sn 1Tn) for Exponential Tn; T − − 1 2 − = Sobolev space L , ([0,T) Rd), Sn UnSn 1 for Uniform(0, 1) Un. → = − paths up to time T , beginning in a compact set, together X S P is a perpetuity. form a weakly compact set. n = n n Compute P infinite time to get to infinity Π 1. | =   101 102 Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics

About Π-paths (II) d Suppose one wishes to connect two points ξ1 and ξ2 in R by a Π-path. Suppose γ > d (turns out to be essential).

We can now apply standard compactness arguments for the Construct small balls around ξ1 and ξ2; Sobolev space L1,2([0,T) Rd): Connect balls by fastest line intersecting both balls; → ` Construct daughter nodes on ` closest to ξ1 and ξ2; Lemma: Π-geodesics exist if Π-paths exist Recurse. Suppose Π is a speed-marked Poisson line process in Rd with 1 Borel-Cantelli, et cetera: establish almost sure existence intensity measure (γ 1)v γ d v d r d θ. If γ > 1 then al- 2 − − of resulting path. most surely Π-geodesics exist between all point pairs ξ1 and d This yields a binary tree representation of the path. Note ξ2 in R if Π-paths exist. that this is unavoidable if d > 2! A similar but more complicated argument almost surely allows simultaneous construction of paths between all d possible pairs ξ1 and ξ2 in R . Exercise: Figure out a way to visualize Π-paths in case d 3. 103 104 = Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics

Construction to connect two points Lower bound on measure of set of lines hitting 2 balls

1 1 First find the fastest line passing reasonably close to both Reduction of hitting set [ball(x1, α r )] [ball(x2, α r )] to − ∩ − source and destination. a smaller hitting set for which the line measure is more easily computable yet which still provides a useful lower bound. Then recurse.

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Now use resulting lower bound About Π-paths (III) d 1 Exponential moments for powers of Π-diameters 1 1 r − µd [ball(ξ1, α− r )] [ball(ξ2, α− r )] κd 1 Kahn (2016), improving on WSK (2017, Theorem 3.6) has ∩ ≥ − 4α2     proved results of which the following is a simple example: and a Borel-Cantelli argument to control the amount of time Consider ball(o,R), a ball of Euclidean radius R. spent traversing recursively defined paths between any pair Let TR be the supremum of the minimum times for a of points. Π-path to pass from one point of ball(o,R) to another;

Lemma: Existence of Π-paths TR sup inf T : = x,y ball(o,R) { Suppose Π is a speed-marked Poisson line process in Rd with ∈ intensity measure 1 1 γ d d d . If then - some Π-path ξ : [0,T] B satisfies ξ(0) x, ξ(T ) y . 2 (γ )v− v r θ γ > d Π → = = } − d paths exist between all point pairs ξ1 and ξ2 in R and no We call TR the Π-diameter of ball(o,R). Π-paths of finite length can reach infinity. For explicit δ > 0, K < , R R ∞ γ 1 Remark: Random metrics for Euclidean space E exp δ T − K < . R R ≤ R ∞ So if γ > d then Π determines a random metric on Rd. h  i

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Simulations (approximate!) of a typical set of routes Are Π-geodesics unique? (I) 2 Suppose now d 2 and γ > 2, and we fix ξ1 and ξ2 R . If = ∈ Π is to generate a network between a finite set of points, then we need to know the Π-geodesic between ξ1 and ξ2 is almost surely unique. Theorem: “line meets line” All non-singleton intersections of a Π-geodesic with lines ` of Π are of the form “line meets line”.

First, reduce to case of ` being fastest line in region, with speed w. Now change focus from high speed v to low “cost”, where csc θ cot θ “cost” . = v − w Case γ 16 where θ is angle of line with `. = Argue that Π-geodesic hits ` using line of finite cost. 109 110 Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics

Are Π-geodesics unique? (II)

So Π-geodesics between ξ1 and ξ2 are made up of countable collection of intervals of lines of Π. Fix a given ` from Π, and consider the set S of such Theorem: Uniqueness of Π-geodesics in planar case intervals lying in `. Suppose Π is a speed-marked Poisson line process in R2 with 1 Consider two different finite collections S1 S and γ ⊂ intensity measure 2 (γ 1)v− d v d r d θ. If γ > 2 then for S2 S, each composed of non-overlapping intervals. − 2 ⊂ any point pair ξ1 and ξ2 in R it is almost surely the case that Probability density argument: the total lengths of S1 and there is just one Π-geodesic between ξ1 and ξ2. S2 have a joint density, unless one is empty. Conditioning on time spent off `, almost surely two Π-paths using S1 and S2 respectively must have different Almost surely there will exist non-unique Π-geodesics! total travel times. Almost surely two Π-geodesics between ξ1 and ξ2 must use the same finite collection of non-overlapping intervals from each ` of Π. But we can reconstruct the Π-geodesic uniquely from the collections of intervals of each line ` in Π.

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Do Π-geodesics have finite mean length? Diagram to prove Pre-SIRSN property A careful construction, together with a Borel-Cantelli Suppose again that d 2 and γ > 2. = argument, shows that re-scaled small perturbations of Techniques for showing existence of Π-paths show finite the following diagram (for suitable re-scaled speeds mean of length of Π-geodesic if lying in a fixed ball. 1 < a < b < c) can be found at all length scales: Could fast geodesics generate long lengths outside balls? (Oxford Cambridge by motorway via London? → or Edinburgh?...) Time spent by Π-geodesic can be bounded above by time spent on a circuit of a “racetrack” construction around ξ1 and ξ2 using fastest lines. We can upper-bound distance travelled outside a ball by using the “idealized path” construction employed above. The resulting perpetuity can be combined with the “racetrack” bound to establish finite mean length. If c > 10b > 59a/3 > 354/3, red segment is only exit. 113 114 Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics Geodesics SIRSN Π-paths Π-geodesics SIRSNs Random Metrics

Kahn (2016) has now produced a proof of the strong SIRSN property for all dimensions d and all γ > d, using the exponential inequality mentioned above for powers of Π-diameters

γ 1 Theorem: Poisson line process SIRSN E exp δ T − K < . R R ≤ R ∞ h  i Suppose Π is a speed-marked Poisson line process in Rd with 1 This can be used to show that Π-geodesics must make intensity measure (γ 1)v γ d v d r d θ. If γ > d then Π is 2 − − substantial re-use of shared lines: for all Π-geodesics a (strong) SIRSN. bridging a suitable annulus, the exponential inequality forces each Π-geodesic to make substantial use of a limited number of “fast” lines. SIRSN follows by using a measure-theoretic version of the pigeonhole principle. Kahn (2016) deploys further ingenious arguments to obtain uniqueness (essential if above argument is to work) and finite mean-length of Π-geodesics in case d > 2.

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Random Metric Spaces Lecture V

Study of metric spaces generated by line processes should Random Flights on the SIRSN form a chapter in the theory of random metric spaces: 1. Vershik (2004) builds random metric spaces out of random distance matrices (compare MDS in statistics); almost all are isometric to Urysohn’s celebrated Wandering about Yorkshire and Durham universal metric space. Not finite-dimensional. Random wandering on a SIRSN 2. Brownian map: limit of random quadrangulations of 2-sphere (eg, Le Gall, 2010). Far from flat. Scattering – an abstract approach 3. Baccelli, Tchoumatchenko, and Zuyev (2000) link to geometric spanners; they exhibit 4 -spanner paths in Application to RRF on SIRSN – a quick sketch π Poisson-Delaunay triangulations. Conclusion

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Wandering about Yorkshire and Durham Random wandering on a SIRSN

Options: (thanks to Banerjee, Croydon for helpful conversations!) 1. , connect successive jumps by Π-geodesics. (Con: Always stopping and starting!) 2. Brownian motion on the line pattern. (Con: Relationship with speed limits is not explicit.) 3. Rayleigh random flight: https://www.openstreetmap.org/ Proceed at top speed along current line; Switch to intersecting lines depending on relative speeds; Tsukayama, Washington Post, 30th January 2013: Choose new direction equi-probably. Estimated annual impact of online maps: $1.6tn. 4. This is SIRSN-RRF: a “randomly broken Π-geodesic”. Estimated growth 30% per year (smartphones!). Can it be speed-neighbourhood-recurrent?

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Scattering – an abstract approach (I) Scattering – an abstract approach (sketch)

How to define a SIRSN-RRF? Proceed at top speed along current line; Switch to intersecting lines depending on relative speeds; Choose new direction equi-probably. Suffices to sample process when it changes speed. Since Π is improper line process, opportunities to change speed are dense along each line! Convenient to take an abstract view: Advantages of axiomatic method “same as the advantages of theft over honest toil” (Russell, 1919, p.71).

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Scattering – an abstract approach (II) Scattering – an abstract approach (III) 5 Adopt “Metropolis-Hastings recipe”: divide state-space Generalization (possibly of wider interest?): into equivalence classes using ω, 1. A discrete-time on countable state-space is set π min κ, κ where we choose κ, κ as positive a = { 0} 0 an abstract scattering process if p ω s for constants belonging to classes of a and a, ab = ab b (symmetric) transmission probabilities ω and set scattering probability sa min 1, κ/κ0 ab = { } (dynamic reversibility is then automatic!).e scattering probabilities sb (set ωa,a 0, sa > 0 for all a). = 6 In case of a suitable total ordering for each “line”, 2. (Consider a matrix (ωab): if row-vectors (ωa ) all lie in ≺ e · transmission probabilities are functions of scattering `1 then the Hahn-Banach theorem can be used to probabilities. characterize whether (ωab) is transmission matrix.) For each a b, there are ωa, summing to 1 with 3. Require dynamically reversibility for measure π: ≺ ± Involution a a preserving measure π, ↔ with π p π p . p ω s ωa, (1 sc) s , a a,b = b b,a a,b = a,b b = +  −  b e a c b 4. Relate to scatter-equivalence classes (“lines”): ≺Y≺ πa/sa e e e e e  e  e and similar for using . if there is a chain a b0, b1,..., bn c with pb,a ωb, e = = − ω > 0, then πa/s πc/s . All follows from choice of the class constants κ bm 1,bm a c e − = 1 (and say equiprobable choice of direction ωa, 2 ). e e e ± = 123 124 Wandering RRF Scattering Application Conclusion References Wandering RRF Scattering Application Conclusion References

Application Dynamical reversibility Let f (x, Π) be bounded, measurable, x an intersection of lines 1, 2 in Π. L L Set f ( 1, 2; Π) f ( 2, 1; Π). Consider the non-symmetric form Define the SIRSN-RRF, sampled at changes in direction, by L L = L L e specifying equilibrium probabilities at intersections of lines. B(f , g) = 1. Scaling invariance: min α α , parameter . π(`1,`2) v1 , v2 α E E E f (Z0; Π) g(Z1; Π) π Z0 x ( 1, 2) Π . = { }   × × x = = L L  α 1≠ 2 Π h i 2. Scattering probability: s min 1, (v2/v1) . LXXL ∈ (`1,`2) = { }   e e  3. Dynamical reversibility: non-symmetric Dirichlet form. Using Campbell-Mecke-Slivnyak theory twice, this can be reduced (taking out 4. Apply Campbell-Slivnyak-Mecke theorem (twice!) to translations, rotations, scale-changes) to the study of identify (translated, rotated, scaled) “environment viewed   from ” via reduced non-symmetric Dirichlet form. E (2) ; 1 f ( 1∗ Π 1∗ )s( ˜0, ∗) ( s( ˜0, )) " L ∪ {L } L L1 − L L 3 Π  Π: separates  5. Resulting log-relative-speed-changes , , . . . form a LX∈  L∈ LY  V1 V2  ˜  origin from 0 3  .  L ∩L  (2) ; s( ˜0, 3)g ( 3 Π 1∗ ) 6. Analyze using RWRE theory (but state-space is not a × L L L ∪ {L } # lattice!). Read off equilibrium distribution from reduced non-symmetric form: at critical α 2(γ 1), the typical log-relative-speed-change X has = − stationary symmetric Laplace distribution. 125 126 Wandering RRF Scattering Application Conclusion References Wandering RRF Scattering Application Conclusion References

Ergodicity (I) (II) Rayleigh random flight (RRF) on Π, unit initial speed. Theorem: Neighborhood-recurrence and ergodicity

The cumulative log-speed-change X V1 ... V can Suppose V1,..., Vn, . . . form a stationary ergodic sequence, n = + + n be expressed using a sum of log-relative-speed-changes with E [V1] 0. Set X V1 ... V . Then for all ε > 0 it is = n = + + n V1,..., Vn, . . . which form a stationary sequence. the case that If the mean log-relative-speed-change µ E [V1] is = P [ Xn X0 ε infinitely often in n] 1 . non-zero then X /n (V1 ... V )/n will converge to | − | ≤ = n = + + n a non-identically-zero limit random variable Proof. (“non-ergodic” part of Birkhoff’s ergodic theorem); 1 Xn/n 0 a.s. so n− sup X1 ,..., Xn 0 a.s. therefore for any ε > 0 there is at least a positive chance → {| | | |} → Set An [ Xm Xn > ε for all m > n]. Birkhoff’s ergodic of Xn eventually never re-visiting ( ε, ε). = | 1 − | − theorem: n− (I [A1] ... I [A ]) p P [A1]. + + n → = If we could show V1,..., Vn, . . . to be ergodic, then the Packing argument: X X > ε on A A if m ≠ n. | m − n| n ∩ m limit equals µ. In this case transience is sure if µ ≠ 0. Deduce 1 1 Remarkably, the converse also holds! (Zero-mean forces n (I [A1] ... I [A ]) (nε) sup X1 ,..., X . − + + n ≤ − {| | | n|} neighborhood recurrence.) This is a consequence of the Let n and deduce p 0. → ∞ = ideas of the Kesten-Spitzer-Whitman range theorem. Complete argument by sub- . . . .

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An argument of Kozlov type Proving ergodicity We only need to show ergodicity of the environment viewed Consider the process Ψ of the (re-scaled, shifted, rotated) from the particle. environment viewed from the RRF particle. Let h be a 1. Given δ > 0, approximate E by E depending only on bounded harmonic function on the (re-scaled, shifted, δ e lines of Π in a finite speed-range [ W,W] (and coming rotated) environment state-space (harmonic with respect to − within a bounded distance R of o): the process Ψ). Then h(Ψn) is almost surely constant in (discrete) time . For consider n P [E∆E ] δ . e δ ≤ 2 2 E (h(Ψ0) h(Ψ1)) 2 E h(Ψ0) 2 E h(Ψ0)h(Ψ1) 0 , − = − = 2. Find a composition of transformations T such that T h i h i h i e e e e moves lines in speed-range [ W,W] to a disjoint where the first step follows from stationarity and the second − 1 speed-range, so that Eδ and T − Eδ are independent. because h(Ψ) is a martingale. Thus P h(Ψ1) h(Ψ0) 1. 2 2 = = 3. Deduce that P [E] P [E] P [E E] P [E] h i 1| − | =1 | ∩ − | ≤ e e 1 e P Eδ T − Eδ P [Eδ] P T − Eδ 4δ 4δ. Thus if E is a shift-invariant event then T − E E (up to | ∩ − | + = = null-sets) for T a composition of any finite sequence of 4. Since δ > 0 is arbitrary, it follows P [E] must be zero or transformations induced by possible moves of the RRF. one. It follows that Ψ, and hence the log-speed-change process, is ergodic. QED

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Concluding Theorem Conclusion: always more questions!

1. So a critical “randomly-broken Π-geodesic” does not halt en route. What about Π-geodesics themselves? Theorem: Transience and neighborhood-recurrence 2. Is critical case space-(neighbourhood)-recurrent? 1. Critical case (α 2(γ 1)): SIRSN-RRF speed is = − speed-point-recurrent? (working on proof . . . ) neighbourhood-recurrent. 3. Above argument should work for critical (non-SIRSN) Π 2. Sub-critical case (α < 2(γ 1)): SIRSN-RRF converges to − for which γ 2 and α 2 (not yet checked). a random limiting point in the plane (trapped by cells of = = tessellation of high-speed lines). 4. “Brownian-like” variations should follow from WSK and Westcott (1987): scale-invariant “Liouville diffusion” 3. Super-critical case (α > 2(γ 1)): SIRSN-RRF disappears − (Berestycki, 2015; Garban, Rhodes, and Vargas, 2013). off to infinity (consider high-speed tessellation). 5. Can anything be done for the high-dimensional SIRSN supplied by case γ > d > 2? 6. Replace lines by long segments? nearly straight fibres?

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Version 1.58 (Fri Jun 22 14:58:57 2018 +0100) ======commit 424bdbf774bbcf936d1ff456c5d84d34bfecf7fd Author: Wilfrid Kendall

Corrected version after conclusion of Edinburgh June 2018 lecture course.

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