Asymptotic Theory for Statistics of Geometric Structures Joe Yukich Universidad Carlos III, October 11, 2019 Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 1 / 26
Introduction · X ⊂ R d random finite point set. · Convex geometry. How many vertices in convex hull of X ? Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 2 / 26
Introduction · X ⊂ R d random finite point set. · Convex geometry. How many vertices in convex hull of X ? · Stochastic geometry. Fix ρ > 0 . At each point of X place a ball of radius ρ . What is volume of the union of such balls? Number of components? Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 2 / 26
Introduction · X ⊂ R d random finite point set. · Convex geometry. How many vertices in convex hull of X ? · Stochastic geometry. Fix ρ > 0 . At each point of X place a ball of radius ρ . What is volume of the union of such balls? Number of components? · Statistical physics. RSA packing. Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 2 / 26
Introduction · X ⊂ R d random finite point set. · Convex geometry. How many vertices in convex hull of X ? · Stochastic geometry. Fix ρ > 0 . At each point of X place a ball of radius ρ . What is volume of the union of such balls? Number of components? · Statistical physics. RSA packing. · Graph and networks. L G ( X ) := length of graph G on X . What is the behavior of L G ( X ) for large X ? Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 2 / 26
· The random variable X has density κ ( x ) if � P ( X ∈ A ) = κ ( x ) dx. A · Theorem (Beardwood, Halton, Hammersley (1959)): X i , 1 ≤ i ≤ n , i.i.d. with density κ ( x ) on [0 , 1] d . Then L MST ( { X 1 , ..., X n } ) � P [0 , 1] d κ ( x ) ( d − 1) /d dx. lim = γ MST ( d ) n ( d − 1) /d n →∞ Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 3 / 26
Introduction Questions pertaining to statistics of geometric structures on random input X ⊂ R d often involve analyzing sums of spatially correlated terms � ξ ( x, X ) , x ∈X where the R -valued score function ξ , defined on pairs ( x, X ) , represents the interaction of x with respect to X . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 4 / 26
Introduction Questions pertaining to statistics of geometric structures on random input X ⊂ R d often involve analyzing sums of spatially correlated terms � ξ ( x, X ) , x ∈X where the R -valued score function ξ , defined on pairs ( x, X ) , represents the interaction of x with respect to X . The sums describe some global feature of the random structure in terms of local contributions ξ ( x, X ) , x ∈ X . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 4 / 26
Introduction Questions pertaining to statistics of geometric structures on random input X ⊂ R d often involve analyzing sums of spatially correlated terms � ξ ( x, X ) , x ∈X where the R -valued score function ξ , defined on pairs ( x, X ) , represents the interaction of x with respect to X . The sums describe some global feature of the random structure in terms of local contributions ξ ( x, X ) , x ∈ X . We give some examples. Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 4 / 26
Random graphs X ⊂ R d finite; let G ( X ) be a graph on X . (a) For x ∈ X , put ξ ( x, X ) := 1 2( sum of lengths of edges in graph incident to x ) . Then � x ∈X ξ ( x, X ) gives the total edge length of G ( X ) . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 5 / 26
Random graphs X ⊂ R d finite; let G ( X ) be a graph on X . (a) For x ∈ X , put ξ ( x, X ) := 1 2( sum of lengths of edges in graph incident to x ) . Then � x ∈X ξ ( x, X ) gives the total edge length of G ( X ) . 1 (b) k ∈ N ; ξ k ( x, X ) = k +1 ( number of k-simplices containing x ) . Then � ξ k ( x, X ) x ∈X gives the number of k -simplices in G ( X ) . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 5 / 26
Random convex hulls · X ⊂ R d finite. Let co( X ) denote the convex hull of X . · For x ∈ X , k ∈ { 0 , 1 , ..., d − 1 } , we put 1 f k ( x, X ) := k +1 (number of k − dimensional faces containing x ) . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 6 / 26
Random convex hulls · X ⊂ R d finite. Let co( X ) denote the convex hull of X . · For x ∈ X , k ∈ { 0 , 1 , ..., d − 1 } , we put 1 f k ( x, X ) := k +1 (number of k − dimensional faces containing x ) . · Total number of k -dimensional faces of co( X ) : � x ∈X f k ( x, X ) . · R´ enyi, Sulanke Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 6 / 26
Continuum percolation X ⊂ R d ; join two points with an edge iff they are distant at most one. ξ comp ( x, X ) := (size of component containing x) − 1 . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 7 / 26
Continuum percolation X ⊂ R d ; join two points with an edge iff they are distant at most one. ξ comp ( x, X ) := (size of component containing x) − 1 . Component count in continuum percolation model on X : � ξ comp ( x, X ) . x ∈X Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 7 / 26
Random sequential adsorption · Unit volume balls B 1 ,n , B 2 ,n ..., arrive sequentially and uniformly at random on the cube [ − n 1 /d 2 , n 1 /d 2 ] d . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 8 / 26
Random sequential adsorption · Unit volume balls B 1 ,n , B 2 ,n ..., arrive sequentially and uniformly at random on the cube [ − n 1 /d 2 , n 1 /d 2 ] d . · The first ball B 1 ,n is packed , and recursively for i = 2 , 3 , . . . , the i -th ball B i,n is packed iff B i,n does not overlap any ball in B 1 ,n , ..., B i − 1 ,n which has already been packed. If not packed, the i -th ball is discarded. Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 8 / 26
Random sequential adsorption · Unit volume balls B 1 ,n , B 2 ,n ..., arrive sequentially and uniformly at random on the cube [ − n 1 /d 2 , n 1 /d 2 ] d . · The first ball B 1 ,n is packed , and recursively for i = 2 , 3 , . . . , the i -th ball B i,n is packed iff B i,n does not overlap any ball in B 1 ,n , ..., B i − 1 ,n which has already been packed. If not packed, the i -th ball is discarded. · X ⊂ R d a temporally marked point set. Define the ‘score’ at ( x, τ x ) ∈ X : � 1 if ball centered at x with arrival time τ x is accepted ξ (( x, τ x ) , X ) := 0 otherwise. Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 8 / 26
Random sequential adsorption · Unit volume balls B 1 ,n , B 2 ,n ..., arrive sequentially and uniformly at random on the cube [ − n 1 /d 2 , n 1 /d 2 ] d . · The first ball B 1 ,n is packed , and recursively for i = 2 , 3 , . . . , the i -th ball B i,n is packed iff B i,n does not overlap any ball in B 1 ,n , ..., B i − 1 ,n which has already been packed. If not packed, the i -th ball is discarded. · X ⊂ R d a temporally marked point set. Define the ‘score’ at ( x, τ x ) ∈ X : � 1 if ball centered at x with arrival time τ x is accepted ξ (( x, τ x ) , X ) := 0 otherwise. Total number of balls accepted: � x ∈X ξ (( x, τ x ) , X ) . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 8 / 26
Poisson input · For purposes of exposition, we consider Poisson input on R d . · By Poisson input, we mean a Poisson point process in R d . The Poisson point process (PPP) on R d is the probabilist’s way of placing points more or less uniformly at random in space. The PPP with rate (intensity) τ is denoted by P τ and has these properties: (i) the number of points that P τ puts in disjoint sets are independent r.v. Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 9 / 26
Poisson input · For purposes of exposition, we consider Poisson input on R d . · By Poisson input, we mean a Poisson point process in R d . The Poisson point process (PPP) on R d is the probabilist’s way of placing points more or less uniformly at random in space. The PPP with rate (intensity) τ is denoted by P τ and has these properties: (i) the number of points that P τ puts in disjoint sets are independent r.v. (ii) the number of points of P τ in the set B is a Poisson r.v. with parameter equal to the product of τ and Lebesque measure of B . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 9 / 26
Dimension estimators P := homogeneous rate one Poisson pt process on R d , x ∈ R d , k ≥ 3 . D j := D j ( x, P ) := dist. between x and its jth nearest neighbor in P . Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 10 / 26
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