A stretched-exponential Pinning Model with heavy-tailed Disorder

Transcription

1 A stretched-exponential Pinning Model with heavy-tailed Disorder Niccolò Torri Université Claude-Bernard - Lyon 1 & Università degli Studi di Milano-Bicocca Roma, October 8, 2014

2 References A. Auffinger and O. Louidor Directed polymers in random environment with heavy tails Comm. on Pure and Applied Math., 64: , 2011 B. Hambly and J. B. Martin Heavy tails in last-passage percolation Probability Theory and Related Fields, 137: , 2007 N. Torri Pinning Model with Heavy Tailed disorder arxiv: , 2014 N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

3 1 Overview 2 Definitions and Results Definition of the Pinning Model Assumptions Results 3 Proofs A sketch of the proof Critical Threshold 4 The directed Polymer in a random Environment with Heavy Tails Introduction Improve the Critical threshold 5 Conclusions and open problems N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

4 What is the Pinning Model? Consider a Point Process τ N 0. τ 0 τ 1 τ 2 τ m-1 τ m-1 τ m 0 N N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

5 What is the Pinning Model? Consider a Point Process τ N 0. τ 0 τ 1 τ 2 τ m-1 τ m-1 τ m 0 N Reference example: 0-level set of a Markov Chain S. S 10 S 12 S 1 S 7 S 9 S 13 S 17 S 18 S 11 S 2 S 4 S14 S 0 S 6 S 8 S 16 S 3 S 5 S 15 τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 = It is useful thought (τ, P) as a random subset of N. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

6 What is the Pinning Model? Consider a Point Process τ N 0. τ 0 τ 1 τ 2 τ m-1 τ m-1 τ m 0 N Fix N and put a weight ω n on each integer smaller than N. τ 0 τ 1 τ 2 τ m-1 τ m-1 τ m... 0 ω 1 ω 2 ω 3 ω 4 ω 5 ω N-5 ω N-4 ω N-3 ω N-2 ω N-1 N The sequence ω = (ω n ) N n=1 is the disorder: at each point s j we give a reward/penalty to visit the point proportional to the value of ω sj. We have an interaction between τ and ω described by a given function Φ ω N ( ). Therefore the Pinning Model is P ω N(τ 1 = s 1, τ 2 = s 2,, τ m = s m ) Φ ω N(s 1,, s m )P(τ 1 = s 1, τ 2 = s 2,, τ m = s m ) N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

7 What do we want to know? We want to understand the behavior of the process τ with respect to the Pinning Model P ω N in the limit N : Will the behavior of (τ, P ω N ) be the same of the original process (τ, P)? In what way will the behavior depend of the disorder? N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

8 What do we want to know? We want to understand the behavior of the process τ with respect to the Pinning Model P ω N in the limit N : Will the behavior of (τ, P ω N ) be the same of the original process (τ, P)? In what way will the behavior depend of the disorder? We will consider the rescaled process τ/n. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

9 2 Definitions and Results Definition of the Pinning Model Assumptions Results

10 Definition of the Pinning Model Let (τ, P) be a Point process starting from 0 consider Let P N be the law of τ/n [0, 1]. τ/n. Consider a probability measure defined by a Radon-Nikodym derivative: I [0, 1] dp ω ( N 1 ) β,h,n (I) exp (βω n h)1 dp n/n I. N n=1 This definition depends of three parameters: h, β and ω = (ω n ) n N a real sequence called disorder. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

11 Definition of the Pinning Model Add a normalization constant to make P a probability: for I [0, 1] dp ω β,h,n (I) = 1 ( N 1 ) 1 I dp N Z ω exp (βω n h)1 n/n I. β,h,n P N be the law of τ/n [0, 1], 1 1 I boundary condition, Z ω β,h,n normalization constant. n=1 N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

12 Definition of the Pinning Model Add a normalization constant to make P a probability: for I [0, 1] dp ω β,h,n (I) = 1 ( N 1 ) 1 I dp N Z ω exp (βω n h)1 n/n I. β,h,n P N be the law of τ/n [0, 1], 1 1 I boundary condition, Z ω β,h,n normalization constant. The Pinning Model is a probability measure on the space of all closed sets of [0, 1] which contain 0, 1. n=1 N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

13 2 Definitions and Results Definition of the Pinning Model Assumptions Results

14 Assumption on the Point Process We consider (τ, P) a Renewal process, that is a Point Process such that τ 0 = 0, (τ j τ j 1 ) j N is an i.i.d. sequence. Let K(n) = P(τ 1 = n), which is a probability on N { }. If K( ) > 0 Def. τ is terminating τ <, P-a.s. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

15 Assumption on the Point Process We consider (τ, P) a Renewal process, that is a Point Process such that τ 0 = 0, (τ j τ j 1 ) j N is an i.i.d. sequence. Let K(n) = P(τ 1 = n), which is a probability on N { }. If K( ) > 0 Def. τ is terminating τ <, P-a.s. We consider a non-terminating renewal process τ s.t. Precisely K(n) = e Cnγ γ (0, 1). 1 subexponential: lim n K(n + k)/k(n) = 1 for any k > 0 and lim n K (2) (n)/k(n) = 2, 2 stretched-exponential: lim n log K(n)/N γ = C, for some C > 0 and γ (0, 1). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

16 Assumption on the Parameters The disorder ω is a fixed (quenched) realization of an i.i.d. sequence of random variables such that P(ω 1 > t) ct α, t. α (0, 1) Moreover we assume ω 1 positives with a continuous distribution. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

17 Assumption on the Parameters The disorder ω is a fixed (quenched) realization of an i.i.d. sequence of random variables such that P(ω 1 > t) ct α, t. α (0, 1) Moreover we assume ω 1 positives with a continuous distribution. β = β N = ˆβN γ 1 α 0 as N. h > 0 fixed. N 1 n=1 (βω n h)1 n/n I N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

18 The role of β and h the role of β is to tune the intensity of the disorder ω. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

19 The role of β and h the role of β is to tune the intensity of the disorder ω. the role of h > 0 is to force away the renewal process: consider the Homogeneous Pinning Model (β = 0), then if I = m P ω 0,h,N(I) = 1 1 I Z ω e ( (m 1)h) m K (N(x i x i 1 )) 0,h,N i=1 = 1 m 1 I e h Z ω e h K (N(x i x i 1 )). 0,h,N i=1 K(n) = e h K(n) = P ω 0,h,N (I) = P cons N (I) : the law of a (rescaled) terminating Renewal Process constrained to visit 1. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

20 The role of β and h the role of β is to tune the intensity of the disorder ω. the role of h > 0 is to force away the renewal process. Homogeneous Pinning Model (β = 0): = P ω 0,h,N (I) = P cons N (I) : the law of a (rescaled) terminating Renewal Process constrained to visit 1. This forces the typical trajectories to have big jumps: h>0 h=0 N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

21 Remark about h > 0 If h > 0, then we can assume to be 0 by replacing the original renewal process with a new terminating one. ( N 1 P ω l β,h,n(i) exp β ω n 1 n/n I) K (N(x i x i 1 )) n=1 i=1 where K is a terminating renewal process. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

22 Remark about h > 0 If h > 0, then we can assume to be 0 by replacing the original renewal process with a new terminating one. ( N 1 P ω l β,h,n(i) exp β ω n 1 n/n I) K (N(x i x i 1 )) n=1 i=1 where K is a terminating renewal process. Keep in mind this formula in the sequel. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

23 2 Definitions and Results Definition of the Pinning Model Assumptions Results

24 Set-up We have a random set τ/n [0, 1], a modification of its law P ω β N,h,N. Goal: study its behavior as N. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

25 Set-up We have a random set τ/n [0, 1], a modification of its law P ω β N,h,N. Goal: study its behavior as N.= Specify the space and the topology. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

26 Set-up We have a random set τ/n [0, 1], a modification of its law P ω β N,h,N. Goal: study its behavior as N.= Specify the space and the topology. We look at τ/n [0, 1] as a random variables on the space of the all closed subsets of [0, 1] which contain 0, 1. We equip this space with the Hausdorff distance: Hausdorff distance: d H (A, B) < ɛ Def. a A, b B : a b < ɛ and vice-versa, interchanging A and B. B A b a ε N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

27 Theorems: Concentration Theorem (Concentration of the Renewal Process) For any N N there exists a set Iβ ω N,N [0, 1] depending only of ω, α and γ s.t. for any fixed ɛ > 0 P ω β N,h,N ( d H (I, I ω β N,N) > ɛ) P 0, Here d H (, ) denotes the Hausdorff distance. ε ε ε ε N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

28 Theorems: Convergence Theorem (Limit Theorem) For any ˆβ > 0 there exists a random closed subset Î [0, 1], function wˆβ, of a suitable continuum disorder w, such that I ω β N,N (d) Îwˆβ, on (X, d H ), the space of all closed sets of [0, 1] which contain 0, 1. The limit set Î depends only of w, α and γ. wˆβ, N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

29 Conclusion: Concentration & Convergence Corollary (Convergence of the Renewal Process) N If β N ˆβN γ 1 α, then τ/n [0, 1] (d) Î wˆβ, in (X, d H ), with respect to the Pinning Model measure P ω β N,h,N. ε ε ε Limit Set Renewal Set Size of the mesh 1/N N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

30 Critical Threshold Theorem (Critical Threshold) Let w the continuum disorder, then there exists a random variable ˆβ w c 1 1 If ˆβ < ˆβ w c then Îwˆβ, {0, 1}, a.e.-w s.t If ˆβ > ˆβ w c then Îwˆβ, {0, 1}, a.e.-w Moreover ˆβ w c > 0 for a.e.-w and for any choice of α, γ (0, 1). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

31 Interpretation of the Critical Threshold If ˆβ < ˆβ c, the disorder is irrelevant : the behavior is like the one of the Homogeneous Pinning Model β = 0. ε ε If ˆβ > ˆβ c different. ε ε ε ε N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

32 3 Proofs A sketch of the proof Critical Threshold

33 Energy-Entropy to visit a set ( N 1 P ω l β,h,n(i) exp β ω n 1 n/n I) K (N(x i x i 1 )) n=1 i=1 Keep in Mind: Any time that we make a jump to visit a point x, we are penalized by e (N( length of the jump))γ, the probability to make the jump, but we gain an Energy given by the disorder on the point x. Case I = {0, x, 1}: we have to make two jumps. Penalization = K(Nx)K(N(1 x)) = e CNγ (x γ +(1 x) γ ) 0 x 1 P ω β,h,n(i) e βωx CNγ (x γ +(1 x) γ ) CN γ (x γ + (1 x) γ ) the Entropy penalization to visit exactly x. ω x the Energy gained by visiting x. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

34 Keep in Mind: Any time that we make a jump to visit a point x, we are penalized by e (N( length of the jump))γ, the probability to make the jump, but we gain an Energy given by the disorder on the point x. i.i.d. structure of the jumps: if I = {x 0 = 0 < x 1 < < x l = 1}, then penalization to visit exactly a fixed set ι is K(Nx 1 )K(N(x 2 x 1 )) K(N(1 x l 1 )) = e CNγ l i=1 (x i x i 1 ) γ ι= { x 0 x 1 x 2 x 3 x 4 x 5 x 6 } x 0 x 1 x 2 x 3 x 4 x 5 x 6 ω 1 ω 2 ω 3 ω 4 ω 5 ω 13 ω 14 ω 15 ω 16 ω 17 l Entropy E N (ι) := CN γ (x i x i 1 ) γ i=1 Energy σ N = l ω xi i=1 ω 6 ω 7 ω 8 ω 9 ω 10 ω 11 ω 12 visit only ι N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

35 The Idea Idea: compare the Energy gained and the Entropy penalization to visit a given set ι.... ω N-5 ω 1 ω 2 ω 3 ω 4 ω 5 ω N-4 ω N-3 ω N-2 ω N-1 The probability to visit (exactly) a set ι P ω β,h,n (ι) βσ N(ι) E N (ι). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

36 The Idea Idea: compare the Energy gained and the Entropy penalization to visit a given set ι.... ω N-5 ω 1 ω 2 ω 3 ω 4 ω 5 ω N-4 ω N-3 ω N-2 ω N-1 Maximize the probability The probability to visit (exactly) a set ι P ω β,h,n (ι) βσ N(ι) E N (ι). I ω β,n = arg max {βσ N (I) E N (I)}. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

37 The Idea Idea: compare the Energy gained and the Entropy penalization to visit a given set ι.... ω N-5 ω 1 ω 2 ω 3 ω 4 ω 5 ω N-4 ω N-3 ω N-2 ω N-1 Maximize the probability The probability to visit (exactly) a set ι P ω β,h,n (ι) βσ N(ι) E N (ι). I ω β,n = arg max {βσ N (I) E N (I)}. Choose β: N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

38 The Idea Idea: compare the Energy gained and the Entropy penalization to visit a given set ι.... ω N-5 ω 1 ω 2 ω 3 ω 4 ω 5 ω N-4 ω N-3 ω N-2 ω N-1 Maximize the probability The probability to visit (exactly) a set ι P ω β,h,n (ι) βσ N(ι) E N (ι). I ω β,n = arg max {βσ N (I) E N (I)}. Choose β: Energy: σ N N 1/α = β Nγ Entropy: E N N γ N 1/α. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

39 The Idea Idea: compare the Energy gained and the Entropy penalization to visit a given set ι.... ω N-5 ω 1 ω 2 ω 3 ω 4 ω 5 ω N-4 ω N-3 ω N-2 ω N-1 Maximize the probability The probability to visit (exactly) a set ι P ω β,h,n (ι) βσ N(ι) E N (ι). I ω β,n = arg max {βσ N (I) E N (I)}. Choose β: Energy: σ N N 1/α = β Nγ Entropy: E N N γ N 1/α. prove concentration around I ω β N,N. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

40 3 Proofs A sketch of the proof Critical Threshold

41 Critical Threshold: the continuum disorder We consider the limit set Î wˆβ,. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

42 Critical Threshold: the continuum disorder We consider the limit set Î wˆβ,. It depends of a continuum disorder w = (M i, Y i ) N. M i = (E E i ) 1/α, α (0, 1), E i E(1) i.i.d. and Y i U([0, 1]) i.i.d. These two sequences are independent. How does it turn out? N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

43 Critical Threshold: the continuum disorder We consider the limit set Î wˆβ,. It depends of a continuum disorder w = (M i, Y i ) N. M i = (E E i ) 1/α, α (0, 1), E i E(1) i.i.d. and Y i U([0, 1]) i.i.d. These two sequences are independent. How does it turn out? We regard the Ordered Statistic of the disorder ω. M (N) i Y (N) i = value of the i-maximum, = position of the i-maximum. ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 ω 8 ω 9 ω 10 ω 11 ω 12 ω 13 ω 14 ω 15 ω 16 ω 17 N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

44 Extreme value theory: (N 1/α M (N) i, Y (N) i ) i (d) (M i, Y i ) i N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

45 Critical Threshold: Energy-Entropy balance We define { Î wˆβ, = arg max ˆβσ (I) E(I)}, I σ = σ (M i, Y i ) is the continuum Energy: σ (I) = i:y i I M i E is the continuum Entropy, a suitable extension of the Entropy (if I is finite E coincide with the original Entropy). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

46 Critical Threshold: Positivity ˆβ c = inf{ ˆβ : Î {0, 1}}. wˆβ, For a.e.-w, ˆβ c > 0 for any choice of α, γ (0, 1). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

47 Critical Threshold: Positivity ˆβ c = inf{ ˆβ : Î {0, 1}}. wˆβ, For a.e.-w, ˆβ c > 0 for any choice of α, γ (0, 1). 1 Given ɛ > 0, then Î wˆβ, [0, ɛ] [1 ɛ, 1] for ˆβ small. ε ε N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

48 Critical Threshold: Positivity ˆβ c = inf{ ˆβ : Î {0, 1}}. wˆβ, For a.e.-w, ˆβ c > 0 for any choice of α, γ (0, 1). 1 Given ɛ > 0, then Î wˆβ, [0, ɛ] [1 ɛ, 1] for ˆβ small. ε ε 2 If ɛ > 0 is too small, x ε = total Energy contained in [0,ε] U [1-ε,1] 0 ε 1-ε Energy(Î wˆβ, ) X ɛ ɛ 1/α and Entropy(Î wˆβ, ) Cɛγ + 1. Then ˆβ Energy(Î wˆβ, ) Entropy(Î wˆβ, ) < 1: impossible because Entropy({0, 1}) = 1. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31 1

49 4 The directed Polymer in a random Environment with Heavy Tails Introduction Improve the Critical threshold

50 The directed Polymer 1 Let s be a dimensional simple random walk starting from 0 and constrained to come back to 0 after N-steps. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

51 The directed Polymer 1 Let s be a dimensional simple random walk starting from 0 and constrained to come back to 0 after N-steps. 2 Take an i.i.d. sequence (ω = {ω i,j } i,j N, P) placed on all integers that can be touched by the walk. This sequence is called environment. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

52 Directed Polymer: Gibbs Measure For a given path s we define the Gibbs measure µ β,n (s) = eβσ N(s) Q β,n, where σ N ( ) = i,j ω i,j1 (si =j) is the Energy and Q β,n is a normalization constant. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

53 Directed Polymer: Gibbs Measure For a given path s we define the Gibbs measure µ β,n (s) = eβσ N(s) Q β,n, where σ N ( ) = i,j ω i,j1 (si =j) is the Energy and Q β,n is a normalization constant. A. Auffinger and O. Louidor (2011): the environment has heavy tails with index α (0, 2). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

54 4 The directed Polymer in a random Environment with Heavy Tails Introduction Improve the Critical threshold

55 The Critical threshold Let σ Continuum Energy E Entropy ˆγ β = arg max {βσ (γ) E(γ)} γ N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

56 The Critical threshold Let σ Continuum Energy E Entropy The critical threshold is ˆγ β = arg max {βσ (γ) E(γ)} γ β c = inf{β > 0 : ˆγ β 0}. Theorem (A. Auffinger and O. Louidor, 2011) For a.e. realization of the continuum disorder, β c > 0 for α [0, 1/3) and β c = 0 for [1/2, 2). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

57 The Critical threshold Let σ Continuum Energy E Entropy The critical threshold is ˆγ β = arg max {βσ (γ) E(γ)} γ β c = inf{β > 0 : ˆγ β 0}. Theorem (A. Auffinger and O. Louidor, 2011) For a.e. realization of the continuum disorder, β c > 0 for α [0, 1/3) and β c = 0 for [1/2, 2). AIM: β c > 0 for α [1/3, 1/2). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

58 Step 1 For any ɛ > 0, ˆγ β < ɛ if β is small enough. ε (x,ε) N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

59 Step 2 If ɛ > 0 is small enough, then we cannot never gain enough Energy to compensate any Entropy cost. KEY: Consider the process X ɛ of the Energy in A ɛ A ε ε Conclusion: σ (ˆγ β ) X ɛ ɛ 1/α and E(ˆγ β ) Cɛ 2γ = ˆβσ (ˆγ β ) < E(ˆγ β ). N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

60 Conclusions and open problems 1 γ = 0, that is when K(n) L(n) n 1+γ. Conjecture: β N ˆβN 1/α log N. 2 α > 1 which is linked with the open problem of A. Auffinger and O. Louidor in the case α > 2. N. Torri (Lyon 1 & Milano-Bicocca) Pinning Model Roma, October 8, / 31

61 Thanks for your attention!

62 Entropy Let us consider the space L 0 = {s : [0, 1] R : s is 1 Lipschitz, s(0) = s(1) = 0} equipped with L -norm, denoted by. For a curve γ L 0 we define its Entropy as E(γ) = 1 0 ( ) d e dx γ(x) dx, where e(x) = 1 2 ((1 + x) log(1 + x) + (1 x) log(1 x)).

63 Energy We introduce the continuous environment σ as σ (γ) = i M i δ Zi (graph(γ)), γ L 0. with graph(γ) = {(x, γ(x)) : x [0, 1])} D = {(x, y) R 2 : y x (1 x)} is the graph of γ, M i = (E E i ) 1/α, α (0, 2) and (Z i ) i N is an i.i.d.-sequence of Uniform(D) r.v. s independent of (M i ) i N.