Distinct sites, common sites and maximal displacement of N random walkers

Transcription

1 Distinct sites, common sites and maximal displacement of N random walkers Anupam Kundu GGI workshop Florence Joint work with Satya N. Majumdar, LPTMS Gregory Schehr, LPTMS

2 Outline 1. N independent walkers 2. N vicious walkers

3 Distinct visited site Distinct site :- Site visited by the walker

4 Distinct visited site Distinct site :- Site visited by the walker

5 Distinct visited site Distinct site :- Site visited by the walker

6 Distinct visited site Distinct site :- Site visited by the walker

7 Distinct visited site

8 Distinct visited site

9 Distinct visited site Distinct site :- Site visited by any walker

10 Distinct visited site

11 Common visited site Common site :- Site visited by all the walkers

12 Common visited site

13 Number of distinct & common sites # of distinct sites visited by N walkers in time step t = # of common sites visited by N walkers in time step t =

14 Number of Distinct sites A. Dvoretzky and P. Erdos (1951) for a single walker in d dimension. B. H. Hughes Later studied by Vineyard, Montroll, Weiss.

15 Number of Distinct sites Larralde et al. : N independent random walkers in d dimension Nature, 355, 423 (1992)

16 Number of Distinct sites Larralde et al. : N independent random walkers in d dimension Nature, 355, 423 (1992) Three different growths of separated by two time scales

17 Number of Distinct sites Larralde et al. : N independent random walkers in d dimension Nature, 355, 423 (1992) Three different growths of separated by two time scales

18 Number of Common sites Majumdar and Tamm - Phys. Rev. E 86, , (2012)

19 Number of Common sites Majumdar and Tamm - PRE (2012)

20 Number of Common sites Majumdar and Tamm - PRE (2012)

21 Number of Common and distinct sites

22 Probability Distributions = Distribution function of the number of distinct sites visited by N walkers in time step t = Distribution function of the number of common sites visited by N walkers in time step t

23 Probability Distributions = Distribution function of the number of distinct sites visited by N walkers in time step t = Distribution function of the number of common sites visited by N walkers in time step t Applications : Territory of animal population of size N Popular tourist place visited by all the tourists in a city Diffusion of proteins along DNA Annealing of defects in crystal Popular hub sites in a multiple user network

24 Probability Distributions = Distribution function of the number of distinct sites visited by N walkers in time step t = Distribution function of the number of common sites visited by N walkers in time step t One dimension Maximum overlap Connection with extreme value statistics : exactly solvable Total # of distinct sites = range or span # of common sites = common range or common span

25 Model N one dimensional t-step Brownian walkers Each of them starts at the origin and have diffusion constants D

26 Scaling All displacements are scaled by Probability distributions take following scaling forms :

27 Scaling All displacements are scaled by Probability distributions take following scaling forms :

28 Range: Single particle

29 Range: Many particles

30 Span Union Span

31 Common span Intersection Common span

32 Span =

33 Span =

34 Common span =

35 Connection with extreme values

36 Connection with extreme values

37 Connection with extreme values The variables Similarly the variables random variables are correlated random variables are also correlated We need joint probability distributions

38 Single particle M, m are correlated random variables

39 Particle inside the box

40 Distribution of the span : N=1

41 Span for N > 2

42 Span for N > 2

43 Common Span for N > 2

44 Cumulative distribution of and

45 Common Span for N > 2

46 Distribution of span & common span

47 Exact Distributions for N=1 Distribution of span or common span N=1 A. K, Majumdar & Schehr, PRL (2013)

48 Exact Distributions for N=1 & N=2 Distribution of span N=1 N=2 Distribution of common span A. K, Majumdar & Schehr, PRL (2013)

49 Distributions : N = 2

50 Exact Distributions for N=1 & N=2 Distribution of span N=1 N=2 Distribution of common span Distribution of span A. K, Majumdar & Schehr, PRL (2013)

51 Distributions : N = 2

52 Distributions Are there any limiting forms of these two distributions for large N?

53 Moments 1 st moment 2 nd moment

54 Moments : Span : Common span :

55 Moments : Span : Random variable x has N independent distribution Common span : Random variable y has N independent distribution

56 Moments : Span : Random variable x has N independent distribution Common span : Random variable y has N independent distribution

57 Distributions : Large N Distribution of the number of distinct sites or the span A. K, Majumdar & Schehr, PRL (2013)

58 Distributions : Large N Distribution of the number of common sites or the common span A. K, Majumdar & Schehr, PRL (2013)

59 Span : are Gumbel variables The variables M i 's are independent, positive random variables

60 are Gumbel variables Span : For large N, both distributed according to Gumbel distribution : For large N, both are of

61 Span : Two ways of creating S Two ways of creating s : Single particle creating Two particles creating s + s + s - s -

62 Span : Two ways of creating S Two ways of creating s : Single particle creating Two particles creating s + s + s - s -

63 Distribution of the span So, when, become independent : where,

64 Distribution of the span So, when, become independent : where, Span : Common Span :

65 Asymptotes : finite N D(x) O C(y) O

66 Asymptotes : finite N Span Common Span A. K, Majumdar & Schehr, PRL, 110, , (2013)

67 What happens when the walkers are interacting?

68 Non-intersection Interaction x 1 (t) time x 2 (t) x 3 (t) x 4 (t) 0 x 1 x 2 x 3 x 4 space Vicious walkers

69 Span in different situations Watermelon without wall time 0 t space Watermelon with wall time t 0 space t s Till survival time space 0 x 1 x 2 x 3 x 4

70 Common Span time t space 0 L 1

71 Span till survival t s time x 2 (t) m N x 1 (t) x 3 (t) x 4 (t) 0 x 1 x 2 x 3 x 4 space Prob. [ Global maximum ]=?

72 time Single Brownian walker : N=1 t f m 1 0 x 1 space

73 time N = 2 particles t s m 2 x 1 (t) x 2 (t) 0 x 1 x 2 space

74 time N = 2 particles in a box t s m 2 x 1 (t) x 2 (t) x 2 0 x 1 x 2 L space L Exit probability 0 x 1 L The two walkers stay non-intersecting inside the box [0, L] Prob.[ ] till the first walker crosses the origin for the first time

75 F=1 F=1 F=0 N = 2 particles in a box x 2 L F=0 F=0 0 L x 1 t s m 2 x 2 F=0 L 2 m 1 0 x 1 x 2 F=0 L 1 x 1

76 Marginal cumulative probabilities

77 N = 2 case t f N=1 m t s x 1 (t) m x 2 (t) 0 x 1 x 2 Kundu, Majumdar, Schehr (2014)

78 N 2 N Non-intersecting walkers : For m x 1 0 x 2 x 3 x 4 Kundu, Majumdar, Schehr (2014) N Non-interacting or independent walkers : m N space x 1 x 2 x 3 x 4 Krapivsky, Majumdar, Rosso, J. Phys. A (2010)

79 N 2 walkers : propagator Start with the N particle propagator in the box [0, L]: = Probability density that particles starting from ( ) reach ( ) inside [0,L] in time t. t y 1 y 2 y 3 y 4 time x 2 (t) x 3 (t) x 4 (t) x 1 (t) 0 x 1 x 2 x 3 x 4 space L

80 N 2 walkers : exit probability Start with the N particle propagator in the box [0, L]: Exit probability : = Probability density that particles starting from ( ) reach ( ) in time t. Slater Determinant Prob.[ ] The N walkers stay non-intersecting inside the box [0, L] till the first walker crosses the origin for the first time

81 N 2 walkers : Distribution Start with the N particle propagator in the box [0, L]: Exit probability : = Probability density that particles starting from ( ) reach ( ) in time t. Slater Determinant Prob.[ ] The N walkers stay non-intersecting inside the box [0, L] till the first walker crosses the origin for the first time

82 time Heuristic argument First passage time probability distribution : t s Fisher 1984 Krattenthaler et al 2000 Bray, Winkler, 2004 m 0 x 1 x 2 x 3 x 4 space Decreases as N increases Independent walkers

83 Heuristic argument First passage time probability distribution : Independent walkers For large N

84 Distribution Independent walkers Kundu, Majumdar, Schehr (2014)

85 Prefactor Independent walkers Where and Krapivsky, Majumdar, Rosso,(2010) The N walkers stay non-intersecting till the Prob.[ ] first walker crosses the origin for the first time Kundu, Majumdar, Schehr (2014)

86 Prefactor Independent walkers Where and Krapivsky, Majumdar, Rosso, J. Phys. A (2010) Kundu, Majumdar, Schehr (2014)

87 E N for large N The constant E N can be computed for any given N. Kundu, Majumdar, Schehr (2014)

88 Remarks & summary : independent walkers Exact distribution of the number of distinct and common sites visited by N independent random walkers. Connection with extreme displacements => Exact limiting distributions for large N :, Walkers moving in a globally bounded potential:,

89 Remarks & summary : Vicious walkers t s Till survival 0 x 1 x 2 x 3 x 4 Watermelon with wall t 0 Schehr, Majumdar, Comtet, Forrester (2013)

90 Remarks & summary : Vicious walkers t Watermelon without wall 0 What about the distribution of the maximum displacement of the 1 st walker?

91 Thank You