SLIDE 1
Universal Record Statistics of Random Walks
GGI Workshop in Advances in Non-Equilibrium Statistical Mechanics
Grégory Schehr, LPTMS (Orsay)
SLIDE 2 Universal Record Statistics of Random Walks
GGI Workshop in Advances in Non-Equilibrium Statistical Mechanics
Grégory Schehr, LPTMS (Orsay)
Collaborators:
- C. Godrèche (IPhT, Saclay)
- S. N. Majumdar (LPTMS, Orsay)
- G. Wergen (Uni. of Cologne)
SLIDE 3
Statement of the problem
random variables (e.g. time series)
n i xi
SLIDE 4
Statement of the problem
random variables (e.g. time series) is a record iff
n i xi
SLIDE 5
Statement of the problem
random variables (e.g. time series) is a record iff
n i xi
Questions: Statistics of the number of records ?
SLIDE 6 Statement of the problem
random variables (e.g. time series) is a record iff Questions: Statistics of the number of records ?
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Statistics of the ages of records ?
SLIDE 7
Some recent applications of records in physics
Avalanches Le Doussal & Wiese ‘09 Financial data Wergen, Bogner & Krug ‘11, Wergen ‘13 Domain wall dynamics Alessandro et al. ‘90 Spin-glasses Sibani ‘07 Global warming Redner & Petersen ‘06, Wergen & Krug ‘10 Evolutionary biology Jain & Krug ‘05 Growing networks Godrèche & Luck ‘08 Random walks Majumdar & Ziff ‘08, Wergen, Majumdar, G. S. ‘12
SLIDE 8
Some recent applications of records in physics
Avalanches Le Doussal & Wiese ‘09 Financial data Wergen, Bogner & Krug ‘11, Wergen ‘13 Domain wall dynamics Alessandro et al. ‘90 Spin-glasses Sibani ‘07 Global warming Redner & Petersen ‘06, Wergen & Krug ‘10 Evolutionary biology Jain & Krug ‘05 Growing networks Godrèche & Luck ‘08 Random walks Majumdar & Ziff ‘08, Wergen, Majumdar, G. S. ‘12
SLIDE 9
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Nber of records
if is a record if is NOT a record
y i xi n k
SLIDE 10
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Nber of records
if is a record if is NOT a record is the proba. that a record is broken at step
y i xi n k
SLIDE 11
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Nber of records
if is a record if is NOT a record is the proba. that a record is broken at step
y i xi n k
SLIDE 12
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Nber of records
if is a record if is NOT a record is the proba. that a record is broken at step
y i xi n k
SLIDE 13
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Nber of records
if is a record if is NOT a record is the proba. that a record is broken at step
Universal !
y i xi n k
SLIDE 14
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF
y i xi n k
SLIDE 15
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Average nber of records
y i xi n k
SLIDE 16
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Average nber of records Variance
y i xi n k
SLIDE 17
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Average nber of records Variance Universal probability distribution
y i xi n k
SLIDE 18
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Average nber of records Variance Universal probability distribution
Stirling numbers: number of permutations of elements with disjoint cycles
y i xi n k
SLIDE 19
Record statistics of i.i.d. random variables
i.i.d. random variables with PDF Average nber of records Variance Universal probability distribution
Stirling numbers: number of permutations of elements with disjoint cycles
Gaussian for large
y i xi n k
SLIDE 20
Record statistics of random walks
where the jumps s are i.i.d. with PDF continuous & symmetric
SLIDE 21
Record statistics of random walks
where the jumps s are i.i.d. with PDF
Including Ordinary random walks
continuous & symmetric
Lévy flights is the Lévy index
SLIDE 22
Record statistics of random walks
where the jumps s are i.i.d. with PDF
Including Ordinary random walks
continuous & symmetric
Lévy flights is the Lévy index
Q: Dependence of records on the jump distribution ?
SLIDE 23
Mean record number of random walks
is the proba. that a record is broken at step
y i xi n k
SLIDE 24
Mean record number of random walks
is the proba. that a record is broken at step
y i xi n k y xi k y xi k ⇐ ⇒
SLIDE 25 Mean record number of random walks
is the proba. that a record is broken at step
y xi k y xi k ⇐ ⇒
- Proba. that the walker stays negative up to step
starting from the origin
SLIDE 26 Mean record number of random walks
is the proba. that a record is broken at step
y xi k y xi k ⇐ ⇒
- Proba. that the walker stays negative up to step
starting from the origin is given by the Sparre Andersen Theorem
SLIDE 27
is given by the Sparre Andersen Theorem
Mean record number of random walks
SLIDE 28
is given by the Sparre Andersen Theorem For symmetric RW
Mean record number of random walks
SLIDE 29 is given by the Sparre Andersen Theorem For symmetric RW
Universal, i.e. independent
- f the jump distribution !
Mean record number of random walks
SLIDE 30 is given by the Sparre Andersen Theorem For symmetric RW
Universal, i.e. independent
- f the jump distribution !
Mean record number of random walks
Majumdar, Ziff `08
SLIDE 31
Record statistics of random walks with a drift
where the jumps s are i.i.d. with PDF continuous & symmetric
RW with a drift
SLIDE 32
Record statistics of random walks with a drift
where the jumps s are i.i.d. with PDF continuous & symmetric
RW with a drift Mean number of records of :
SLIDE 33
Record statistics of random walks with a drift
where the jumps s are i.i.d. with PDF continuous & symmetric
RW with a drift Mean number of records of : (Generalized) Sparre Andersen theorem
SLIDE 34
Record statistics of random walks with a drift
RW with a drift ,
c µ
1 2
I III IV V II
Majumdar, G. S., Wergen `12
SLIDE 35
Record statistics of random walks with a drift
RW with a drift ,
c µ
1 2
I III IV V II
Majumdar, G. S., Wergen `12
SLIDE 36
Record statistics of random walks with a drift
RW with a drift ,
c µ
1 2
I III IV V II
Majumdar, G. S., Wergen `12
SLIDE 37
Record statistics of random walks with a drift
RW with a drift ,
c µ
1 2
I III IV V II
Majumdar, G. S., Wergen `12
SLIDE 38
Record statistics of random walks with a drift
RW with a drift ,
c µ
1 2
I III IV V II
Majumdar, G. S., Wergen `12
SLIDE 39
Record statistics of random walks with a drift
RW with a drift ,
c µ
1 2
I III IV V II
Majumdar, G. S., Wergen `12
SLIDE 40
Record statistics of random walks with a drift
RW with a drift ,
c µ
1 2
I III IV V II
What about the full distribution of ?
Majumdar, G. S., Wergen `12
SLIDE 41 Renewal approach to records of RW
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Joint distribution of ?
SLIDE 42 Renewal approach to records of RW
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Joint distribution of ? RW is a Markov process independent except for the global constraint are
SLIDE 43 Renewal approach to records of RW
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Joint distribution of ? RW is a Markov process independent except for the global constraint are are identical RW is translationally invariant s while has different statistics
SLIDE 44 Two main objects: Persistence (or survival) probability
Distribution of first-passage time (from below) Joint distribution of ?
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Renewal approach to records of RW
SLIDE 45 Joint distribution of
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Renewal approach to records of RW
survival proba. first passage proba.
SLIDE 46
- Proba. distribution of the number of records
with
SLIDE 47
- Proba. distribution of the number of records
with Generating function w.r.t. the number of steps
SLIDE 48
- Proba. distribution of the number of records
with Generating function w.r.t. the number of steps
(for symmetric jumps)
SLIDE 49
- Proba. distribution of the number of records
By ``inverting’’ the GF (for symmetric jumps):
Majumdar, Ziff `08
SLIDE 50
- Proba. distribution of the number of records
By ``inverting’’ the GF (for symmetric jumps): For :
Majumdar, Ziff `08
SLIDE 51
- Proba. distribution of the number of records
By ``inverting’’ the GF (for symmetric jumps): For : RW with a drift
c µ
1 2
I III IV V II
Majumdar, Ziff `08 Majumdar, G. S., Wergen `12
SLIDE 52
- Proba. distribution of the number of records
By ``inverting’’ the GF (for symmetric jumps): For : RW with a drift
c µ
1 2
I III IV V II
Majumdar, Ziff `08 Majumdar, G. S., Wergen `12
SLIDE 53
- Proba. distribution of the number of records
By ``inverting’’ the GF (for symmetric jumps): For : RW with a drift
c µ
1 2
I III IV V II
e.g. for
Majumdar, Ziff `08 Majumdar, G. S., Wergen `12
SLIDE 54 Statistics of the ages of records
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
SLIDE 55 Statistics of the ages of records
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Typical age of a record:
SLIDE 56 Statistics of the ages of records
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Typical age of a record:
What about the longest or shortest age of a record ?
SLIDE 57 Statistics of the ages of records
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Typical age of a record:
What about the longest or shortest age of a record ? What is the proba. that the current record is the oldest one ?
Godrèche, Majumdar, G. S., `14
SLIDE 58 Statistics of the ages of records
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Typical age of a record:
What about the longest or shortest age of a record ? What is the proba. that the current record is the oldest one ?
Godrèche, Majumdar, G. S., `14
SLIDE 59 Statistics of the ages of records
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
= ?
SLIDE 60 Statistics of the ages of records
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
= ?
SLIDE 61 Statistics of the ages of records
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
= ?
SLIDE 62
Statistics of the ages of records
SLIDE 63
Statistics of the ages of records
SLIDE 64
Statistics of the ages of records
SLIDE 65
Statistics of the ages of records
with
SLIDE 66
Statistics of the ages of records
with
Generating function
SLIDE 67
Statistics of the ages of records
with
Generating function
SLIDE 68
Statistics of the ages of records
SLIDE 69
Statistics of the ages of records For symmetric RW
SLIDE 70
Statistics of the ages of records For symmetric RW One finds
SLIDE 71 Statistics of the ages of records
0.625 0.626 0.627 0.628 10 20 30 40 50 60 70 80 90 100
Q (n) n 0.626508...
Godrèche, Majumdar, G. S.,`14
SLIDE 72 Statistics of the ages of records
0.625 0.626 0.627 0.628 10 20 30 40 50 60 70 80 90 100
Q (n) n 0.626508...
Godrèche, Majumdar, G. S.,`14
SLIDE 73 Statistics of the ages of records
0.625 0.626 0.627 0.628 10 20 30 40 50 60 70 80 90 100
Q (n) n 0.626508...
Godrèche, Majumdar, G. S.,`14 Pitman, Yor, `97 Godrèche, Majumdar, G. S., `09
SLIDE 74 New observable...new universal constant
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
SLIDE 75 New observable...new universal constant
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
SLIDE 76 New observable...new universal constant
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
SLIDE 77 New observable...new universal constant
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6
n
Godrèche, Majumdar, G. S.,`14
SLIDE 78 Conclusions
Exact results for records of strongly correlated time series Universal records statistics for (symmetric) RWs Extension to multiparticle systems
Wergen, Majumdar, G. S. `12
Extension to Continuous Time Random Walks (CTRWs)
see arXiv:1305.0639 for a short review
SLIDE 79 Conclusions
Exact results for records of strongly correlated time series Universal records statistics for (symmetric) RWs Extension to multiparticle systems
Wergen, Majumdar, G. S. `12
Extension to Continuous Time Random Walks (CTRWs)
High sensitivity to the definition of the age of the last record
see arXiv:1305.0639 for a short review
SLIDE 80 Sensitivity to the definition of the age of the last record
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6 τ6 = 7
n
Godrèche, Majumdar, G. S., `14
SLIDE 81 Sensitivity to the definition of the age of the last record
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6 τ6 = 7
n
Godrèche, Majumdar, G. S., `14
SLIDE 82 Sensitivity to the definition of the age of the last record
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6 τ6 = 7
n
Godrèche, Majumdar, G. S., `14
SLIDE 83 Sensitivity to the definition of the age of the last record
An = 3
i xi
τ1 = 4 τ2 = 3 τ3 = 2 τ4 = 5 τ5 = 6 τ6 = 7
n
Godrèche, Majumdar, G. S., `14
