Universal Record Statistics of Random Walks GGI Workshop in Advances - PowerPoint PPT Presentation

Universal Record Statistics of Random Walks GGI Workshop in Advances in Non-Equilibrium Statistical Mechanics Grégory Schehr, LPTMS (Orsay)

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)

Statement of the problem random variables (e.g. time series) x i i n

Statement of the problem random variables (e.g. time series) x i is a record iff i n

Statement of the problem random variables (e.g. time series) x i is a record iff i n Questions: ? Statistics of the number of records

Statement of the problem random variables (e.g. time series) x i is a record iff τ 5 = 6 τ 4 = 5 τ 3 = 2 τ 2 = 3 A n = 3 τ 1 = 4 i n Questions: ? Statistics of the number of records ? Statistics of the ages of records

Some recent applications of records in physics Domain wall dynamics Alessandro et al. ‘90 Evolutionary biology Jain & Krug ‘05 Global warming Redner & Petersen ‘06, Wergen & Krug ‘10 Spin-glasses Sibani ‘07 Random walks Majumdar & Ziff ‘08, Wergen, Majumdar, G. S. ‘12 Growing networks Godrèche & Luck ‘08 Avalanches Le Doussal & Wiese ‘09 Financial data Wergen, Bogner & Krug ‘11, Wergen ‘13

Some recent applications of records in physics Domain wall dynamics Alessandro et al. ‘90 Evolutionary biology Jain & Krug ‘05 Global warming Redner & Petersen ‘06, Wergen & Krug ‘10 Spin-glasses Sibani ‘07 Random walks Majumdar & Ziff ‘08, Wergen, Majumdar, G. S. ‘12 Growing networks Godrèche & Luck ‘08 Avalanches Le Doussal & Wiese ‘09 Financial data Wergen, Bogner & Krug ‘11, Wergen ‘13

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Nber of records x i y if is a record i k n if is NOT a record

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Nber of records x i y if is a record i k n if is NOT a record is the proba. that a record is broken at step

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Nber of records x i y if is a record i k n if is NOT a record is the proba. that a record is broken at step

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Nber of records x i y if is a record i k n if is NOT a record is the proba. that a record is broken at step

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Nber of records x i y if is a record i k n if is NOT a record is the proba. that a record is broken at step Universal !

Record statistics of i.i.d. random variables i.i.d. random variables with PDF x i y i k n

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Average nber of records x i y i k n

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Average nber of records x i y Variance i k n

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Average nber of records x i y Variance i k n Universal probability distribution

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Average nber of records x i y Variance i k n Stirling numbers: Universal probability distribution number of permutations of elements with disjoint cycles

Record statistics of i.i.d. random variables i.i.d. random variables with PDF Average nber of records x i y Variance i k n Stirling numbers: Universal probability distribution number of permutations of elements with disjoint cycles Gaussian for large

Record statistics of random walks where the jumps s are i.i.d. with PDF continuous & symmetric

Record statistics of random walks where the jumps s are i.i.d. with PDF continuous & symmetric Including Ordinary random walks Lévy flights is the Lévy index

Record statistics of random walks where the jumps s are i.i.d. with PDF continuous & symmetric Including Ordinary random walks Lévy flights is the Lévy index Q: Dependence of records on the jump distribution ?

Mean record number of random walks x i y is the proba. that a record is broken at step i k n

Mean record number of random walks x i y is the proba. that a record is broken at step i k n x i k y ⇐ ⇒ y k x i

Mean record number of random walks x i k y ⇐ ⇒ y is the proba. that a record is k broken at step x i Proba. that the walker stays negative up to step starting from the origin

Mean record number of random walks x i k y ⇐ ⇒ y is the proba. that a record is k broken at step x i Proba. that the walker stays negative up to step starting from the origin is given by the Sparre Andersen Theorem

Mean record number of random walks is given by the Sparre Andersen Theorem

Mean record number of random walks is given by the Sparre Andersen Theorem For symmetric RW

Mean record number of random walks is given by the Sparre Andersen Theorem For symmetric RW Universal, i.e. independent of the jump distribution !

Mean record number of random walks is given by the Sparre Andersen Theorem For symmetric RW Universal, i.e. independent of the jump distribution ! Majumdar, Ziff `08

Record statistics of random walks with a drift where the jumps s are i.i.d. with PDF continuous & symmetric RW with a drift

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 :

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

Record statistics of random walks with a drift , RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 I c

Record statistics of random walks with a drift , RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 I c

Record statistics of random walks with a drift , RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 I c

Record statistics of random walks with a drift , RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 I c

Record statistics of random walks with a drift , RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 I c

Record statistics of random walks with a drift , RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 I c

Record statistics of random walks with a drift , RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 ? What about the full distribution of I c

Renewal approach to records of RW x i Joint distribution of τ 5 = 6 τ 4 = 5 ? τ 3 = 2 τ 2 = 3 A n = 3 τ 1 = 4 i n

Renewal approach to records of RW x i Joint distribution of τ 5 = 6 τ 4 = 5 ? τ 3 = 2 τ 2 = 3 A n = 3 τ 1 = 4 i n RW is a Markov process are independent except for the global constraint

Renewal approach to records of RW x i Joint distribution of τ 5 = 6 τ 4 = 5 ? τ 3 = 2 τ 2 = 3 A n = 3 τ 1 = 4 i n RW is a Markov process are independent except for the global constraint s are identical RW is translationally invariant while has different statistics

Renewal approach to records of RW x i Joint distribution of τ 5 = 6 τ 4 = 5 ? τ 3 = 2 τ 2 = 3 A n = 3 τ 1 = 4 i n Two main objects: Persistence (or survival) probability indep. of Distribution of first-passage time (from below) indep. of

Renewal approach to records of RW x i Joint distribution of τ 5 = 6 τ 4 = 5 τ 3 = 2 τ 2 = 3 A n = 3 τ 1 = 4 i n first passage proba. survival proba.

Proba. distribution of the number of records with

Proba. distribution of the number of records with Generating function w.r.t. the number of steps

Proba. distribution of the number of records with Generating function w.r.t. the number of steps (for symmetric jumps)

Proba. distribution of the number of records By ``inverting’’ the GF (for symmetric jumps): Majumdar, Ziff `08

Proba. distribution of the number of records By ``inverting’’ the GF (for symmetric jumps): Majumdar, Ziff `08 For :

Proba. distribution of the number of records By ``inverting’’ the GF (for symmetric jumps): Majumdar, Ziff `08 For : RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 I c

Proba. distribution of the number of records By ``inverting’’ the GF (for symmetric jumps): Majumdar, Ziff `08 For : RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III II 1 I c

Proba. distribution of the number of records By ``inverting’’ the GF (for symmetric jumps): Majumdar, Ziff `08 For : RW with a drift Majumdar, G. S., Wergen `12 µ IV 2 V III e.g. for II 1 I c

Statistics of the ages of records sym. RW x i τ 5 = 6 τ 4 = 5 τ 3 = 2 τ 2 = 3 A n = 3 τ 1 = 4 i n