A statistical physics approach to the Sine β process Myl` ene Ma¨ ıda U. Lille, Laboratoire Paul Painlev´ e Joint work with David Dereudre, Adrien Hardy (U. Lille) and Thomas Lebl´ e (Courant Institute-NYU) CIRM, Luminy - April 2019
Outline of the talk 2
Outline of the talk ◮ One-dimensional log-gases and the Sine β process 2
Outline of the talk ◮ One-dimensional log-gases and the Sine β process ◮ Dobrushin-Lanford-Ruelle (DLR) equations for the Sine β process 2
Outline of the talk ◮ One-dimensional log-gases and the Sine β process ◮ Dobrushin-Lanford-Ruelle (DLR) equations for the Sine β process ◮ Applications of the DLR equations 2
Outline of the talk ◮ One-dimensional log-gases and the Sine β process ◮ Dobrushin-Lanford-Ruelle (DLR) equations for the Sine β process ◮ Applications of the DLR equations ◮ Perspectives 2
Log-gases Configuration γ = { x 1 , . . . , x n } of n points in R (or U ) 3
Log-gases Configuration γ = { x 1 , . . . , x n } of n points in R (or U ) The energy of the configuration is n H n ( γ ) := 1 � � − log | x i − x j | + n V ( x i ) , 2 i � = j i =1 with a confining potential V ( x ) . 3
Log-gases Configuration γ = { x 1 , . . . , x n } of n points in R (or U ) The energy of the configuration is n H n ( γ ) := 1 � � − log | x i − x j | + n V ( x i ) , 2 i � = j i =1 with a confining potential V ( x ) . V ,β the Gibbs measure on R n or U n associated to this We denote by P n energy : 1 e − β d P n 2 H n ( x 1 ,..., x n ) d x 1 . . . d x n V ,β ( x 1 , . . . , x n ) = Z n V ,β 3
Log-gases Configuration γ = { x 1 , . . . , x n } of n points in R (or U ) The energy of the configuration is n H n ( γ ) := 1 � � − log | x i − x j | + n V ( x i ) , 2 i � = j i =1 with a confining potential V ( x ) . V ,β the Gibbs measure on R n or U n associated to this We denote by P n energy : 1 e − β d P n 2 H n ( x 1 ,..., x n ) d x 1 . . . d x n V ,β ( x 1 , . . . , x n ) = Z n V ,β On R , if V ( x ) = x 2 / 2 and β > 0 , we recover the G β E (tridiagonal model). 3
Log-gases Configuration γ = { x 1 , . . . , x n } of n points in R (or U ) The energy of the configuration is n H n ( γ ) := 1 � � − log | x i − x j | + n V ( x i ) , 2 i � = j i =1 with a confining potential V ( x ) . V ,β the Gibbs measure on R n or U n associated to this We denote by P n energy : 1 e − β d P n 2 H n ( x 1 ,..., x n ) d x 1 . . . d x n V ,β ( x 1 , . . . , x n ) = Z n V ,β On R , if V ( x ) = x 2 / 2 and β > 0 , we recover the G β E (tridiagonal model). (On U , if V = 0 , we recover the C β E (pentadiagonal model)). 3
Microscopic behavior of the log-gas 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. Then Nakano showed that the two are the same, called Sine β process. 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. Then Nakano showed that the two are the same, called Sine β process. ◮ The proofs based on tridiagonal/pentadiagonal matricial model 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. Then Nakano showed that the two are the same, called Sine β process. ◮ The proofs based on tridiagonal/pentadiagonal matricial model ◮ The description of the process goes through “a coupled family of stochastic differential equations driven by a two-dimensional Brownian motion” (Brownian carousel) : 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. Then Nakano showed that the two are the same, called Sine β process. ◮ The proofs based on tridiagonal/pentadiagonal matricial model ◮ The description of the process goes through “a coupled family of stochastic differential equations driven by a two-dimensional Brownian motion” (Brownian carousel) : 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. Then Nakano showed that the two are the same, called Sine β process. ◮ The proofs based on tridiagonal/pentadiagonal matricial model ◮ The description of the process goes through “a coupled family of stochastic differential equations driven by a two-dimensional Brownian motion” (Brownian carousel) : d α λ ( t ) = λβ 4 d t + ℜ (( e i α λ ( t ) − 1) d Z t ) , α λ (0) = 0 , 4 e − β t The number of points of Sine β in [0 , λ ] is α λ ( ∞ ) / (2 π ) . 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. Then Nakano showed that the two are the same, called Sine β process. ◮ The proofs based on tridiagonal/pentadiagonal matricial model ◮ The description of the process goes through “a coupled family of stochastic differential equations driven by a two-dimensional Brownian motion” (Brownian carousel) : d α λ ( t ) = λβ 4 d t + ℜ (( e i α λ ( t ) − 1) d Z t ) , α λ (0) = 0 , 4 e − β t The number of points of Sine β in [0 , λ ] is α λ ( ∞ ) / (2 π ) . ◮ Some properties obtained via the SDE description by Valk´ o, Vir´ ag, Holcomb, Paquette... 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. Then Nakano showed that the two are the same, called Sine β process. ◮ The proofs based on tridiagonal/pentadiagonal matricial model ◮ The description of the process goes through “a coupled family of stochastic differential equations driven by a two-dimensional Brownian motion” (Brownian carousel) : d α λ ( t ) = λβ 4 d t + ℜ (( e i α λ ( t ) − 1) d Z t ) , α λ (0) = 0 , 4 e − β t The number of points of Sine β in [0 , λ ] is α λ ( ∞ ) / (2 π ) . ◮ Some properties obtained via the SDE description by Valk´ o, Vir´ ag, Holcomb, Paquette... ◮ Valk´ o-Vir´ ag recently showed that the process can also be seen as the spectrum of a random differential operator 4
Microscopic behavior of the log-gas ◮ Valk´ o-Vir´ ag and Killip-Stoiciu independently showed existence of a limit point process for zoomed G β E and C β E respectively. Then Nakano showed that the two are the same, called Sine β process. ◮ The proofs based on tridiagonal/pentadiagonal matricial model ◮ The description of the process goes through “a coupled family of stochastic differential equations driven by a two-dimensional Brownian motion” (Brownian carousel) : d α λ ( t ) = λβ 4 d t + ℜ (( e i α λ ( t ) − 1) d Z t ) , α λ (0) = 0 , 4 e − β t The number of points of Sine β in [0 , λ ] is α λ ( ∞ ) / (2 π ) . ◮ Some properties obtained via the SDE description by Valk´ o, Vir´ ag, Holcomb, Paquette... ◮ Valk´ o-Vir´ ag recently showed that the process can also be seen as the spectrum of a random differential operator ◮ Universality with respect to V obtained (Bourgade-Erd¨ os-Yau-Lin/Bekerman-Figalli-Guionnet) 4
“Physical” description of the Sine β process ? 5
“Physical” description of the Sine β process ? We started with 1 e − β 2 H n ( x 1 ,..., x n ) d x 1 . . . d x n d P n V ,β ( x 1 , . . . , x n ) = Z n V ,β We look at the rescaled configuration γ n := � n i =1 δ nx i . 5
“Physical” description of the Sine β process ? We started with 1 e − β 2 H n ( x 1 ,..., x n ) d x 1 . . . d x n d P n V ,β ( x 1 , . . . , x n ) = Z n V ,β We look at the rescaled configuration γ n := � n i =1 δ nx i . As n goes to infinity, we may expect 5
“Physical” description of the Sine β process ? We started with 1 e − β 2 H n ( x 1 ,..., x n ) d x 1 . . . d x n d P n V ,β ( x 1 , . . . , x n ) = Z n V ,β We look at the rescaled configuration γ n := � n i =1 δ nx i . As n goes to infinity, we may expect ◮ γ n → C infinite configuration 5
“Physical” description of the Sine β process ? We started with 1 e − β 2 H n ( x 1 ,..., x n ) d x 1 . . . d x n d P n V ,β ( x 1 , . . . , x n ) = Z n V ,β We look at the rescaled configuration γ n := � n i =1 δ nx i . As n goes to infinity, we may expect ◮ γ n → C infinite configuration ◮ H n ( γ n ) converges to some function H ( C ) 5
“Physical” description of the Sine β process ? We started with 1 e − β 2 H n ( x 1 ,..., x n ) d x 1 . . . d x n d P n V ,β ( x 1 , . . . , x n ) = Z n V ,β We look at the rescaled configuration γ n := � n i =1 δ nx i . As n goes to infinity, we may expect ◮ γ n → C infinite configuration ◮ H n ( γ n ) converges to some function H ( C ) ◮ the limiting process may satisfy dSine β ( C ) = 1 Z exp( − β H ( C )) d Π( C ) , with Π the Poisson process. 5
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