Free propagator via path integral � x ( t )= x � t � − 1 � x 2 ( τ ) d τ • G ( x , t | x 0 , 0) = D x ( τ ) exp ˙ 4 D x (0)= x 0 0 1 • identifying the action of a free quantum particle with mass m = 2 D • G ( x , t | x 0 , 0) = � x | e − ˆ p 2 H t | x 0 � where ˆ H ≡ ˆ 2 m ≡ − D ∂ 2 x • In the eigenbasis of ˆ H ψ E ( x ) = E ψ E ( x ) (Schr¨ odinger equation) • � x | e − ˆ � H t | x 0 � = ψ E ( x ) ψ ∗ E ( x 0 ) e − Et E S.N. Majumdar The Wonderful World of Brownian Functionals
Free propagator via path integral � x ( t )= x � t � − 1 � x 2 ( τ ) d τ • G ( x , t | x 0 , 0) = D x ( τ ) exp ˙ 4 D x (0)= x 0 0 1 • identifying the action of a free quantum particle with mass m = 2 D • G ( x , t | x 0 , 0) = � x | e − ˆ p 2 H t | x 0 � where ˆ H ≡ ˆ 2 m ≡ − D ∂ 2 x • In the eigenbasis of ˆ H ψ E ( x ) = E ψ E ( x ) (Schr¨ odinger equation) • � x | e − ˆ � H t | x 0 � = ψ E ( x ) ψ ∗ E ( x 0 ) e − Et E • For free particle, (Schr¨ odinger equation) reads: 2 π e ikx with E = Dk 2 − D ∂ 2 1 x ψ E ( x ) = E ψ E ( x ) → ψ k ( x ) = √ S.N. Majumdar The Wonderful World of Brownian Functionals
Free propagator via path integral � x ( t )= x � t � − 1 � x 2 ( τ ) d τ • G ( x , t | x 0 , 0) = D x ( τ ) exp ˙ 4 D x (0)= x 0 0 1 • identifying the action of a free quantum particle with mass m = 2 D • G ( x , t | x 0 , 0) = � x | e − ˆ p 2 H t | x 0 � where ˆ H ≡ ˆ 2 m ≡ − D ∂ 2 x • In the eigenbasis of ˆ H ψ E ( x ) = E ψ E ( x ) (Schr¨ odinger equation) • � x | e − ˆ � H t | x 0 � = ψ E ( x ) ψ ∗ E ( x 0 ) e − Et E • For free particle, (Schr¨ odinger equation) reads: 2 π e ikx with E = Dk 2 − D ∂ 2 1 x ψ E ( x ) = E ψ E ( x ) → ψ k ( x ) = √ � dk • Free propagator: G ( x , t | x 0 , 0) = � x | e − ˆ 2 π e ik ( x − x 0 ) e − Dk 2 t H t | x 0 � = S.N. Majumdar The Wonderful World of Brownian Functionals
Free propagator via path integral � x ( t )= x � t � − 1 � x 2 ( τ ) d τ • G ( x , t | x 0 , 0) = D x ( τ ) exp ˙ 4 D x (0)= x 0 0 1 • identifying the action of a free quantum particle with mass m = 2 D • G ( x , t | x 0 , 0) = � x | e − ˆ p 2 H t | x 0 � where ˆ H ≡ ˆ 2 m ≡ − D ∂ 2 x • In the eigenbasis of ˆ H ψ E ( x ) = E ψ E ( x ) (Schr¨ odinger equation) • � x | e − ˆ � H t | x 0 � = ψ E ( x ) ψ ∗ E ( x 0 ) e − Et E • For free particle, (Schr¨ odinger equation) reads: 2 π e ikx with E = Dk 2 − D ∂ 2 1 x ψ E ( x ) = E ψ E ( x ) → ψ k ( x ) = √ � dk • Free propagator: G ( x , t | x 0 , 0) = � x | e − ˆ 2 π e ik ( x − x 0 ) e − Dk 2 t H t | x 0 � = 1 − 1 4 Dt ( x − x 0 ) 2 � � = 4 π Dt exp √ S.N. Majumdar The Wonderful World of Brownian Functionals
Brownian functional • Brownian path x ( τ ) propagating from x 0 at τ = 0 to x ( t ) = x at τ = t S.N. Majumdar The Wonderful World of Brownian Functionals
Brownian functional • Brownian path x ( τ ) propagating from x 0 at τ = 0 to x ( t ) = x at τ = t � t • Brownian functional: T = U ( x ( τ )) d τ ) where U ( x ) → arbitrary 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Brownian functional • Brownian path x ( τ ) propagating from x 0 at τ = 0 to x ( t ) = x at τ = t � t • Brownian functional: T = U ( x ( τ )) d τ ) where U ( x ) → arbitrary 0 • Clearly T → random variable–varies from path to path S.N. Majumdar The Wonderful World of Brownian Functionals
Brownian functional • Brownian path x ( τ ) propagating from x 0 at τ = 0 to x ( t ) = x at τ = t � t • Brownian functional: T = U ( x ( τ )) d τ ) where U ( x ) → arbitrary 0 • Clearly T → random variable–varies from path to path Q: What is its probability distribution P ( T | x , t , x 0 ) ? S.N. Majumdar The Wonderful World of Brownian Functionals
Brownian functional • Brownian path x ( τ ) propagating from x 0 at τ = 0 to x ( t ) = x at τ = t � t • Brownian functional: T = U ( x ( τ )) d τ ) where U ( x ) → arbitrary 0 • Clearly T → random variable–varies from path to path Q: What is its probability distribution P ( T | x , t , x 0 ) ? � t • Example: U ( x ) = x → T = 0 x ( τ ) d τ → area under the path x (τ) T=Area under the curve x x 0 t 0 τ S.N. Majumdar The Wonderful World of Brownian Functionals
Brownian functional: Feynman-Kac formula S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 � ∞ P ( T | x , t , x 0 ) e − sT dT • Taking Laplace transform: ˜ P ( x , t | x 0 , s ) = 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 � ∞ P ( T | x , t , x 0 ) e − sT dT • Taking Laplace transform: ˜ P ( x , t | x 0 , s ) = 0 � x ( t )= x � t � t D x ( τ ) e − 1 x 2 ( τ ) d τ − s • ˜ 0 ˙ 0 U ( x ( τ )) d τ P ( x , t | x 0 , s ) ∝ 4 D x (0)= x 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 � ∞ P ( T | x , t , x 0 ) e − sT dT • Taking Laplace transform: ˜ P ( x , t | x 0 , s ) = 0 � x ( t )= x � t � t D x ( τ ) e − 1 x 2 ( τ ) d τ − s 0 U ( x ( τ )) d τ = � x | e − ˆ • ˜ 0 ˙ Ht | x 0 � P ( x , t | x 0 , s ) ∝ 4 D x (0)= x 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 � ∞ P ( T | x , t , x 0 ) e − sT dT • Taking Laplace transform: ˜ P ( x , t | x 0 , s ) = 0 � x ( t )= x � t � t D x ( τ ) e − 1 x 2 ( τ ) d τ − s 0 U ( x ( τ )) d τ = � x | e − ˆ • ˜ 0 ˙ Ht | x 0 � P ( x , t | x 0 , s ) ∝ 4 D x (0)= x 0 p 2 where ˆ H ≡ ˆ 2 m + sU ( x ) ≡ − D ∂ 2 x + sU ( x ) S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 � ∞ P ( T | x , t , x 0 ) e − sT dT • Taking Laplace transform: ˜ P ( x , t | x 0 , s ) = 0 � x ( t )= x � t � t D x ( τ ) e − 1 x 2 ( τ ) d τ − s 0 U ( x ( τ )) d τ = � x | e − ˆ • ˜ 0 ˙ Ht | x 0 � P ( x , t | x 0 , s ) ∝ 4 D x (0)= x 0 p 2 where ˆ H ≡ ˆ 2 m + sU ( x ) ≡ − D ∂ 2 x + sU ( x ) Functional sU ( x ) → quantum potential in Shr¨ odinger equation S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 � ∞ P ( T | x , t , x 0 ) e − sT dT • Taking Laplace transform: ˜ P ( x , t | x 0 , s ) = 0 � x ( t )= x � t � t D x ( τ ) e − 1 x 2 ( τ ) d τ − s 0 U ( x ( τ )) d τ = � x | e − ˆ • ˜ 0 ˙ Ht | x 0 � P ( x , t | x 0 , s ) ∝ 4 D x (0)= x 0 p 2 where ˆ H ≡ ˆ 2 m + sU ( x ) ≡ − D ∂ 2 x + sU ( x ) Functional sU ( x ) → quantum potential in Shr¨ odinger equation odinger equation): [ − D ∂ 2 1 . Solve (Schr¨ x + sU ( x )] ψ E ( x ) = E ψ E ( x ) S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 � ∞ P ( T | x , t , x 0 ) e − sT dT • Taking Laplace transform: ˜ P ( x , t | x 0 , s ) = 0 � x ( t )= x � t � t D x ( τ ) e − 1 x 2 ( τ ) d τ − s 0 U ( x ( τ )) d τ = � x | e − ˆ • ˜ 0 ˙ Ht | x 0 � P ( x , t | x 0 , s ) ∝ 4 D x (0)= x 0 p 2 where ˆ H ≡ ˆ 2 m + sU ( x ) ≡ − D ∂ 2 x + sU ( x ) Functional sU ( x ) → quantum potential in Shr¨ odinger equation odinger equation): [ − D ∂ 2 1 . Solve (Schr¨ x + sU ( x )] ψ E ( x ) = E ψ E ( x ) P ( x , t | x 0 , s ) = � x | e − ˆ ˜ � H t | x 0 � = ψ E ( x ) ψ ∗ E ( x 0 ) e − Et 2 . E S.N. Majumdar The Wonderful World of Brownian Functionals
Feynman-Kac formula: main idea � t • Prob. distribution of T = 0 U ( x ( τ )) d τ is: � x ( t )= x � t � � � t x 2 ( τ ) d τ δ D x ( τ ) e − 1 0 ˙ P ( T | x , t , x 0 ) ∝ T − U ( x ( τ )) d τ 4 D x (0)= x 0 0 � ∞ P ( T | x , t , x 0 ) e − sT dT • Taking Laplace transform: ˜ P ( x , t | x 0 , s ) = 0 � x ( t )= x � t � t D x ( τ ) e − 1 x 2 ( τ ) d τ − s 0 U ( x ( τ )) d τ = � x | e − ˆ • ˜ 0 ˙ Ht | x 0 � P ( x , t | x 0 , s ) ∝ 4 D x (0)= x 0 p 2 where ˆ H ≡ ˆ 2 m + sU ( x ) ≡ − D ∂ 2 x + sU ( x ) Functional sU ( x ) → quantum potential in Shr¨ odinger equation odinger equation): [ − D ∂ 2 1 . Solve (Schr¨ x + sU ( x )] ψ E ( x ) = E ψ E ( x ) P ( x , t | x 0 , s ) = � x | e − ˆ ˜ � H t | x 0 � = ψ E ( x ) ψ ∗ E ( x 0 ) e − Et 2 . E 3 . Finally invert the Laplace transform ˜ P ( x , t | x 0 , s ) w.r.t s S.N. Majumdar The Wonderful World of Brownian Functionals
Examples of Brownian functionals � t • Residence (Occupation) time: T = θ ( x ( τ )) d τ → U ( x ) = θ ( x ) 0 P ( T , t | x 0 = 0) = 1 1 ( L´ evy, ’39, Lamperti, ’57 ) π � T ( t − T ) important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ... (Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....) S.N. Majumdar The Wonderful World of Brownian Functionals
Examples of Brownian functionals � t • Residence (Occupation) time: T = θ ( x ( τ )) d τ → U ( x ) = θ ( x ) 0 P ( T , t | x 0 = 0) = 1 1 ( L´ evy, ’39, Lamperti, ’57 ) π � T ( t − T ) important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ... (Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....) • Finance: stock price S ( τ ) = e − β x ( τ ) where x ( τ ) → Brownian motion ( Black-Scholes model, 1973 ) S.N. Majumdar The Wonderful World of Brownian Functionals
Examples of Brownian functionals � t • Residence (Occupation) time: T = θ ( x ( τ )) d τ → U ( x ) = θ ( x ) 0 P ( T , t | x 0 = 0) = 1 1 ( L´ evy, ’39, Lamperti, ’57 ) π � T ( t − T ) important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ... (Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....) • Finance: stock price S ( τ ) = e − β x ( τ ) where x ( τ ) → Brownian motion ( Black-Scholes model, 1973 ) � t 0 e − β x ( τ ) d τ → U ( x ) = e − β x ( M. Yor, ... ) Integrated stock price: T = S.N. Majumdar The Wonderful World of Brownian Functionals
Examples of Brownian functionals � t • Residence (Occupation) time: T = θ ( x ( τ )) d τ → U ( x ) = θ ( x ) 0 P ( T , t | x 0 = 0) = 1 1 ( L´ evy, ’39, Lamperti, ’57 ) π � T ( t − T ) important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ... (Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....) • Finance: stock price S ( τ ) = e − β x ( τ ) where x ( τ ) → Brownian motion ( Black-Scholes model, 1973 ) � t 0 e − β x ( τ ) d τ → U ( x ) = e − β x ( M. Yor, ... ) Integrated stock price: T = T → partition function of the Sinai model in a box [0 , t ] S.N. Majumdar The Wonderful World of Brownian Functionals
Examples of Brownian functionals � t • Residence (Occupation) time: T = θ ( x ( τ )) d τ → U ( x ) = θ ( x ) 0 P ( T , t | x 0 = 0) = 1 1 ( L´ evy, ’39, Lamperti, ’57 ) π � T ( t − T ) important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ... (Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....) • Finance: stock price S ( τ ) = e − β x ( τ ) where x ( τ ) → Brownian motion ( Black-Scholes model, 1973 ) � t 0 e − β x ( τ ) d τ → U ( x ) = e − β x ( M. Yor, ... ) Integrated stock price: T = T → partition function of the Sinai model in a box [0 , t ] • Other examples: “Brownian functionals in physics and computer science” [review by S.M. in Current Science, (2005), cond-mat/0510064 ] S.N. Majumdar The Wonderful World of Brownian Functionals
Fluctuating (1 + 1) -dimensional interfaces • Fluctuating steps on crystals such as Si, Al(111),.. • diploar chains of non-magnetic particles in a ferrofluid ... S.N. Majumdar The Wonderful World of Brownian Functionals
Fluctuating (1 + 1) -dimensional interface Fluctuating Interface height H(x,t) x L 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Fluctuating (1 + 1) -dimensional interface Fluctuating Interface height H(x,t) x L 0 • Fluctuating interfaces → very well studied x H + λ ( ∂ x H ) 2 + η ( x , t ) ( Kardar, Parisi, Zhang, 1986 ) ∂ t H = ∂ 2 S.N. Majumdar The Wonderful World of Brownian Functionals
Fluctuating (1 + 1) -dimensional interface Fluctuating Interface height H(x,t) x L 0 • Fluctuating interfaces → very well studied x H + λ ( ∂ x H ) 2 + η ( x , t ) ( Kardar, Parisi, Zhang, 1986 ) ∂ t H = ∂ 2 • λ = 0 → Linear interface ( Hammersley, ’66, Edwards-Wilkinson, ’81 ) S.N. Majumdar The Wonderful World of Brownian Functionals
• Relevant object that measures fluctuations � L h ( x , t ) = H ( x , t ) − 1 H ( x , t ) dx → relative height L 0 S.N. Majumdar The Wonderful World of Brownian Functionals
• Relevant object that measures fluctuations � L h ( x , t ) = H ( x , t ) − 1 H ( x , t ) dx → relative height L 0 � L By definition: h ( x , t ) dx = 0 0 S.N. Majumdar The Wonderful World of Brownian Functionals
• Relevant object that measures fluctuations � L h ( x , t ) = H ( x , t ) − 1 H ( x , t ) dx → relative height L 0 � L By definition: h ( x , t ) dx = 0 0 • Width of the interface: � h 2 ( x , t ) � ∼ t β for ξ ( t ) ∼ t 1 / z << L � w ( t , L ) = (growing regime) ∼ L 1 / 2 for ξ ( t ) ∼ t 1 / z >> L (steady state) S.N. Majumdar The Wonderful World of Brownian Functionals
• Relevant object that measures fluctuations � L h ( x , t ) = H ( x , t ) − 1 H ( x , t ) dx → relative height L 0 � L By definition: h ( x , t ) dx = 0 0 • Width of the interface: � h 2 ( x , t ) � ∼ t β for ξ ( t ) ∼ t 1 / z << L � w ( t , L ) = (growing regime) ∼ L 1 / 2 for ξ ( t ) ∼ t 1 / z >> L (steady state) • β = 1 / 3 and z = 3 / 2 (for KPZ, λ > 0) • β = 1 / 4 and z = 2 (for EW, λ = 0) S.N. Majumdar The Wonderful World of Brownian Functionals
• Relevant object that measures fluctuations � L h ( x , t ) = H ( x , t ) − 1 H ( x , t ) dx → relative height L 0 � L By definition: h ( x , t ) dx = 0 0 • Width of the interface: � h 2 ( x , t ) � ∼ t β for ξ ( t ) ∼ t 1 / z << L � w ( t , L ) = (growing regime) ∼ L 1 / 2 for ξ ( t ) ∼ t 1 / z >> L (steady state) • β = 1 / 3 and z = 3 / 2 (for KPZ, λ > 0) • β = 1 / 4 and z = 2 (for EW, λ = 0) • Full distribution of w 2 ( t , L ) = h 2 ( x , t ) → in the steady state → Exact solution ( G. Foltin, K. Oerding, Z. Racz, R.L. Workman, R.K.P. Zia, 1994) S.N. Majumdar The Wonderful World of Brownian Functionals
Extreme Relative Height Recent interest in Extreme height fluctuations (Rare Events) h m = max 0 ≤ x ≤ L { h ( x , t ) } → maximal relative height (MRH) S.N. Majumdar The Wonderful World of Brownian Functionals
Extreme Relative Height Recent interest in Extreme height fluctuations (Rare Events) h m = max 0 ≤ x ≤ L { h ( x , t ) } → maximal relative height (MRH) In particular, in the steady state regime t >> L z where the heights at different points are strongly correlated ( ξ ( t ) >> L ) Q: Prob. distr. of h m : P ( h m , L ) =? S.N. Majumdar The Wonderful World of Brownian Functionals
S.N. Majumdar The Wonderful World of Brownian Functionals
Exact solution via Path Integral method M h(x) h m L 0 x • h m = max 0 ≤ x ≤ L { h ( x , t ) } → maximal relative height • Cumulative distribution of h m in the steady state ( t 1 / z >> L ) F ( M , L ) = Prob [ h m ≤ M , L ] = ⇒ Prob . that the path stays below the level M over x ∈ [0 , L ] S.N. Majumdar The Wonderful World of Brownian Functionals
Path Integral with Constraint • Joint distribution of relative heights (steady state) for (1 + 1)-dim. KPZ interfaces: � L � � 2 � − 1 � ∂ h Prob . [ { h ( x } ] ∝ exp dx 2 ∂ x 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Path Integral with Constraint • Joint distribution of relative heights (steady state) for (1 + 1)-dim. KPZ interfaces: � L �� L � � 2 � � − 1 � ∂ h Prob . [ { h ( x } ] ∝ exp × δ h ( x ) dx dx 2 ∂ x 0 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Path Integral with Constraint • Joint distribution of relative heights (steady state) for (1 + 1)-dim. KPZ interfaces: � L �� L � � 2 � � − 1 � ∂ h Prob . [ { h ( x } ] ∝ exp × δ h ( x ) dx dx 2 ∂ x 0 0 • Cumulative distribution F ( M , L ) ≡ Prob . that the path stays below M S.N. Majumdar The Wonderful World of Brownian Functionals
Path Integral with Constraint • Joint distribution of relative heights (steady state) for (1 + 1)-dim. KPZ interfaces: � L �� L � � 2 � � − 1 � ∂ h Prob . [ { h ( x } ] ∝ exp × δ h ( x ) dx dx 2 ∂ x 0 0 • Cumulative distribution F ( M , L ) ≡ Prob . that the path stays below M � M � h ( L )= h 0 �� L � � L 0 ( ∂ x h ) 2 dx × δ D h ( x ) e − 1 F ( M , L ) ∝ h ( x ) dx dh 0 2 −∞ h (0)= h 0 0 � × “ θ ( M − h ( x )) ” 0 ≤ x ≤ L S.N. Majumdar The Wonderful World of Brownian Functionals
Path Integral with Constraint • Joint distribution of relative heights (steady state) for (1 + 1)-dim. KPZ interfaces: � L �� L � � 2 � � − 1 � ∂ h Prob . [ { h ( x } ] ∝ exp × δ h ( x ) dx dx 2 ∂ x 0 0 • Cumulative distribution F ( M , L ) ≡ Prob . that the path stays below M � M � h ( L )= h 0 �� L � � L 0 ( ∂ x h ) 2 dx × δ D h ( x ) e − 1 F ( M , L ) ∝ h ( x ) dx dh 0 2 −∞ h (0)= h 0 0 � × “ θ ( M − h ( x )) ” 0 ≤ x ≤ L • Identify x ≡ τ and change variable M − h ( x ) ≡ x ( τ ): � ∞ � x ( L )= x 0 �� L � � L 0 ( ∂ τ x ) 2 d τ × δ D x ( τ ) e − 1 F ( M , L ) ∝ x ( τ ) d τ − ML dx 0 2 0 x (0)= x 0 0 � × “ θ ( x ( τ )) ” 0 ≤ τ ≤ L S.N. Majumdar The Wonderful World of Brownian Functionals
Path Integral with Constraint • Joint distribution of relative heights (steady state) for (1 + 1)-dim. KPZ interfaces: � L �� L � � 2 � � − 1 � ∂ h Prob . [ { h ( x } ] ∝ exp × δ h ( x ) dx dx 2 ∂ x 0 0 • Cumulative distribution F ( M , L ) ≡ Prob . that the path stays below M � M � h ( L )= h 0 �� L � � L 0 ( ∂ x h ) 2 dx × δ D h ( x ) e − 1 F ( M , L ) ∝ h ( x ) dx dh 0 2 −∞ h (0)= h 0 0 � × “ θ ( M − h ( x )) ” 0 ≤ x ≤ L • Identify x ≡ τ and change variable M − h ( x ) ≡ x ( τ ): � ∞ � x ( L )= x 0 �� L � � L 0 ( ∂ τ x ) 2 d τ × δ D x ( τ ) e − 1 F ( M , L ) ∝ x ( τ ) d τ − ML dx 0 2 0 x (0)= x 0 0 � × “ θ ( x ( τ )) ” 0 ≤ τ ≤ L • Now apply Feynman-Kac technique S.N. Majumdar The Wonderful World of Brownian Functionals
Path Integral with Constraint • Joint distribution of relative heights (steady state) for (1 + 1)-dim. KPZ interfaces: � L �� L � � 2 � � − 1 � ∂ h Prob . [ { h ( x } ] ∝ exp × δ h ( x ) dx dx 2 ∂ x 0 0 • Cumulative distribution F ( M , L ) ≡ Prob . that the path stays below M � M � h ( L )= h 0 �� L � � L 0 ( ∂ x h ) 2 dx × δ D h ( x ) e − 1 F ( M , L ) ∝ h ( x ) dx dh 0 2 −∞ h (0)= h 0 0 � × “ θ ( M − h ( x )) ” 0 ≤ x ≤ L • Identify x ≡ τ and change variable M − h ( x ) ≡ x ( τ ): � ∞ � x ( L )= x 0 �� L � � L 0 ( ∂ τ x ) 2 d τ × δ D x ( τ ) e − 1 F ( M , L ) ∝ x ( τ ) d τ − ML dx 0 2 0 x (0)= x 0 0 � × “ θ ( x ( τ )) ” 0 ≤ τ ≤ L • Now apply Feynman-Kac technique → take Laplace transform S.N. Majumdar The Wonderful World of Brownian Functionals
Exact Results � ∞ � ∞ F ( M , L ) e − sM L dM ∝ � H L � dx 0 � x 0 | e − ˆ e − ˆ H L | x 0 � = Tr • 0 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Exact Results � ∞ � ∞ F ( M , L ) e − sM L dM ∝ � H L � dx 0 � x 0 | e − ˆ e − ˆ H L | x 0 � = Tr • 0 0 with ˆ H ≡ − 1 2 ∂ 2 x + V ( x ) where the quantum potential V ( x ) = s x for x > 0 = ∞ for x < 0 S.N. Majumdar The Wonderful World of Brownian Functionals
Exact Results � ∞ � ∞ F ( M , L ) e − sM L dM ∝ � H L � dx 0 � x 0 | e − ˆ e − ˆ H L | x 0 � = Tr • 0 0 with ˆ H ≡ − 1 2 ∂ 2 x + V ( x ) where the quantum potential V ( x ) = s x for x > 0 = ∞ for x < 0 Schr¨ odinger eq. in a traingular potential → exactly solvable S.N. Majumdar The Wonderful World of Brownian Functionals
Exact Results � ∞ � ∞ F ( M , L ) e − sM L dM ∝ � H L � dx 0 � x 0 | e − ˆ e − ˆ H L | x 0 � = Tr • 0 0 with ˆ H ≡ − 1 2 ∂ 2 x + V ( x ) where the quantum potential V ( x ) = s x for x > 0 = ∞ for x < 0 Schr¨ odinger eq. in a traingular potential → exactly solvable • Prob. density of h m : P ( h m , L ) = ∂ M F ( M , L ) | M = h m S.N. Majumdar The Wonderful World of Brownian Functionals
Exact Results � ∞ � ∞ F ( M , L ) e − sM L dM ∝ � H L � dx 0 � x 0 | e − ˆ e − ˆ H L | x 0 � = Tr • 0 0 with ˆ H ≡ − 1 2 ∂ 2 x + V ( x ) where the quantum potential V ( x ) = s x for x > 0 = ∞ for x < 0 Schr¨ odinger eq. in a traingular potential → exactly solvable • Prob. density of h m : P ( h m , L ) = ∂ M F ( M , L ) | M = h m � � 1 h m = ⇒ P ( h m , L ) = L f where √ √ L S.N. Majumdar The Wonderful World of Brownian Functionals
Exact Results � ∞ � ∞ F ( M , L ) e − sM L dM ∝ � H L � dx 0 � x 0 | e − ˆ e − ˆ H L | x 0 � = Tr • 0 0 with ˆ H ≡ − 1 2 ∂ 2 x + V ( x ) where the quantum potential V ( x ) = s x for x > 0 = ∞ for x < 0 Schr¨ odinger eq. in a traingular potential → exactly solvable • Prob. density of h m : P ( h m , L ) = ∂ M F ( M , L ) | M = h m � � 1 h m = ⇒ P ( h m , L ) = L f where √ √ L � ∞ ∞ √ e − α k 2 − 1 / 3 s 2 / 3 f ( x ) e − s x dx = s � 2 π 0 k =1 where α k ’s → zeroes of Airy function Ai(z) [S.M. and A. Comtet, PRL, 92 , 225501 (2004); J. Stat. Phys. 119, 777 (2005)] S.N. Majumdar The Wonderful World of Brownian Functionals
Exact Results � ∞ � ∞ F ( M , L ) e − sM L dM ∝ � H L � dx 0 � x 0 | e − ˆ e − ˆ H L | x 0 � = Tr • 0 0 with ˆ H ≡ − 1 2 ∂ 2 x + V ( x ) where the quantum potential V ( x ) = s x for x > 0 = ∞ for x < 0 Schr¨ odinger eq. in a traingular potential → exactly solvable • Prob. density of h m : P ( h m , L ) = ∂ M F ( M , L ) | M = h m � � 1 h m = ⇒ P ( h m , L ) = L f where √ √ L � ∞ ∞ √ e − α k 2 − 1 / 3 s 2 / 3 f ( x ) e − s x dx = s � 2 π 0 k =1 where α k ’s → zeroes of Airy function Ai(z) [S.M. and A. Comtet, PRL, 92 , 225501 (2004); J. Stat. Phys. 119, 777 (2005)] • The Laplace transform can be inverted → exact asymptotics S.N. Majumdar The Wonderful World of Brownian Functionals
The scaling function f ( x ) √ ∞ f ( x ) = 2 6 e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 b 2 / 3 � k / 27 k x 10 / 3 k =1 S.N. Majumdar The Wonderful World of Brownian Functionals
The scaling function f ( x ) √ ∞ f ( x ) = 2 6 e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 b 2 / 3 � k / 27 k x 10 / 3 k =1 U ( a , b , z ) → Kummer’s function and α k → zeroes of Airy function Ai ( z ) S.N. Majumdar The Wonderful World of Brownian Functionals
The scaling function f ( x ) √ ∞ f ( x ) = 2 6 e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 b 2 / 3 � k / 27 k x 10 / 3 k =1 U ( a , b , z ) → Kummer’s function and α k → zeroes of Airy function Ai ( z ) � − 2 α 3 � 81 α 9 / 2 x − 5 exp 8 Asymptotics: f ( x ) ≈ as x → 0 ( α 1 = 2 . 3381 .. ) 1 1 27 x 2 √ √ π x 2 e − 6 x 2 ≈ 72 6 as x → ∞ S.N. Majumdar The Wonderful World of Brownian Functionals
The scaling function f ( x ) √ ∞ f ( x ) = 2 6 e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 b 2 / 3 � k / 27 k x 10 / 3 k =1 U ( a , b , z ) → Kummer’s function and α k → zeroes of Airy function Ai ( z ) � − 2 α 3 � 81 α 9 / 2 x − 5 exp 8 Asymptotics: f ( x ) ≈ as x → 0 ( α 1 = 2 . 3381 .. ) 1 1 27 x 2 √ √ π x 2 e − 6 x 2 ≈ 72 6 as x → ∞ S.N. Majumdar The Wonderful World of Brownian Functionals
Airy distribution function √ ∞ e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 f ( x ) = 2 6 b 2 / 3 � k / 27 k x 10 / 3 k =1 S.N. Majumdar The Wonderful World of Brownian Functionals
Airy distribution function √ ∞ e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 f ( x ) = 2 6 b 2 / 3 � k / 27 k x 10 / 3 k =1 − 2 α 3 x − 5 exp � � 81 α 9 / 2 8 Asymptotics: f ( x ) ≈ 1 as x → 0 ( α 1 = 2 . 3381 .. ) 1 27 x 2 √ √ π x 2 e − 6 x 2 ≈ 72 6 as x → ∞ S.N. Majumdar The Wonderful World of Brownian Functionals
Airy distribution function √ ∞ e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 f ( x ) = 2 6 b 2 / 3 � k / 27 k x 10 / 3 k =1 − 2 α 3 x − 5 exp � � 81 α 9 / 2 8 Asymptotics: f ( x ) ≈ 1 as x → 0 ( α 1 = 2 . 3381 .. ) 1 27 x 2 √ √ π x 2 e − 6 x 2 ≈ 72 6 as x → ∞ • A number of discrete SOS models → same f ( x ) ( G. Schehr & S.M, 2006 ) S.N. Majumdar The Wonderful World of Brownian Functionals
Airy distribution function √ ∞ e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 f ( x ) = 2 6 b 2 / 3 � k / 27 k x 10 / 3 k =1 − 2 α 3 x − 5 exp � � 81 α 9 / 2 8 Asymptotics: f ( x ) ≈ 1 as x → 0 ( α 1 = 2 . 3381 .. ) 1 27 x 2 √ √ π x 2 e − 6 x 2 ≈ 72 6 as x → ∞ • A number of discrete SOS models → same f ( x ) ( G. Schehr & S.M, 2006 ) • Amazingly, the same f ( x ) appears in a number of problems in graph theory and computer science: • total path length in rooted planar trees ( Takacs, ’94 ) • connected components in a random graph ( Flajolet, Knuth, Pittel, ’89 ) . . . f ( x ) → Airy distribution function ( Janson & Louchard, 2007 ) S.N. Majumdar The Wonderful World of Brownian Functionals
Airy distribution function √ ∞ e − b k / x 2 U ( − 5 / 6 , 4 / 3 , b k / x 2 ) , b k = 2 α 3 f ( x ) = 2 6 b 2 / 3 � k / 27 k x 10 / 3 k =1 − 2 α 3 x − 5 exp � � 81 α 9 / 2 8 Asymptotics: f ( x ) ≈ 1 as x → 0 ( α 1 = 2 . 3381 .. ) 1 27 x 2 √ √ π x 2 e − 6 x 2 ≈ 72 6 as x → ∞ • A number of discrete SOS models → same f ( x ) ( G. Schehr & S.M, 2006 ) • Amazingly, the same f ( x ) appears in a number of problems in graph theory and computer science: • total path length in rooted planar trees ( Takacs, ’94 ) • connected components in a random graph ( Flajolet, Knuth, Pittel, ’89 ) . . . f ( x ) → Airy distribution function ( Janson & Louchard, 2007 ) • Other measures of extreme fluctuations → studied recently ( Burkhardt, Gy¨ cz, 2007, Rambeau & Schehr, 2009 ) orgyi, Moloney, Ra´ S.N. Majumdar The Wonderful World of Brownian Functionals
First-passage Brownian Functional Brownian motion till its first−passage through 0 two different realizations x (τ) x 0 t f t f 0 τ dx • Brownian motion: d τ = η ( τ ) where η ( τ ) → Gaussian white noise S.N. Majumdar The Wonderful World of Brownian Functionals
First-passage Brownian Functional Brownian motion till its first−passage through 0 two different realizations x (τ) x 0 t f t f 0 τ dx • Brownian motion: d τ = η ( τ ) where η ( τ ) → Gaussian white noise • The process, starting at fixed x 0 , stops when the particle hits x = 0 t f → first-passage time → random variable S.N. Majumdar The Wonderful World of Brownian Functionals
First-passage Brownian Functional Brownian motion till its first−passage through 0 two different realizations x (τ) x 0 t f t f 0 τ dx • Brownian motion: d τ = η ( τ ) where η ( τ ) → Gaussian white noise • The process, starting at fixed x 0 , stops when the particle hits x = 0 t f → first-passage time → random variable � t f • first-passage functional: T = U ( x ( τ )) d τ ; Question : P ( T | x 0 ) ? 0 S.N. Majumdar The Wonderful World of Brownian Functionals
First-passage Brownian Functional Brownian motion till its first−passage through 0 two different realizations x (τ) x 0 t f t f 0 τ dx • Brownian motion: d τ = η ( τ ) where η ( τ ) → Gaussian white noise • The process, starting at fixed x 0 , stops when the particle hits x = 0 t f → first-passage time → random variable � t f • first-passage functional: T = U ( x ( τ )) d τ ; Question : P ( T | x 0 ) ? 0 • Ex: (i) If U ( x ) = 1, T = t f � t f (ii) If U ( x ) = x , T = 0 x ( τ ) d τ → area till first-passage time S.N. Majumdar The Wonderful World of Brownian Functionals
Queueing theory: busy period l n = l n−1 + η n �� �� �� �� �� �� � � � � � � � � � � � � queue �� �� �� �� �� �� length ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� � � l n 6 � � 4 5 5 6 3 4 4 4 5 6 2 3 3 3 4 5 6 1 2 2 2 3 4 5 6 busy period time n • Queue length: l n = l n − 1 + η n η n = 1 if a new customer arrives = − 1 when a customer gets served S.N. Majumdar The Wonderful World of Brownian Functionals
Queueing theory: busy period l n = l n−1 + η n �� �� �� �� �� �� � � � � � � � � � � � � queue �� �� �� �� �� �� length ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� � � l n 6 � � 4 5 5 6 3 4 4 4 5 6 2 3 3 3 4 5 6 1 2 2 2 3 4 5 6 busy period time n • Queue length: l n = l n − 1 + η n η n = 1 if a new customer arrives = − 1 when a customer gets served • T ≡ total waiting time of all customers during the busy period ≡ area under the curve S.N. Majumdar The Wonderful World of Brownian Functionals
Queueing theory: busy period l n = l n−1 + η n �� �� �� �� �� �� � � � � � � � � � � � � queue �� �� �� �� �� �� length ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� � � l n 6 � � 4 5 5 6 3 4 4 4 5 6 2 3 3 3 4 5 6 1 2 2 2 3 4 5 6 busy period time n • Queue length: l n = l n − 1 + η n η n = 1 if a new customer arrives = − 1 when a customer gets served • T ≡ total waiting time of all customers during the busy period ≡ area under the curve • Continuum limit: l n → x ( τ ) → dx d τ = η ( τ ) → Brownian motion S.N. Majumdar The Wonderful World of Brownian Functionals
Queueing theory: busy period l n = l n−1 + η n �� �� �� �� �� �� � � � � � � � � � � � � queue �� �� �� �� �� �� length ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� � � l n 6 � � 4 5 5 6 3 4 4 4 5 6 2 3 3 3 4 5 6 1 2 2 2 3 4 5 6 busy period time n • Queue length: l n = l n − 1 + η n η n = 1 if a new customer arrives = − 1 when a customer gets served • T ≡ total waiting time of all customers during the busy period ≡ area under the curve • Continuum limit: l n → x ( τ ) → dx d τ = η ( τ ) → Brownian motion � t f T ≡ A = 0 x ( τ ) d τ → area till the first-passage time; P ( A | x 0 ) = ? S.N. Majumdar The Wonderful World of Brownian Functionals
First-passage Brownian functional Brownian motion till its first−passage through 0 x (τ) • dx d τ = η ( τ ); x (0) = x 0 x 0 + ∆ x � t f • T = 0 U ( x ( τ )) d τ x 0 • P ( T | x 0 ) =? t f 0 τ ∆τ S.N. Majumdar The Wonderful World of Brownian Functionals
First-passage Brownian functional Brownian motion till its first−passage through 0 x (τ) • dx d τ = η ( τ ); x (0) = x 0 x 0 + ∆ x � t f • T = 0 U ( x ( τ )) d τ x 0 • P ( T | x 0 ) =? t f 0 τ ∆τ � ∞ P ( T | x 0 ) e − s T dT • Laplace transform: ˜ P s ( x 0 ) = 0 S.N. Majumdar The Wonderful World of Brownian Functionals
First-passage Brownian functional Brownian motion till its first−passage through 0 x (τ) • dx d τ = η ( τ ); x (0) = x 0 x 0 + ∆ x � t f • T = 0 U ( x ( τ )) d τ x 0 • P ( T | x 0 ) =? t f 0 τ ∆τ � ∞ � tf P ( T | x 0 ) e − s T dT ≡ � e − s • Laplace transform: ˜ U ( x ( τ )) d τ � P s ( x 0 ) = 0 0 S.N. Majumdar The Wonderful World of Brownian Functionals
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