Scaling and Universality in Probability Francesco Caravenna - PowerPoint PPT Presentation

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A probabilistic reformulation Key observation: Riemann sum is . . . integral w.r.t. µ t k � � R ( ϕ, t ) = ϕ ( t i ) p i = ϕ d µ t [0 , 1] i =1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 7 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A probabilistic reformulation Key observation: Riemann sum is . . . integral w.r.t. µ t k � � R ( ϕ, t ) = ϕ ( t i ) p i = ϕ d µ t [0 , 1] i =1 Theorem If mesh( t ( n ) ) → 0 and ϕ : [0 , 1] → R is continuous, then � � ϕ d µ t ( n ) − − − − − → ϕ d λ ( ⋆ ) n →∞ [0 , 1] [0 , 1] with λ := Lebesgue measure (probability) on [0 , 1] Francesco Caravenna Scaling and Universality in Probability June 14, 2016 7 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A probabilistic reformulation Key observation: Riemann sum is . . . integral w.r.t. µ t k � � R ( ϕ, t ) = ϕ ( t i ) p i = ϕ d µ t [0 , 1] i =1 Theorem If mesh( t ( n ) ) → 0 and ϕ : [0 , 1] → R is continuous, then � � ϕ d µ t ( n ) − − − − − → ϕ d λ ( ⋆ ) n →∞ [0 , 1] [0 , 1] with λ := Lebesgue measure (probability) on [0 , 1] ◮ Scaling Limit: convergence of µ t ( n ) toward λ Francesco Caravenna Scaling and Universality in Probability June 14, 2016 7 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A probabilistic reformulation Key observation: Riemann sum is . . . integral w.r.t. µ t k � � R ( ϕ, t ) = ϕ ( t i ) p i = ϕ d µ t [0 , 1] i =1 Theorem If mesh( t ( n ) ) → 0 and ϕ : [0 , 1] → R is continuous, then � � ϕ d µ t ( n ) − − − − − → ϕ d λ ( ⋆ ) n →∞ [0 , 1] [0 , 1] with λ := Lebesgue measure (probability) on [0 , 1] ◮ Scaling Limit: convergence of µ t ( n ) toward λ ◮ Universality: the limit λ is the same, for any choice of t ( n ) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 7 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Weak convergence ◮ E is a Polish space (complete separable metric space), e.g. [0 , 1] , C ([0 , 1]) := { continuous f : [0 , 1] → R } , . . . Francesco Caravenna Scaling and Universality in Probability June 14, 2016 8 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Weak convergence ◮ E is a Polish space (complete separable metric space), e.g. [0 , 1] , C ([0 , 1]) := { continuous f : [0 , 1] → R } , . . . ◮ ( µ n ) n ∈ N , µ are probabilities on E Francesco Caravenna Scaling and Universality in Probability June 14, 2016 8 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Weak convergence ◮ E is a Polish space (complete separable metric space), e.g. [0 , 1] , C ([0 , 1]) := { continuous f : [0 , 1] → R } , . . . ◮ ( µ n ) n ∈ N , µ are probabilities on E Definition (weak convergence of probabilities) We say that µ n converges weakly to µ (notation µ n ⇒ µ ) if � � ϕ d µ n − − − − − → ϕ d µ n →∞ E E for every ϕ ∈ C b ( E ) := { continuous and bounded ϕ : E → R } Francesco Caravenna Scaling and Universality in Probability June 14, 2016 8 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Weak convergence ◮ E is a Polish space (complete separable metric space), e.g. [0 , 1] , C ([0 , 1]) := { continuous f : [0 , 1] → R } , . . . ◮ ( µ n ) n ∈ N , µ are probabilities on E Definition (weak convergence of probabilities) We say that µ n converges weakly to µ (notation µ n ⇒ µ ) if � � ϕ d µ n − − − − − → ϕ d µ n →∞ E E for every ϕ ∈ C b ( E ) := { continuous and bounded ϕ : E → R } [ Analysts call this weak- ∗ convergence; note that µ n , µ ∈ C b ( E ) ∗ ] Francesco Caravenna Scaling and Universality in Probability June 14, 2016 8 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A useful reformulation µ n ( A ) → µ ( A ) for all meas. A ⊆ E ? Francesco Caravenna Scaling and Universality in Probability June 14, 2016 9 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A useful reformulation ◮ µ n ⇒ µ does not imply µ n ( A ) → µ ( A ) for all meas. A ⊆ E Francesco Caravenna Scaling and Universality in Probability June 14, 2016 9 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A useful reformulation ◮ µ n ⇒ µ does not imply µ n ( A ) → µ ( A ) for all meas. A ⊆ E Example � 1 n , 2 � µ n = uniform probability on n , . . . , 1 µ n ⇒ λ (Lebesgue) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 9 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A useful reformulation ◮ µ n ⇒ µ does not imply µ n ( A ) → µ ( A ) for all meas. A ⊆ E Example � 1 n , 2 � µ n = uniform probability on n , . . . , 1 A := Q ∩ [0 , 1] µ n ⇒ λ (Lebesgue) 1 = µ n ( A ) � − → λ ( A ) = 0 but Francesco Caravenna Scaling and Universality in Probability June 14, 2016 9 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A useful reformulation ◮ µ n ⇒ µ does not imply µ n ( A ) → µ ( A ) for all meas. A ⊆ E Example � 1 n , 2 � µ n = uniform probability on n , . . . , 1 A := Q ∩ [0 , 1] µ n ⇒ λ (Lebesgue) 1 = µ n ( A ) � − → λ ( A ) = 0 but ◮ Weak convergence means µ n ( A ) → µ ( A ) for “nice” A ⊆ E Francesco Caravenna Scaling and Universality in Probability June 14, 2016 9 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A useful reformulation ◮ µ n ⇒ µ does not imply µ n ( A ) → µ ( A ) for all meas. A ⊆ E Example � 1 n , 2 � µ n = uniform probability on n , . . . , 1 A := Q ∩ [0 , 1] µ n ⇒ λ (Lebesgue) 1 = µ n ( A ) � − → λ ( A ) = 0 but ◮ Weak convergence means µ n ( A ) → µ ( A ) for “nice” A ⊆ E Theorem µ n ⇒ µ iff µ n ( A ) → µ ( A ) ∀ meas. A ⊆ E with µ ( ∂ A ) = 0 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 9 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A useful reformulation ◮ µ n ⇒ µ does not imply µ n ( A ) → µ ( A ) for all meas. A ⊆ E Example � 1 n , 2 � µ n = uniform probability on n , . . . , 1 A := Q ∩ [0 , 1] µ n ⇒ λ (Lebesgue) 1 = µ n ( A ) � − → λ ( A ) = 0 but ◮ Weak convergence means µ n ( A ) → µ ( A ) for “nice” A ⊆ E Theorem µ n ⇒ µ iff µ n ( A ) → µ ( A ) ∀ meas. A ⊆ E with µ ( ∂ A ) = 0 ◮ Weak convergence links measurable and topological structures Francesco Caravenna Scaling and Universality in Probability June 14, 2016 9 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Rest of the talk Three interesting examples of weak convergence, leading to ◮ Brownian motion ◮ Schramm-L¨ owner Evolution (SLE) ◮ Continuum disordered pinning models Francesco Caravenna Scaling and Universality in Probability June 14, 2016 10 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Rest of the talk Three interesting examples of weak convergence, leading to ◮ Brownian motion ◮ Schramm-L¨ owner Evolution (SLE) ◮ Continuum disordered pinning models Common mathematical structure ◮ A Polish space E Francesco Caravenna Scaling and Universality in Probability June 14, 2016 10 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Rest of the talk Three interesting examples of weak convergence, leading to ◮ Brownian motion ◮ Schramm-L¨ owner Evolution (SLE) ◮ Continuum disordered pinning models Common mathematical structure ◮ A Polish space E ◮ A sequence of discrete probabilities µ n (easy) on E Francesco Caravenna Scaling and Universality in Probability June 14, 2016 10 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Rest of the talk Three interesting examples of weak convergence, leading to ◮ Brownian motion ◮ Schramm-L¨ owner Evolution (SLE) ◮ Continuum disordered pinning models Common mathematical structure ◮ A Polish space E ◮ A sequence of discrete probabilities µ n (easy) on E ◮ A “continuum” probability µ (difficult!) such that µ n ⇒ µ Francesco Caravenna Scaling and Universality in Probability June 14, 2016 10 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Outline 1. Weak Convergence of Probability Measures 2. Brownian Motion 3. A glimpse of SLE 4. Scaling Limits in presence of Disorder Francesco Caravenna Scaling and Universality in Probability June 14, 2016 11 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := � ⊆ C ([0 , 1]) 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := � ⊆ C ([0 , 1]) � 1 n Case n = 40 0 1 1 n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := � ⊆ C ([0 , 1]) � 1 n Case n = 40 0 1 1 n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := � ⊆ C ([0 , 1]) � 1 n Case n = 40 0 1 1 n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := � ⊆ C ([0 , 1]) � 1 n Case n = 40 0 1 1 n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := � ⊆ C ([0 , 1]) � 1 n Case n = 40 0 1 1 n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := � ⊆ C ([0 , 1]) � 1 n Case n = 40 0 1 1 n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := piecewise linear f : [0 , 1] → R with � i +1 � i � � 1 � � ± ⊆ C ([0 , 1]) f (0) = 0 and f = f n n n Case n = 40 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := piecewise linear f : [0 , 1] → R with � i +1 � i � � 1 � � ± ⊆ C ([0 , 1]) f (0) = 0 and f = f n n n | E n | = 2 n Case n = 40 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := piecewise linear f : [0 , 1] → R with � i +1 � i � � 1 � � ± ⊆ C ([0 , 1]) f (0) = 0 and f = f n n n | E n | = 2 n Case n = 40 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := piecewise linear f : [0 , 1] → R with � i +1 � i � � 1 � � ± ⊆ C ([0 , 1]) f (0) = 0 and f = f n n n | E n | = 2 n Case n = 40 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion � � ◮ E := C ([0 , 1]) = continuous f : [0 , 1] → R (with � · � ∞ ) � ◮ E n := piecewise linear f : [0 , 1] → R with � i +1 � i � � 1 � � ± ⊆ C ([0 , 1]) f (0) = 0 and f = f n n n | E n | = 2 n √ slope( f ) = ±√ n ∆ f = ± ∆ t � Case n = 40 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 12 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion Let µ n be the probability on C ([0 , 1]) which is uniform on E n : 1 � µ n ( · ) = 2 n δ f ( · ) f ∈ E n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 13 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion Let µ n be the probability on C ([0 , 1]) which is uniform on E n : 1 � µ n ( · ) = 2 n δ f ( · ) f ∈ E n Theorem (Donsker) The sequence ( µ n ) n ∈ N converges weakly on C ([0 , 1]): µ n ⇒ µ Francesco Caravenna Scaling and Universality in Probability June 14, 2016 13 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion Let µ n be the probability on C ([0 , 1]) which is uniform on E n : 1 � µ n ( · ) = 2 n δ f ( · ) f ∈ E n Theorem (Donsker) The sequence ( µ n ) n ∈ N converges weakly on C ([0 , 1]): µ n ⇒ µ The limiting probability µ on C ([0 , 1]) is called Wiener measure Francesco Caravenna Scaling and Universality in Probability June 14, 2016 13 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion Let µ n be the probability on C ([0 , 1]) which is uniform on E n : 1 � µ n ( · ) = 2 n δ f ( · ) f ∈ E n Theorem (Donsker) The sequence ( µ n ) n ∈ N converges weakly on C ([0 , 1]): µ n ⇒ µ The limiting probability µ on C ([0 , 1]) is called Wiener measure ◮ Deep result! Francesco Caravenna Scaling and Universality in Probability June 14, 2016 13 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion Let µ n be the probability on C ([0 , 1]) which is uniform on E n : 1 � µ n ( · ) = 2 n δ f ( · ) f ∈ E n Theorem (Donsker) The sequence ( µ n ) n ∈ N converges weakly on C ([0 , 1]): µ n ⇒ µ The limiting probability µ on C ([0 , 1]) is called Wiener measure ◮ Deep result! ◮ Wiener measure is the law of Brownian motion Francesco Caravenna Scaling and Universality in Probability June 14, 2016 13 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder From random walk to Brownian motion Let µ n be the probability on C ([0 , 1]) which is uniform on E n : 1 � µ n ( · ) = 2 n δ f ( · ) f ∈ E n Theorem (Donsker) The sequence ( µ n ) n ∈ N converges weakly on C ([0 , 1]): µ n ⇒ µ The limiting probability µ on C ([0 , 1]) is called Wiener measure ◮ Deep result! ◮ Wiener measure is the law of Brownian motion ◮ Wiener measure is a “natural” probability on C ([0 , 1]) (like Lebesgue for [0 , 1]) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 13 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Reminders (II). Random variables and their laws A random variable (r.v.) is a measurable function X : Ω → E [ where (Ω , A , P) is some abstract probability space ] Francesco Caravenna Scaling and Universality in Probability June 14, 2016 14 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Reminders (II). Random variables and their laws A random variable (r.v.) is a measurable function X : Ω → E [ where (Ω , A , P) is some abstract probability space ] ◮ X describes a random element of E Francesco Caravenna Scaling and Universality in Probability June 14, 2016 14 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Reminders (II). Random variables and their laws A random variable (r.v.) is a measurable function X : Ω → E [ where (Ω , A , P) is some abstract probability space ] The law (or distribution) µ X of X is a probability on E µ X ( A ) = P( X − 1 ( A )) = P( X ∈ A ) for A ⊆ E ◮ X describes a random element of E Francesco Caravenna Scaling and Universality in Probability June 14, 2016 14 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Reminders (II). Random variables and their laws A random variable (r.v.) is a measurable function X : Ω → E [ where (Ω , A , P) is some abstract probability space ] The law (or distribution) µ X of X is a probability on E µ X ( A ) = P( X − 1 ( A )) = P( X ∈ A ) for A ⊆ E ◮ X describes a random element of E ◮ µ X describes the values taken by X and the resp. probabilities Francesco Caravenna Scaling and Universality in Probability June 14, 2016 14 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Reminders (II). Random variables and their laws A random variable (r.v.) is a measurable function X : Ω → E [ where (Ω , A , P) is some abstract probability space ] The law (or distribution) µ X of X is a probability on E µ X ( A ) = P( X − 1 ( A )) = P( X ∈ A ) for A ⊆ E ◮ X describes a random element of E ◮ µ X describes the values taken by X and the resp. probabilities Instead of a probability µ on E , it is often convenient to work with a random variable X with law µ Francesco Caravenna Scaling and Universality in Probability June 14, 2016 14 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Reminders (II). Random variables and their laws A random variable (r.v.) is a measurable function X : Ω → E [ where (Ω , A , P) is some abstract probability space ] The law (or distribution) µ X of X is a probability on E µ X ( A ) = P( X − 1 ( A )) = P( X ∈ A ) for A ⊆ E ◮ X describes a random element of E ◮ µ X describes the values taken by X and the resp. probabilities Instead of a probability µ on E , it is often convenient to work with a random variable X with law µ When E = C ([0 , 1]), a r.v. X is a stochastic process Francesco Caravenna Scaling and Universality in Probability June 14, 2016 14 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Reminders (II). Random variables and their laws A random variable (r.v.) is a measurable function X : Ω → E [ where (Ω , A , P) is some abstract probability space ] The law (or distribution) µ X of X is a probability on E µ X ( A ) = P( X − 1 ( A )) = P( X ∈ A ) for A ⊆ E ◮ X describes a random element of E ◮ µ X describes the values taken by X and the resp. probabilities Instead of a probability µ on E , it is often convenient to work with a random variable X with law µ When E = C ([0 , 1]), a r.v. X = ( X t ) t ∈ [0 , 1] is a stochastic process Francesco Caravenna Scaling and Universality in Probability June 14, 2016 14 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Simple random walk Let us build a stochastic process X ( n ) with law µ n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 15 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Simple random walk Let us build a stochastic process X ( n ) with law µ n Fair coin tossing: independent random variables Y 1 , Y 2 , . . . with P( Y i = +1) = P( Y i = − 1) = 1 2 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 15 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Simple random walk Let us build a stochastic process X ( n ) with law µ n Fair coin tossing: independent random variables Y 1 , Y 2 , . . . with P( Y i = +1) = P( Y i = − 1) = 1 2 Simple random walk: S 0 := 0 S n := Y 1 + Y 2 + . . . + Y n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 15 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Simple random walk Let us build a stochastic process X ( n ) with law µ n Fair coin tossing: independent random variables Y 1 , Y 2 , . . . with P( Y i = +1) = P( Y i = − 1) = 1 2 Simple random walk: S 0 := 0 S n := Y 1 + Y 2 + . . . + Y n √ Diffusive rescaling: space ∝ time X ( n ) ( t ) := linear interpol. of S nt √ n t ∈ [0 , 1] Francesco Caravenna Scaling and Universality in Probability June 14, 2016 15 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Simple random walk Let us build a stochastic process X ( n ) with law µ n Fair coin tossing: independent random variables Y 1 , Y 2 , . . . with P( Y i = +1) = P( Y i = − 1) = 1 2 Simple random walk: S 0 := 0 S n := Y 1 + Y 2 + . . . + Y n √ Diffusive rescaling: space ∝ time X ( n ) ( t ) := linear interpol. of S nt √ n t ∈ [0 , 1] The law of X ( n ) (r.v. in C ([0 , 1])) is µ n uniform probab. on E n Francesco Caravenna Scaling and Universality in Probability June 14, 2016 15 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Simple random walk Let us build a stochastic process X ( n ) with law µ n Fair coin tossing: independent random variables Y 1 , Y 2 , . . . with P( Y i = +1) = P( Y i = − 1) = 1 2 Simple random walk: S 0 := 0 S n := Y 1 + Y 2 + . . . + Y n √ Diffusive rescaling: space ∝ time X ( n ) ( t ) := linear interpol. of S nt √ n t ∈ [0 , 1] The law of X ( n ) (r.v. in C ([0 , 1])) is µ n uniform probab. on E n Donsker: The law of simple random walk, diffusively rescaled, converges weakly to the law of Brownian motion Francesco Caravenna Scaling and Universality in Probability June 14, 2016 15 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder General random walks Instead of coin tossing, take independent random variables Y i with a generic law, with zero mean and finite variance (say 1) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 16 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder General random walks Instead of coin tossing, take independent random variables Y i with a generic law, with zero mean and finite variance (say 1) Define random walk S n and its diffusive rescaling X ( n ) ( t ) as before Francesco Caravenna Scaling and Universality in Probability June 14, 2016 16 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder General random walks Instead of coin tossing, take independent random variables Y i with a generic law, with zero mean and finite variance (say 1) Define random walk S n and its diffusive rescaling X ( n ) ( t ) as before P( Y i = +2) = 1 P( Y i = − 1) = 2 E.g. 3 , 3 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 16 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder General random walks Instead of coin tossing, take independent random variables Y i with a generic law, with zero mean and finite variance (say 1) Define random walk S n and its diffusive rescaling X ( n ) ( t ) as before P( Y i = +2) = 1 P( Y i = − 1) = 2 E.g. 3 , 3 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 16 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder General random walks Instead of coin tossing, take independent random variables Y i with a generic law, with zero mean and finite variance (say 1) Define random walk S n and its diffusive rescaling X ( n ) ( t ) as before P( Y i = +2) = 1 P( Y i = − 1) = 2 E.g. 3 , 3 0 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 16 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder General random walks Instead of coin tossing, take independent random variables Y i with a generic law, with zero mean and finite variance (say 1) Define random walk S n and its diffusive rescaling X ( n ) ( t ) as before P( Y i = +2) = 1 P( Y i = − 1) = 2 E.g. 3 , 3 0 1 The law µ n of X ( n ) is a (non uniform!) probability on C ([0 , 1]) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 16 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Universality of Brownian motion Theorem (Donsker) µ n ⇒ µ := Wiener measure The law of any RW (zero mean, finite variance) diffusively rescaled converges weakly to the law of Brownian motion (Wiener measure) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 17 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Universality of Brownian motion Theorem (Donsker) µ n ⇒ µ := Wiener measure The law of any RW (zero mean, finite variance) diffusively rescaled converges weakly to the law of Brownian motion (Wiener measure) Universality: µ n ( A ) − → µ ( A ) ∀ A ⊆ C ([0 , 1]) with µ ( ∂ A ) = 0 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 17 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Universality of Brownian motion Theorem (Donsker) µ n ⇒ µ := Wiener measure The law of any RW (zero mean, finite variance) diffusively rescaled converges weakly to the law of Brownian motion (Wiener measure) Universality: µ n ( A ) − → µ ( A ) ∀ A ⊆ C ([0 , 1]) with µ ( ∂ A ) = 0 Example (Feller I, Chapter III) ◮ U + ( f ) := Leb { t ∈ [0 , 1] : f ( t ) > 0 } = { amount of time in which f > 0 } Francesco Caravenna Scaling and Universality in Probability June 14, 2016 17 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Universality of Brownian motion Theorem (Donsker) µ n ⇒ µ := Wiener measure The law of any RW (zero mean, finite variance) diffusively rescaled converges weakly to the law of Brownian motion (Wiener measure) Universality: µ n ( A ) − → µ ( A ) ∀ A ⊆ C ([0 , 1]) with µ ( ∂ A ) = 0 Example (Feller I, Chapter III) ◮ U + ( f ) := Leb { t ∈ [0 , 1] : f ( t ) > 0 } = { amount of time in which f > 0 } ◮ A := � � f : U + ( f ) ≥ 0 . 95 or U + ( f ) ≤ 0 . 05 ⊆ C ([0 , 1]) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 17 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Universality of Brownian motion Theorem (Donsker) µ n ⇒ µ := Wiener measure The law of any RW (zero mean, finite variance) diffusively rescaled converges weakly to the law of Brownian motion (Wiener measure) Universality: µ n ( A ) − → µ ( A ) ∀ A ⊆ C ([0 , 1]) with µ ( ∂ A ) = 0 Example (Feller I, Chapter III) ◮ U + ( f ) := Leb { t ∈ [0 , 1] : f ( t ) > 0 } = { amount of time in which f > 0 } ◮ A := � � f : U + ( f ) ≥ 0 . 95 or U + ( f ) ≤ 0 . 05 ⊆ C ([0 , 1]) Then µ n ( A ) → µ ( A ) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 17 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Universality of Brownian motion Theorem (Donsker) µ n ⇒ µ := Wiener measure The law of any RW (zero mean, finite variance) diffusively rescaled converges weakly to the law of Brownian motion (Wiener measure) Universality: µ n ( A ) − → µ ( A ) ∀ A ⊆ C ([0 , 1]) with µ ( ∂ A ) = 0 Example (Feller I, Chapter III) ◮ U + ( f ) := Leb { t ∈ [0 , 1] : f ( t ) > 0 } = { amount of time in which f > 0 } ◮ A := � � f : U + ( f ) ≥ 0 . 95 or U + ( f ) ≤ 0 . 05 ⊆ C ([0 , 1]) Then µ n ( A ) → µ ( A ) ≃ 0 . 29 . Francesco Caravenna Scaling and Universality in Probability June 14, 2016 17 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Universality of Brownian motion Theorem (Donsker) µ n ⇒ µ := Wiener measure The law of any RW (zero mean, finite variance) diffusively rescaled converges weakly to the law of Brownian motion (Wiener measure) Universality: µ n ( A ) − → µ ( A ) ∀ A ⊆ C ([0 , 1]) with µ ( ∂ A ) = 0 Example (Feller I, Chapter III) ◮ U + ( f ) := Leb { t ∈ [0 , 1] : f ( t ) > 0 } = { amount of time in which f > 0 } ◮ A := � � f : U + ( f ) ≥ 0 . 95 or U + ( f ) ≤ 0 . 05 ⊆ C ([0 , 1]) Then µ n ( A ) → µ ( A ) ≃ 0 . 29 . Random walk has a chance of 29% of spending 95% or more of its time on the same side! Francesco Caravenna Scaling and Universality in Probability June 14, 2016 17 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Universality of Brownian motion Theorem (Donsker) µ n ⇒ µ := Wiener measure The law of any RW (zero mean, finite variance) diffusively rescaled converges weakly to the law of Brownian motion (Wiener measure) Universality: µ n ( A ) − → µ ( A ) ∀ A ⊆ C ([0 , 1]) with µ ( ∂ A ) = 0 Example (Feller I, Chapter III) ◮ U + ( f ) := Leb { t ∈ [0 , 1] : f ( t ) > 0 } = { amount of time in which f > 0 } ◮ A := � � f : U + ( f ) ≥ 0 . 99 or U + ( f ) ≤ 0 . 01 ⊆ C ([0 , 1]) Then µ n ( A ) → µ ( A ) ≃ 0 . 13 . Random walk has a chance of 13% of spending 99% or more of its time on the same side! Francesco Caravenna Scaling and Universality in Probability June 14, 2016 17 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 81% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 97% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 71% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 62% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 0% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 29% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 95% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 20% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 4% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Some sample paths of the SRW U_N/N = 15% 50 0 −50 0 200 400 600 800 1000 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 18 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Outline 1. Weak Convergence of Probability Measures 2. Brownian Motion 3. A glimpse of SLE 4. Scaling Limits in presence of Disorder Francesco Caravenna Scaling and Universality in Probability June 14, 2016 19 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A glimpse of SLE Even the simplest randomness (coin tossing) can lead to interesting models, such as random walks and Brownian motion Francesco Caravenna Scaling and Universality in Probability June 14, 2016 20 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A glimpse of SLE Even the simplest randomness (coin tossing) can lead to interesting models, such as random walks and Brownian motion Brownian motion is at the heart of Schramm-L¨ owner Evolution (SLE), one of the greatest achievements of modern probability [Fields Medal awarded to W. Werner (2006) and S. Smirnov (2010)] Francesco Caravenna Scaling and Universality in Probability June 14, 2016 20 / 33

Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A glimpse of SLE Even the simplest randomness (coin tossing) can lead to interesting models, such as random walks and Brownian motion Brownian motion is at the heart of Schramm-L¨ owner Evolution (SLE), one of the greatest achievements of modern probability [Fields Medal awarded to W. Werner (2006) and S. Smirnov (2010)] We present an instance of SLE, which emerges as the scaling limit of percolation (spatial version of coin tossing) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 20 / 33