Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Scaling and Universality in Probability Francesco Caravenna Universit` a degli Studi di Milano-Bicocca Luxembourg ∼ June 14, 2016 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 1 / 33
Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Overview A more expressive (but less fancy) title would be Convergence of Discrete Probability Models to a Universal Continuum Limit This is a key topic of classical and modern probability theory I will present a (limited) selection of representative results, in order to convey the main ideas and give the flavor of the subject Francesco Caravenna Scaling and Universality in Probability June 14, 2016 2 / 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 3 / 33
Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Reminders (I). Probability spaces Fix a set Ω. A probability P is a map from subsets of Ω to [0 , 1] s.t. �� � = � P(Ω) = 1 , P i ∈ N A i i ∈ N P( A i ) for disjoint A i [ P is only defined on a subclass ( σ -algebra) A of “measurable” subsets of Ω ] (Ω , A , P) is an abstract probability space. We will be “concrete”: � � Metric space E , “Borel σ -algebra” , Probability µ ◮ Integral � E ϕ d µ for bounded and continuous ϕ : E → R ◮ Discrete probability µ = � i p i δ x i with x i ∈ E , p i ∈ [0 , 1] � E ϕ d µ := � i p i ϕ ( x i ) Francesco Caravenna Scaling and Universality in Probability June 14, 2016 4 / 33
Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder Riemann sums and integral on [0 , 1] ◮ Partition t = ( t 0 , t 1 , . . . , t k ) of [0 , 1] 0 = t 0 < t 1 < . . . < t k = 1 ( k ∈ N ) ◮ Riemann sum of a function ϕ : [0 , 1] → R relative to t k � R ( ϕ, t ) := ϕ ( t i ) ( t i − t i − 1 ) i =1 Theorem Let t ( n ) be partitions with t ( n ) − t ( n ) mesh( t ( n ) ) := max � � − − − → 0 i i − 1 1 ≤ i ≤ k n n →∞ If ϕ : [0 , 1] → R is continuous, then � 1 R ( ϕ, t ( n ) ) − − − − − → ϕ ( x ) d x n →∞ 0 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 5 / 33
Weak Convergence Brownian Motion A glimpse of SLE Scaling Limits with Disorder A probabilistic reformulation � � Partition t = t 0 , t 1 , . . . , t k discrete probability µ t on [0 , 1] � k � µ t ( · ) := p i δ t i ( · ) where p i := t i − t i − 1 i =1 Uniform partition � 1 0 , 1 n , 2 n , 2 � � � t = n , . . . , 1 µ t = uniform probability on n , . . . , 1 � 1 5 0 t 1 t 2 t 3 t 4 1 Francesco Caravenna Scaling and Universality in Probability June 14, 2016 6 / 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 } , . . . ◮ ( 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 ⇒ 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) but 1 = µ n ( A ) � − → λ ( A ) = 0 ◮ 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 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 � · � ∞ ) � ◮ E n := piecewise linear f : [0 , 1] → R with � i +1 � i � � � � 1 f (0) = 0 and f = f ± ⊆ C ([0 , 1]) 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 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 ] 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 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) 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
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