Putting loops back on ◮ A be a set, z ∈ A . p Markov chain on A ◮ Take independently: ◮ A loop-erased walk from z to ∂A outputting η ◮ A realization of the loop soup with intensity 1 outputting a collection of unrooted loops ℓ 1 , ℓ 2 , . . . ordered by the time that they occurred. ◮ For each loop ℓ that intersects η choose the first point on η , say η j that ℓ hits. ◮ Choose a rooted representative of ℓ that is rooted at η j and add it to the curve. (If more than one choice, choose randomly.) ◮ The curve one gets has the distribution of the MC from z to ∂A . 20 / 106
Brownian Loop Measure/Soup (L-Werner) ◮ Scaling limit of random walk loop ◮ Rooted (Brownian) loop measure in R d : choose ( z, t, ˜ γ ) according to ( Lebesgue ) × 1 dt (2 πt ) d/ 2 × ( Brownian bridge of time 1) . t and output √ γ ( s ) = z + t ˜ γ ( s/t ) , 0 ≤ s ≤ t. ◮ (Unrooted) Brownian loop measure: rooted loop measure “forgetting the root”. ◮ Poissonian realizations are called Brownian loop soup. 21 / 106
◮ The measure of loops restricted to a bounded domain is infinite because of small loops. ◮ Measure of loops of diameter ≥ ǫ in a bounded domain is finite. ◮ If d = 2 , then the Brownian loop measure (on unrooted loops) is conformally invariant: if f : D → f ( D ) is a conformal transformation and f ◦ γ is defined with change of parametrization, then for every set of curves V , µ f ( D ) ( V ) = µ D { γ : f ◦ γ ∈ V } . ◮ True for unrooted loops but not true for rooted loops. 22 / 106
Convergence of Random Walk Soup ◮ Consider (simple) random walk measure on Z 2 scaled to N − 1 Z 2 . ◮ Scale the paths using Brownian scaling but do not scale the measure. ◮ The limit is Brownian loop measure in a strong sense. (L-Trujillo Ferreras). ◮ Given a bounded, simply connected domain D , we can couple the Brownian soup and the random walk soup with scaling N − 1 such that, except for an event of probability O ( N − α ) , the loops of time duration at least N − β are very close. ◮ A version for all loops, viewing the soup as a field, in preparation (L-Panov). 23 / 106
Loop soups and Gaussian Free Field ◮ Let A be a finite set with real-valued, symmetric, integrable weight q . Let G = ( I − Q ) − 1 be the Green’s function which is positive definite. ◮ If q is a positive weight, G has all nonnegative entries. However, negative q allow for G to have some negative entries. ◮ The corresponding (discrete) Gaussian free field (with Dirichlet boundary conditions) is a centered multivariate normal Z x , x ∈ A with covariance matrix G . ◮ (Le Jan) Use the random walk loop soup to sample from Z 2 x / 2 . ◮ (Lupu) If Q is positive, find way to add signs to get Z x . 24 / 106
Discrete time version of isomorphism theorem ◮ Consider the loop soup at intensity 1 / 2 . For each configuration of loops, let N x denote the number of times that vertex x is visited. ◮ The random walk loop measure gives a measure on possible values { N x : x ∈ A } . ◮ Take independent Gamma processes Γ x ( t ) of rate 1 at each x ∈ A and let T x = Γ x ( 1 2 + N x ) . ◮ Theorem: { T x : x ∈ A } has the same distribution as { Z 2 x / 2 : x ∈ A } . ◮ As an example, if q ≡ 0 , so that there are no loops then N ≡ 0 , and { T x : x ∈ A } are independent Γ( 1 2 ) , that is, have the distribution of Y 2 / 2 where Y is a standard normal. 25 / 106
Proof of Isomorphism Theorem ◮ Just check it. ◮ (L-Perlman) Using Laplace transform adapting proof of Le Jan. Does not need positive weights. ◮ Can give a direct proof at intensity 1 / 2 using a combinatorial graph identity and get the joint distribution of T x and the current (local time on undirected edges). ◮ (L-Panov) Direct proof with intensity 1 for the sum of two indpendent copies (or for | Z | 2 for a complex field Z = X + iY ). Uses an easier combinatorial identity. ◮ Intensity λ is related to central charge c of conformal field theory, λ = ± c 2 . 26 / 106
Part 2 (One-sided) LERW in Z d , d ≥ 2 ◮ ( d ≥ 3) Take simple random walk (SRW) and erase loops chronologically. This gives an infinite self-avoiding path. ◮ We get the same measure by starting with SRW conditioned to never return to the origin. ◮ The latter definition extends to d = 2 by using SRW “conditioned to never return to 0”, more precisely, tilted by the potential kernel (Green’s function). ◮ This is equivalent to other natural definitions such as take SRW stopped when it reaches distance R , erase loops, and take the (local) limit of measure as R → ∞ . 27 / 106
LERW as the Laplacian Random Walk ◮ Start with ˆ S 0 = 0 . ◮ Given [ ˆ S 0 , . . . , ˆ S n ] = η = [ x 0 , . . . , x n ] choose x n +1 among nearest neighbors of x n using distribution c φ where ◮ φ = φ η is the unique function that vanishes on η ; is (discrete) harmonic on Z d \ η and has asymptotics φ ( z ) → 1 , d ≥ 3 , φ ( z ) ∼ 2 π log | z | , d = 2 . ◮ Could also consider Laplacian- b walk where we use c φ b with b � = 1 but this is much more difficult and very little is known about. 28 / 106
Basic idea for understanding LERW ◮ If the number of points in the first n steps of the walk remaining after loop-erasure is f ( n ) then S f ( n ) | 2 = | S n | 2 ≍ n, S m | 2 ≍ f − 1 ( m ) . | ˆ | ˆ ◮ The point S n is not erased if and only if LE ( S [0 , n ]) ∩ S [ n + 1 , ∞ ) = ∅ . Hence, f ( n ) ≍ n P { LE ( S [0 , n ]) ∩ S [ n + 1 , ∞ ) = ∅} . 29 / 106
Critical Exponent ◮ Let S 1 , S 2 , . . . be independent SRWs and n = min { t : | S j T j t | ≥ e n } . ω j n = S j [1 , T j η j n = LE ( S j [0 , T j n ] , n ]) . ◮ Interested in n = ∅} ≈ e − ξn . p 1 , 1 ( n ) = P { η 1 n ∩ ω 2 ˆ This should be comparable to e − 2 n f ( e 2 n ) of previous slide. ◮ Fractal dimension of LERW should be 2 − ξ . 30 / 106
Similar problem — SRW intersection exponent p 1 ,k ( n ) = P { ω 1 n ∩ [ ω 2 n ∪ · · · ∪ ω k +1 ] = ∅} . n ◮ d = 4 is critical dimension for intersections of two-dimensional sets. ◮ If d ≥ 5 , p 1 ,k ( ∞ ) > 0 . ◮ Using relation with harmonic measure, we can show � e n ( d − 4) d < 4 p 1 , 2 ( n ) ≍ n − 1 d = 4 . ◮ Cauchy-Schwarz gives e n ( d − 4) � � e n ( d − 4) / 2 d < 4 � p 1 , 1 ( n ) � n − 1 n − 1 / 2 d = 4 . 31 / 106
◮ For d = 4 , “mean-field behavior” holds, that is p 1 , 1 ( n ) ≍ [ p 1 , 2 ( n )] 1 / 2 ≍ n − 1 / 2 . ◮ For d < 4 , mean-field behavior does not hold. In fact, p 1 , 1 ( n ) ∼ c e − ξn where ξ = ξ d (1 , 1) ∈ ( 4 − d 2 , 4 − d ) is the Brownian intersection exponent. ◮ For d = 2 , ξ = 5 / 4 . Proved by L-Schramm-Werner using Schramm-Loewner evolution (SLE). ◮ For d = 3 , ξ is not known and may never be known exactly. Numerically ξ ≈ . 58 and rigorously 1 / 2 < ξ < 1 . 32 / 106
◮ ˆ S infinite LERW obtained from SRW S ; X , independent SRW, started distance R = e n away T n = min { j : | X j | ≥ e n } . ◮ Long range intersection 1 , d < 4 P { X [ T n , T n +1 ] ∩ ˆ n − 1 , S � = ∅} ≍ d = 4 e (4 − d ) n d > 4 . ◮ Two exact exponents — third moment and three-arm exponent. Both obtained by considering the event S [ T n , T n +1 ] ∩ ˆ S � = ∅ and considering the “first” intersection. ◮ The difference comes from whether one takes the first on S or the first on X . 33 / 106
Let S 1 , S 2 , . . . be independent simple random walk starting at the origin and η j n = LE ( S j [0 , T j ω j n = S j [1 , T j n ]) , n ] . Third moment estimate � n − 1 , d = 4 P { η 1 n ∩ ( ω 2 n ∪ ω 3 n ∪ ω 4 n ) = ∅} ≍ e ( d − 4) n , d < 4 . Three-arm estimate � n − 1 , d = 4 P { η 1 n ∩ ( ω 2 n ∪ ω 3 n ) = ∅ , η 2 n ∩ ω 3 n = ∅} ≍ e ( d − 4) n , d < 4 . 34 / 106
◮ Let Z n = P { η 1 n ∩ ω 2 n = ∅ | η 1 n } . We are interested in P { η 1 n ∩ ω 2 n = ∅} = E [ Z n ] . ◮ The third moment estimate tells us � n − 1 , d = 4 E [ Z 3 n ] ≍ e ( d − 4) n , d < 4 . n − 1 � � n − 1 / 3 , d = 4 � E [ Z n ] � e ( d − 4) n e ( d − 4) n/ 3 , d < 4 . ◮ Mean-field or non-multifractal behavior would be E [ Z λ n ] ≍ E [ Z n ] λ . ◮ Basic principle: Mean-field behavior holds at the critical dimension d = 4 but not below the critical dimension. 35 / 106
Part 3 Slowly recurrent set in Z d ◮ Let A ⊂ Z d , d ≥ 2 and let X be a simple random walk starting at the origin with stopping times T n = min { j : | X j | ≥ e n } . Let E n be the event E n = { X [ T n − 1 , T n ] ∩ A � = ∅} . ◮ A is recurrent if X visits A infintely often, that is, if P { E n i.o. } = 1 . This is equivalent to (Wiener’s test) ∞ � P ( E n ) = ∞ . n =1 It is slowly recurrent if also P ( E n ) → 0 . Mostly interested in sets with P ( E n ) ≍ 1 /n . 36 / 106
Examples of slowly recurrent sets ◮ A single point in Z 2 . ◮ Line or a half-line in Z 3 A = { ( j, 0 , 0) : j ∈ Z } , A + = { ( j, 0 , 0) : j ∈ Z + } . ◮ A simple random walk path A = S [0 , ∞ ) in Z 4 . ◮ A loop-erased walk A = ˆ S [0 , ∞ ) in Z 4 . ◮ The intersection of two simple random walk paths in Z 3 . 37 / 106
Basic Idea for Slowly Recurrent Sets E n = { X [ T n − 1 , T n ] ∩ A � = ∅} . V n = P { X [1 , T n ] ∩ A = ∅} = P ( E c 1 ∩ · · · ∩ E c n ) . n P ( E c � P ( V n ) = j | V j − 1 ) . j =1 ◮ Although P ( V n − 1 ) is small it is asymptotic to P ( V n − log n ) . Hence P ( E n | V n − 1 ) ∼ P ( E n | V n − log n ) . ◮ The distribution of X ( T n − 1 ) given V n − log n is almost the same as the unconditional distribution. Hence, P ( E n | V n − log n ) ∼ P ( E n ) . ◮ More precisely, find summable δ n such that P ( E n | V n − 1 ) = P ( E n ) + O ( δ n ) . 38 / 106
Suppose that P ( E j ) = α j j . Then, n P ( E c � P ( V n ) = j | V j − 1 ) j =1 n � � 1 − α j � = j + O ( δ j ) j =1 n α j � ∼ c exp − . j j =1 39 / 106
If X 1 , . . . , X k are independent simple random walks and V j n = { X j [1 , T j n ] ∩ A = ∅} , then n kα j n ) k ∼ c ′ exp P ( V 1 n ∩ · · · ∩ V k n ) = P ( V 1 � − . j j =1 40 / 106
Example: line A and half-line A + in Z 3 � 1 P ( E n ) = 1 � n + O , n 2 � 1 n ) = 1 � P ( E + 2 n + O n 2 P ( V n ) ∼ c n, n ) ∼ c ′ � P ( V + √ n ≍ P ( V n ) . 41 / 106
(Not quite precise) description of LERW in Z 4 ◮ ˆ S [0 , ∞ ) infinite LERW in Z 4 . ◮ Let Γ n be ˆ S [0 , ∞ ) from the first visit to {| z | > e n − 1 } to the first visit to {| z | > e n } (almost the same as LE ( S [ T n − 1 , T n )) . ◮ Let X t be an independent simple random walk and let K n be the event that X intersects Γ n . P ( K n ) = H (Γ n ) = Y n n , where Y n has a limit distribution. ◮ Let ( K 1 ∪ · · · ∪ K n ) c | ˆ � � Z n = P S . 42 / 106
◮ If the events K n were independent we would have n � � 1 − Y j � Z n = . j j =1 ◮ 4- d LERW has the same behavior as the toy problem where Y 1 , Y 2 . . . . are independent, nonnegative random variables (with an exponential moment). ◮ n � 1 − Y j − µ � � Z n = c n . j j =1 where µ = E [ Y j ] and n � � 1 − µ � ∼ C µ n − µ . c n = j j =1 43 / 106
◮ There exists a random variable Z such that with probability one n →∞ n µ Z n . Z = lim ◮ The convergence is in every L p . Indeed, � p n � 1 − Y j − µ Z p c p � = n n j j =1 n � 1 − p ( Y j − µ ) � c p � + O ( j − 2 ) = n j j =1 n � � 1 − p ( Y j − µ ) c n − pµ � ∼ . j j =1 44 / 106
Theorem (L-Sun-Wu) Let S, X be independent simple random walks starting at the origin in Z 4 and let ˆ S denote the loop-erasure of S . Let T n be the first time that X reaches {| z | ≥ e n } , and let Z n = P { X [1 , T n ] ∩ ˆ S [0 , ∞ ) = ∅ | S [0 , ∞ ) } . Then the limit n →∞ n 1 / 3 Z n Z = lim exists with probability one and in L p for all p . In particular, E [ Z p n ] ∼ c p n − p/ 3 . ◮ The third-moment estimate tells us that E [ Z 3 n ] ≍ n − 1 which allows us to determine the exponent 1 / 3 . 45 / 106
Combining with earlier results: ◮ S simple random walk in Z 4 with loop-erasure ˆ S . ◮ Define σ ( k ) = max { n : S ( n ) = ˆ S ( k ) } . That is, ˆ S ( k ) = S ( σ ( k )) . ◮ There exists c such that σ ( k ) ∼ c k (log k ) 1 / 3 . ◮ Let ˆ S ( tn ) W ( n ) = n (log n ) 1 / 3 , 0 ≤ t ≤ 1 . t � Then W ( n ) converges to a Brownian motion. ◮ For d ≥ 5 , holds without log correction. 46 / 106
Part 4 Two dimensions and conformal invariance ◮ Associate to each finite z = x + iy ∈ Z + i Z , S z , the closed square of side length 1 centered at z . ◮ If A ⊂ Z 2 , there is the associated domain � � � S z int . z ∈ A 47 / 106
◮ Take D ⊂ C a bounded (simply) connected domain containing the origin. ◮ For each N , let A N be the connected component of { z ∈ Z 2 : S z ⊂ ND } . containing the origin. If D is simply connected, then so is A . We write D N ⊂ D for the domain associated to N − 1 A N . ◮ If z, w ∈ ∂D are distinct, we write z N , w N for appropraite boundary points (edges) in ∂A N so that N − 1 z N ∼ z, N − 1 w N ∼ w . ◮ Take simple random walk from z N to w N in A N and erase loops. 48 / 106
Main questions ◮ Let η = [ η 0 , . . . , η n ] denote a loop-erased random walk from z N to w N in A N . ◮ Find fractal dimension d such that typically n ≍ N d . ◮ Consider the scaled path γ N ( t ) = N − 1 η ( tN d ) , 0 ≤ t ≤ n/N d . What measure on paths on D does this converge to? ◮ Reasonable to expect the limit to be conformally invariant: the limit of simple random walk is c.i. and “loop-erasing” seems conformally invariant since it depends only on the ordering of the points. 49 / 106
Possible approaches ◮ Start by trying to find d directly. ◮ Assume that the limit is conformally invariant and see what possible limits there are. Determine which one has to be LERW limit. Then try to justify it. ◮ Both techniques work and both use conformal invariance. ◮ We will first consider the direct method looking at the discrete process. 50 / 106
◮ If A is a finite, simply connected subset of Z + i Z containing the origin with corresponding domain D A , let f = f A be a conformal transformation from D A to the unit disk with f (0) = 0 . (Riemann mapping theorem) ◮ Associate to each boundary edge of ∂ e A , the corresponding point z on ∂D A which is the midpoint of the edge. ◮ Define θ z ∈ [0 , π ) by f ( z ) = e 2 iθ z ◮ The conformal radius of A (with respect to the origin) is defined to be r A (0) = | f ′ (0) | − 1 . It is comparable to dist(0 , ∂A ) (Koebe 1 / 4 -theorem) 51 / 106
Theorem (Beneˇ s-L-Viklund) There exists ˆ c, u > 0 such that if A is a finite simply connected subset of Z 2 and z, w ∈ ∂ e A , then the probability that loop-erased random walk from z to w in A goes through the origin is sin 3 | θ z − θ w | + O ( r − u c r − 3 / 4 � � ˆ A ) . A ◮ The constant ˆ c is lattice dependent and the proof does not determine it. We could give a value of u that works but we do not know the optimum value. ◮ The exponents 3 / 4 and 3 are universal. ◮ The estimate is uniform over all A with no smoothness assumptions on ∂A (this is important for application). ◮ A weaker version was proved by Kenyon (2000) and the proof uses an important idea from his paper. 52 / 106
◮ Let H A (0 , z ) be the Poisson kernel. ◮ H ∂A ( z, w ) the boundary Poisson kernel. This is also the total mass of the loop-erased measure. ◮ (Kozdron-L): H ∂A ( z, w ) = c ′ H A (0 , z ) H A (0 , w ) [1 + O ( r − u A )] . sin 2 ( θ z − θ w ) ◮ We prove that the ˆ p A measure of paths from z to w that go through the origin is asymptotic to � p A ( η ) ∼ ˆ η : z → w, 0 ∈ η c ∗ H A (0 , z ) H A (0 , w ) sin | θ z − θ w | r − 3 / 4 . A 53 / 106
Fomin’s identity (two path case) ◮ Let A be a bounded set and z 1 , w 1 , z 2 , w 2 distinct points on ∂A . Let H q ˆ q ( ω 1 ) q ( ω 2 ) , � A ( z 1 ↔ w 1 , z 2 ↔ w 2 ) = ω 1 ,ω 2 where the sum is over all paths ω j : z j → w j in A such that ω 2 ∩ LE ( ω 1 ) = ∅ . ◮ ˆ H q q ( η 1 ) q ( η 2 ) F q � A ( z 1 ↔ w 1 , z 2 ↔ w 2 ) = η ( A ) . η =( η 1 ,η 2 ) where the sum is over all nonintersecting pairs of SAWs η = ( η 1 , η 2 ) with η j : z j → w j . 54 / 106
Theorem (Fomin) H q ˆ A ( z 1 ↔ w 1 , z 2 ↔ w 2 ) − ˆ H q A ( z 1 ↔ w 2 , z 2 ↔ w 1 ) = H q A ( z 1 , w 1 ) H q A ( z 2 , w 2 ) − H q A ( z 1 , w 2 ) H q A ( z 2 , w 1 ) . ◮ Gives LERW quantities in terms of random walk quantities ◮ Generalization of Karlin-MacGregor formula for Markov chains. ◮ There is an n -path version giving a determinantal identity. ◮ If A is simply connected then at most one term on the left-hand side is nonzero. 55 / 106
◮ Consider a slightly different quantity � Λ A ( z, w ) = Λ A, + ( z, w ) + Λ A, − ( z, w ) = p A ( η ) ˆ η : z → w, 01 ∈ η where the sum is over all paths whose loop-erasure uses the edge � 01 or its reversal � 10 . ◮ Λ A, + ( z, w ) = 1 4 F 01 ( A ) ˆ H A ′ ( z ↔ 0 , w ↔ 1) , Λ A, − ( z, w ) = 1 4 F 01 ( A ) ˆ H A ′ ( z ↔ 1 , w ↔ 0) , where A ′ = A \ { 0 , 1 } . ◮ Fomin’s identity gives an expression for the difference of the right-hand side in terms of Poisson kernels. 56 / 106
Negative weights (zipper) ◮ Take a path (zipper) on the dual lattice starting at 1 2 − i 2 going to the right. ◮ Let q be the measure that equals p except if an edge crosses the zipper q ( n, n − i ) = − p ( n, n − i ) = − 1 4 , n > 0 . Λ q A ( z, w ) = Λ q A, + ( z, w ) + Λ q � A, − ( z, w ) = q A ( η ) ˆ η : z → w, 01 ∈ η A, + ( z, w ) = 1 Λ q 4 F q 01 ( A ) ˆ H q A ′ ( z ↔ 0 , w ↔ 1) , A, − ( z, w ) = 1 01 ( A ) ˆ Λ q 4 F q H q A ′ ( z ↔ 1 , w ↔ 0) , where A ′ = A \ { 0 , 1 } . 57 / 106
◮ Fomin’s identity gives � � q A ( η ) − ˆ q A ( η ) = ˆ η : z → w, � η : z → w, � 01 ∈ η 10 ∈ η 1 4 F q 01 [ H q A ′ ( z, 0) H q A ′ ( w, 1) − H q A ′ ( z, 1) H q A ′ ( w, 0)] . ◮ q A ( η ) = q ( η ) F q ˆ η ( A ) . ◮ Two topological facts: first, (with appropriate order of z, w ): � � 01 ∈ η p A ( η ) , q A ( η ) = 10 ∈ η. . � − p A ( η ) , 58 / 106
◮ Second, if ℓ is a loop then q ( ℓ ) = ± p ( ℓ ) where the sign is negative iff ℓ has odd winding number about 1 2 − i 2 . Any loop with odd winding number intersects every SAW from z to w in A using 01 . ◮ Therefore, F η ( A ) = F q η ( A ) exp { 2 m ( O A ) } , where O A is the set of loops in A with odd winding number about 1 2 − i 2 . 59 / 106
Main combinatorial Identity � � Λ A ( z, w ) = p A ( η ) + ˆ p A ( η ) ˆ η : z → w, � η : z → w, � 01 ∈ η 10 ∈ η � � = exp { 2 m ( O A ) } q A ( η ) − ˆ q A ( η ) ˆ η : z → w, � η : z → w, � 01 ∈ η 10 ∈ η = F q 01 ( A ) e 2 m ( O A ) × 4 [ H q A ′ ( z, 0) H q A ′ ( w, 1) − H q A ′ ( z, 1) H q A ′ ( w, 0)] . ◮ Here, A ′ = A \ { 0 , 1 } and O A is the set of loops in A with odd winding number about 1 − i 2 . ◮ m = m p is the usual random walk loop measure. 60 / 106
The proof then boils down to three estimates: ◮ F q 0 , 1 ( A ) = c 1 + O ( r − u A ) . ◮ m ( O A ) = log r A + c 2 + O ( r − u A ) . 8 e 2 m ( O A ) = c 3 r 1 / 4 � 1 + O ( r − u � A ) . A ◮ H q A ′ ( z, 0) H q A ′ ( w, 1) − H q A ′ ( z, 1) H q A ′ ( w, 0) = � � c 4 r − 1 | sin( θ z − θ w ) | + O ( r − u A H A (0 , z ) H A (0 , w ) A ) . ◮ The first one is easiest (although takes some argument). ◮ The others strongly use conformal invariance of Brownian motion. 61 / 106
Loops with odd winding number ◮ First consider A n = C n = {| z | < e n } . Let O n = O A n . ◮ O n \ O n − 1 is the set of loops in C n of odd winding number that are not contained in C n − 1 . Macroscopic loops. ◮ Consider Brownian loops in C n of odd winding number about the origin that do not lie in C n − 1 . The measure is independent of n (conformal invariance) and a calculation shows the value is 1 / 8 . ◮ Using coupling with random walk measure, show m ( O n ) − m ( O n − 1 ) = m ( O n \ O n − 1 ) = 1 8 + O ( e − un ) . 62 / 106
◮ m ( O n ) = n 8 + c 2 + O ( e − un ) . ◮ For more general A with e n ≤ r A ≤ e n +1 first approximate by C n − 4 and then attach the last piece. Uses strongly conformal invariance of Brownian measure. 63 / 106
◮ H q A ′ (0 , z ) = H A ′ (0 , z ) E [( − 1) J ] , where the expectation is with respect to an h -process from 0 to z in A ′ and J is the number of times the process crosses the zipper. ◮ Example: A = { x + iy : | x | , | y | < n } , z = − n, w = n . H q A ′ (0 , z ) is the measure of paths starting at 0 , leaving A at z , and not returning to the positive axis. ◮ Paths that return to the postive axis “from above” cancel with those that return “from below”. ◮ H q A ′ (0 , z ) ∼ c n − 1 / 2 . ◮ Combine this discrete cancellation with macroscropic comparisons to Brownian motion. 64 / 106
Part 5 Continuous limit: Schramm-Loewner evolution ( SLE ) ◮ Family of probability measures { µ D ( z, w ) } on simple curves γ : (0 , t γ ) → D from z to w in D . ◮ Supported on curves of fractal dimension 5 4 = 2 − 3 4 . . ◮ Suppose f : D → f ( D ) is a conformal transformation. Define f ◦ γ to be the image of γ parametrized so that the time to traverse f ( γ [ r, s ]) is � s r | f ′ ( γ ( t )) | 5 / 4 dt. 65 / 106
◮ Conformal invariance: f ◦ µ D ( z, w ) = µ f ( D ) ( f ( z ) , f ( w )) . ◮ Here f ◦ µ is the pull-back f ◦ µ ( V ) = µ { γ : f ◦ γ ∈ V } . 66 / 106
◮ Domain Markov property: in the probability measure µ D ( z, w ) , suppose that an initial segment γ [0 , t ] is observed. Then the distribution of the remainder of the path is µ D \ γ [0 ,t ] ( γ ( t ) , w ) . Figure: Domain Markov property (M. Jahangoshahi) 67 / 106
Theorem (Schramm, ...) There is a unique family of measures satisfying the above properties, the (chordal) Schramm-Loewner evolution with parameter 2 ( SLE 2 ) with natural parametrization. ◮ SLE κ exists for other values of κ but the curves have different fractal dimension. ◮ Schramm only considered simply connected domains. In general, extending to multiply connected is difficult but κ = 2 is special where it is more straightforward. 68 / 106
Definition of SLE 2 ◮ g t : H \ γ (0 , t ] → H g t γ (t) 0 Ut ◮ Reparametrize (by capacity) and then g t satisfies 1 ∂ t g t ( z ) = , g 0 ( z ) = z. g t ( z ) − U t where U t is a standard Brownian motion. ◮ Extend to simply connected domains by conformal invariance. For other domains use the (generalized) restriction property. 69 / 106
(Generalized) restriction property ◮ If D ⊂ D ′ , the Radon-Nikodym derivative dµ D ( z, w ) dµ D ′ ( z, w ) ( γ ) is proportional to e − L where L is the measure of loops in D ′ that intersect both γ and D ′ \ D . (Conformally invariant) w D �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� D’\D �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� γ �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� z 70 / 106
SLE Green’s function ◮ Suppose D is a simply connected domain containing the origin and γ : z → w is an SLE 2 path. ◮ There exists c ∗ such that P { dist(0 , γ ) ≤ r } ∼ c ∗ r 3 / 4 sin 3 | θ z − θ w | , r ↓ 0 . ◮ More generally for SLE κ with κ < 8 , κ − 1 | θ z − θ w | , P { dist(0 , γ ) ≤ r } ∼ c ∗ ( κ ) r 1 − κ 8 8 sin r ↓ 0 . 71 / 106
Parametrization ◮ The SLE path is parametrized by (half-plane) capacity so that g t ( z ) = z + 1 z + O ( | z | − 2 ) , z → ∞ . This is singular with respect to the “natural parametrization”. ◮ How does one parametrize a (5 / 4) -dimensional fractal curve? ◮ Hausdorff (5 / 4) -measure is zero. ◮ Hausdorff measure with “gauge function” might be possible but too difficult for SLE paths. 72 / 106
Minkowski content ◮ Let γ t = γ [0 , t ] . ◮ (L-Rezaei) With probability one, r ↓ 0 r − 3 / 4 Area( { z : dist( z, γ t ) ≤ r } ) Cont 5 / 4 ( γ t ) = lim exists, is continuous and strictly increasing in t . ◮ Natural parametrization: Cont 5 / 4 ( γ t ) = t. ◮ Chordal SLE with the natural parametrization is the measure on curves with properties described before. 73 / 106
Convergence result (L - Viklund) ◮ D a bounded, analytic domain containing the origin with distinct boundary points a, b . ◮ For each N , let A be the connected component containing the origin of all z ∈ Z 2 such that S z ⊂ N · D . where S z is the closed square centered at z of side length 1 . ◮ Let a N , b N ∈ ∂ e A N with a N /N → a, b N /N → b . ◮ Let µ N be the probability measure on paths obtained as follows: ◮ Take LERW from a N to b N in A N . Write such a path as η = [ a − , a + , η 2 , . . . , η k − 1 , b + , b − ] . 74 / 106
◮ Scale the path η by scaling space by N − 1 and time by c N − 5 / 4 . Use linear interpolation to make this a continuous path. This defines the probability measure µ N . ◮ Define a metric ρ ( γ 1 , γ 2 ) on paths γ j : [ s j , t j ] → C , � � | γ 2 ( α ( t )) − γ 1 ( t ) | inf sup | α ( t ) − t | + sup . s 1 ≤ t ≤ t 1 s 1 ≤ t ≤ t 1 where the infimum is over all increasing homeomorphisms α : [ s 1 , t 1 ] → [ s 2 , t 2 ] . ◮ Let p denote the corresponding Prokhorov metric. 75 / 106
Theorem (L-Viklund) As n → ∞ , µ N → µ in the Prokhorov metric. ◮ Convergence for curves modulo parametrization (and in capacity parametrization) was proved by L-Schramm-Werner. ◮ The new part is the convergence in the natural parametrization. 76 / 106
Part 6 Two-sided loop-erased random walk ◮ The infinite two-sided loop-erased random walk (two-sided LERW) is the limit measure of the “middle” of a LERW. ◮ Probability measure on pairs of nonintersecting infinite self-avoiding starting at the origin. ◮ Straightforward to construct if d ≥ 5 . ◮ This construction can be adapted for d = 4 using results of L-Sun-Wu. It will not work for d = 2 , 3 . ◮ New result constructs the process for d = 2 and d = 3 . 77 / 106
Constructing two-sided LERW for d ≥ 4 ◮ Start with independent simple random walks starting at the origin S, X . ◮ Erase loops from S giving the (one-sided) LERW ˆ S [0 , ∞ ) . Reverse time so that it goes from time −∞ to 0 . ◮ Tilt the measure on ˆ S by ˜ Z := Z/ E [ Z ] , where Z = P { X [1 , ∞ ) ∩ ˆ S [0 , ∞ ) = ∅ | ˆ S } , d ≥ 5 , n →∞ n 1 / 3 P { X [1 , T n ] ∩ ˆ S [0 , ∞ ) = ∅ | ˆ Z = lim S } , d = 4 . ◮ If d ≥ 5 , ˜ Z is bounded. If d = 4 , it is not bounded but has all moments. 78 / 106
◮ Given ˆ S , choose X as random walk conditioned to avoid ˆ S [0 , ∞ ) . For d = 4 , one does an h -process with harmonic function n →∞ n 1 / 3 P x { X [1 , T n ] ∩ ˆ Z x = lim S [0 , ∞ ) = ∅} . ◮ Erase loops from X to give the “future” of the two-sided LERW. ◮ Uses reversibility of (the distribution of) LERW. ◮ If d < 4 , the marginal distribution of one path is not absolutely continuous with respect to one-sided measure so this does not work. 79 / 106
Notation ◮ C n = { z ∈ Z d : | z | < e n } . ◮ W n is the set of SAWs η starting at the origin, ending in ∂C n and otherwise in C n . ◮ A n is the set of ordered pairs η = ( η 1 , η 2 ) ∈ W 2 n such that η 1 ∩ η 2 = { 0 } . ◮ A n ( a, b ) is the set of such η such that η 1 ends at a and η 2 ends at b . ◮ By considering ( η 1 ) R ⊕ η 2 , we see there is a natural bijection between A n ( a, b ) and the set of SAWs from a to b in C n that go through the origin. 80 / 106
◮ Similarly, we can define A n ( a, b ; A ) for SAWs from a to b in A . ◮ The loop-erased measure on A n ( a, b ; A ) is the measure p A ( η ) = p ( η ) F η ( A ) = (2 d ) −| η | F η ( A ) . ˆ Can normalize to make it a probability measure. Same probability measure if we use A ) = (2 d ) −| η | F η ( ˆ p A ( η ) = p ( η ) F η ( ˆ ˆ ˆ A ) , A = A \ { 0 } . ◮ If C k ⊂ A , then this measure induces a probability measure P A,a → b,k on A k . 81 / 106
Theorem p k on A k such For each k , there exists a probability measure ˆ that if k < n , C n ⊂ A , a, b ∈ ∂ e A with A ( a, b ; A ) nonempty, then for all η ∈ A k , � � 1 + O ( e u ( k − n ) ) P A,a → b,k ( η ) = ˆ p k ( η ) . More precisely, there exist c, u such that for all such k, A, a, b and all η ∈ A k , � � P A,a → b,k ( η ) �� � � � ≤ c e u ( k − n ) . � log � � � p k ( η ) ˆ � ◮ The measures ˆ p k are easily seen to be consistent and this gives the two-sided LERW. 82 / 106
Slightly different setup ◮ Let η 1 , η 2 be independent infinite LERW stopped when then reach ∂C n . This gives a measure µ n × µ n on W 2 n . µ n ( η ) = (2 d ) −| η | F η (ˆ Z d ) Es η ( z ) , where z is the endpoint of η . ◮ Note This is not the same as ”stop a simple random walk when it reaches ∂C n and then erase loops” which would give measure (2 d ) −| η | F η ( C n ) . ◮ Given η j , the remainder of the infinite LERW walk is obtained by: ◮ Take simple random walk starting at the end of η j conditioned to never return to η j ◮ Erase loops. 83 / 106
Tilted measure ν n ◮ Obtain ν n by tilting µ n × µ n by 1 { η ∈ A n } exp {− L n ( η ) } , where L n = L n ( η ) is the loop measure of loops in ˆ C n that intersect both η 1 and η 2 . ◮ This is to compensate for “double counting” of loop terms. ◮ If d = 2 , restrict to loops that do not disconnect 0 from ∂C n (any disconnecting loop intersects all η 1 , η 2 and hence does not contribute to the probability measure). ◮ If C n +1 ⊂ A , then P A,a → b,n ≪ ν # n . 84 / 106
SLE analogue (L- Kozdron, Lind, Werness, Jahangoshahi, Healey,...) ◮ Natural measure on multiple SLE κ paths κ ≤ 4 can be obtained from starting with k independent SLE κ paths γ = ( γ 1 , . . . , γ k ) and tilting by k c = (3 κ − 8)(6 − κ ) c � Y ( γ ) = I exp L j , , 2 2 κ j =2 where L j is the Brownian loop measure of loops that hit at least j of the paths and I is the indicator that the paths are disjoint. ◮ The case k = 2 is sometimes called two-sided radial SLE κ . The scaling limit of two-sided LERW in Z 2 is two-sided SLE 2 . 85 / 106
Coupling ◮ Let γ ∈ A k and ν # n ( · | γ ) the conditional distribution given that the initial configuration is γ . ◮ Challenge: Couple ν # n and ν # n ( · | γ ) so that, except for an event of probability O ( e − u ( n − k ) ) , the paths agree from their first visit to C k +( n − k ) / 2 onward. ◮ Given this, ν n +1 ( A n +1 ) = ν n +1 ( A n +1 ; γ ) � � 1 + O ( e − u ( n − k ) ) ν n ( A n ) ν n ( A n ; γ ) ν n +1 ( A n +1 ; γ ) = ν n ( A n ; γ ) � 1 + O ( e − u ( n − k ) ) � ν n +1 ( A n +1 ) ν n ( A n ) 86 / 106
◮ Fix (large) n and γ k , ˜ γ k ∈ A k with k < n . ◮ Couple Markov chains γ k , γ k +1 , . . . , γ n and γ k , ˜ ˜ γ k +1 , . . . , ˜ γ n so they have the distribution of the beginning of the paths under ν # n . ◮ Write γ j = r ˜ γ j if the paths agree from their first visit to ∂C j − r to ∂C j . ◮ Suppose we can show the following: ◮ For every j < ∞ can find ρ j > 0 such that given any ( γ k , ˜ γ k ) we can couple so that with probability at least ρ j , γ k + j = j − 2 ˜ γ k + j . ◮ If γ k = j ˜ γ k , then we can couple the next step such that, except perhaps on an event of probability O ( e − βj ) , γ k +1 = j +1 ˜ γ k +1 . 87 / 106
◮ Then there exists c, u such that for any γ k , ˜ γ k , γ n } ≥ 1 − ce − u ( n − k ) . P { γ n = ( n − k ) / 2 ˜ ◮ Does not give a good estimate on u . ◮ Same basic strategy used for other problems, e..g, the measure of Brownian motion “at a random cut point”. ◮ The hard work is showing that the conditions on previous slide hold. ◮ We discuss some of the ingredients of the proof. 88 / 106
“Obvious” fact about simple random walk ◮ Let η ∈ W n and S a simple random walk starting at z , the endpoint of η . ◮ Let τ = τ r = min { j : | S j − z ] ≥ r } ◮ Lemma: there exists uniform ρ > 0 such that P {| S τ | ≥ e n + r 3 | S [1 , τ ] ∩ η = ∅} ≥ ρ. ◮ If there were no conditioning this would follow from central limit theorem. Conditioning should only increase the probability so it is“obvious”. ◮ Important to know that there exists ρ that works for all n, η, r . ◮ Various versions have been proved by L, Masson, Shiraishi ◮ Brownian motion version is easier — then careful approximation of BM by random walk. 89 / 106
◮ Corollary: the probability that simple random walk starting at z conditioned to avoid η enters C n − k is less than c e − k . ◮ This obviously holds for the loop-erasure as well. ◮ For d ≥ 3 we use transience of the simple walk: the probability that a RW starting outside C n reaches C n − k is O ( e ( d − 2)( k − n ) ) . ◮ For d = 2 we use the Beurling estimate (Kesten). The probability a random walk starting at C n reaches C n − k and then returns to ∂C n without intersecting η is O ( e k − n ) . ◮ One of the reasons to use “infinite LERW when it reaches ∂C n ” rather than “loop erasure of RW when it reaches ∂C n ” is to use this fact. 90 / 106
Estimating loop measure ◮ If d ≥ 3 , the loop measure of loops that intersect both ∂C n and C n − k is O ( e ( d − 2)( k − n ) ) . ◮ If d = 2 , the loop measure of loops that intersect both ∂C n and C n − k and do not disconnect the origin from ∂C n is O ( e ( k − n ) / 2 ) . ◮ This uses the disconnection exponent for d = 2 RW: the probability that a RW starting next to the origin reaches C n without disconnecting the origin is comparable to O ( e − n/ 4 ) . (L-Puckette, L-Schramm-Werner) ◮ For d = 2 focus on nondisconnecting loops. Loops that disconnect intersect all SAWs and hence do not affect the normalized probability measure on SAWs weighted by a loop term. 91 / 106
Separation lemma ◮ Let η = ( η 1 , η 2 ) ∈ A n . ◮ Consider all ˜ η 1 , ˜ η 2 ) ∈ A n +1 that extend η . η = (˜ ◮ If we tilt by the loop term e − L n +1 there is a positive probability ρ (independent of n , η ) that the endpoints of ˜ η are separated. ◮ First proved for nonintersecting Brownian motions. ◮ An analogue of (parabolic) boundary Harnack principle — if one conditions a Brownian motion to stay in a domain for a while, then the path gets away from the boundary. ◮ This is a key step in coupling η n , γ n with positive probability. ◮ There is also a version for LERW in A from x to y (in ∂A ) conditioned to go through the origin. 92 / 106
Original theorem ◮ Let A ⊃ C n +1 and x, y distinct boundary points. ◮ Consider LERW from x to y in A conditioned so that paths go through the origin. ◮ Let λ A = λ A,x,y be the probability measure obtained by truncating to paths ∈ A n . Consider Y ( η ) = dλ A ( η ) . dλ n ◮ The distribution of Y depends on A, x, y ; however ◮ Y is uniformly bounded. ◮ If η = k γ , then Y ( η ) = Y ( γ ) + O ( e − k/ 2 ) . 93 / 106
Part 7 The distribution in Z 2 = Z + i Z (with F. Viklund, C. Beneˇ s) ◮ The distribution of two-sided LERW for d = 2 is closely related to potential theory with zipper (signed weights) or in double covering of Z 2 . q ( e ) = − p ( e ) = − 1 4 , e = { x − i, x } , x > 0 , q ( e ) = p ( e ) = 1 4 , other e . ◮ ∆ denotes the usual random walk Laplacian and ∆ q the corresponding operator for q : �� � ∆ q f ( z ) = − f ( z ) . q ( z, w ) f ( w ) w 94 / 106
Fundamental solutions ◮ The fundamental solution a ( z ) of ∆ is the potential kernel which is a discrete harmonic approximation of log | z | . ◮ The fundamental solutions of ∆ q are discrete q -harmonic approximations of real and imaginary parts of √ z : ◮ Let S be a simple random walk, σ R = min { j : | S j | ≥ R } , τ + = min { j ≥ 0 : S j ∈ { 0 , 1 , 2 , . . . }} . R →∞ R 1 / 2 P z { σ R < τ + } . u ( z ) = lim 95 / 106
u ( z ) = 0 , z ∈ { 0 , 1 , 2 , 3 , . . . } , u ( x + iy ) = u ( x − iy ) , ∆ u ( z ) = 0 , z �∈ { 0 , 1 , 2 , . . . } , ∆ q u ( z ) = 0 , z � = 0 . ◮ If f ( z ) = | z | 1 / 2 sin( θ z / 2) , then � u ( z ) − 4 � ≤ c f ( z ) � � � � π f ( z ) � � | z | ◮ Define the “conjugate” function v by v ( − x + iy ) = ± u ( x + iy ) , where the sign is chosen to be negative on { Im( z ) < 0 } . 96 / 106
z ∈ { 0 , − 1 , − 2 , − 3 , . . . } , v ( z ) = 0 , v ( x + iy ) = − v ( x − iy ) , y > 0 , ∆ v ( z ) = 0 , z �∈ { 0 , − 1 , − 2 , . . . } , ∆ q v ( z ) = 0 , z � = 0 . ◮ If g ( z ) = | z | 1 / 2 cos( θ z / 2) , then � v ( z ) − 4 � ≤ c | g ( z ) | � � � π g ( z ) � � � | z | 97 / 106
◮ If η is a finite set of vertices containing the origin, � a η ( z ) = a ( z ) − H Z 2 \ η ( z, w ) a ( w ) . w ∈ η ◮ H Z 2 \ η ( z, w ) is the Poisson kernel � H Z 2 \ η ( z, w ) = p ( ω ) , ω : z → w where the sum is over all nearest neighbor paths from z to w , otherwise in Z 2 \ η . ◮ Then a η satisfies a η ≡ 0 on η and ∆ a η ( z ) = 0 , z �∈ η, a η ( z ) ∼ 2 π log | z | , z → ∞ . 98 / 106
◮ H q � u η ( z ) = u ( z ) − Z 2 \ η ( z, w ) u ( w ) . w ∈ η H q � v η ( z ) = v ( z ) − Z 2 \ η ( z, w ) v ( w ) . w ∈ η ◮ H q Z 2 \ η ( z, w ) is the Poisson kernel H q � Z 2 \ η ( z, w ) = q ( ω ) , ω : z → w where the sum is over all nearest neighbor paths from z to w , otherwise in Z 2 \ η . ◮ Then u η , v η satisfies u η , v η ≡ 0 on η and ∆ q u η ( z ) = ∆ q v η ( z ) = 0 , z �∈ η, u η ( z ) = u ( z ) + o (1) , v η ( z ) = v ( z ) + o (1) , z → ∞ . 99 / 106
Let η = [ η 0 = 0 , η 1 = 1 , . . . , η k ] be a SAW starting with [0 , 1] . ◮ The probability that the one-sided LERW traverses η is 4 − k F η (ˆ Z 2 ) ∆ a η ( η k ) . ◮ There exists c such that the probability that the two-sided LERW traverses η is p ( η ) := c 4 − k F q η (ˆ Z 2 ) det M η , ˆ � ∆ q v η ( η k ) ∆ q u η (0) � M η = . ∆ q u η ( η k ) ∆ q v η (0) ◮ Follows from Fomin’s identity using the weight q (as done in BLV) and being able to take the limit (using the recent result). 100 / 106
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