LARGE DEVIATIONS FOR THE WIENER SAUSAGE
Frank den Hollander Leiden University, The Netherlands
Large Deviation Theory in Statistical Physics, ICTS Bangalore, 30 + 31 August 2017 § OUTLINE
The focus in these pedagogical lectures is on the Wiener sausage:
[ Wt = B1(βs), t ≥ 0. s∈[0,t]
d Here,( βs)s≥0 is Brownian motion on R , and B1(x) is the d closed ball of radius1 centred at x ∈ R .
Lecture 1: Basic facts, LLN, CLT, LDP
Lecture 2: Sketch of the proof of the LDP: skeleton approach LECTURE 1
Basic facts, LLN, CLT, LDP § WIENER SAUSAGE
d Let β = (βs)s≥0 be Brownian motion on R , i.e., the continuous-time Markov process with generator∆ (the Laplace operator). Write Px to denote the law of β starting from x, and put P = P0.
The Wiener sausage at time t is the random set [ Wt = B1(βs), t ≥ 0, s∈[0,t] i.e., the Brownian motion drags around a ball of radius1, which traces out a sausage-like environment.
Mark Kac 1974 The Wiener sausage is an important object because it is one of the simplest examples of a non-Markovian functional of Brownian motion. It plays a key role in the study of various stochastic phenomena:
– heat conduction – trapping in random media – spectral properties of random Schr¨odingeroperators – Bose-Einstein condensation
We will look at two specific quantities:
Vt = Vol(Wt), Ct = Cap(Wt).
Here, Vol stands for volume and Cap for capacity. Only the case d ≥ 2 is interesting. I The volume plays a role in trapping phenomena. Write d PPPα to denote the law of a Poisson Point Process on R with intensity α ∈ (0, ∞). Place balls of radius1 around the points. Let τ be the first time that β hits a ball. Then −αV PPPα(τ > t) = e t, t ≥ 0.
The capacity plays a role in hitting phenomena. Let¯ I τWt be the first time an auxiliary (!) Brownian motion β¯ hits Wt. Then C lim d−2 ¯ ¯ = t 0 3 |x| Ex τWt < ∞ , t ≥ , d ≥ , |x|→∞ κd with κd = Cap(B1(0)). Other set functions are of interest too:
perimeter, moment of inertia, principal Dirichlet eigenvalue, heat content, torsional rigidity.
Many have the property that they are either minimal or maximal when the set is a ball, which is analytically helpful.
In what follows we recall some basic facts, and state LLNs and CLTs. After that we focus on LDPs. It will turn out that there is a remarkable dependence on the dimension d.
NOTE: By Brownian scaling, it is trivial to transfer all the properties to be described below to the Wiener sausage with radius r ∈ (0, ∞) rather than r = 1. § VOLUME
The following strong LLN holds: 1 d ≥ 3: limt→∞ t Vt = κd P − a.s. Spitzer 1964 where 4πd/2 κ = Cap(B (0)) = . d 1 d−2 Γ( 2 )
The existence of the limit follows from the subaddivity property
d Vol(A ∪ B) ≤ Vol(A) + Vol(B) ∀A, B ⊂ R Borel. The identification of the limit requires potential theory. The case d = 2 is critical: log t limt→∞ t Vt = κ2 = 4π P − a.s. Le Gall 1988
The CLT holds with t2/ log4 t, d = 2, Var(Vt) t log t, d = 3, t, d ≥ 4, Spitzer 1964, Le Gall 1988
The limit law is Gaussian for d ≥ 3 and non-Gaussian for d = 2. § CAPACITY
d The (Newtonian) capacity of a Borel set A ⊂ R can be defined through the variational formula
1 Z Z = inf µ(dx) µ(dy) Gd(x, y), Cap(A) µ∈P(A) A A where P(A) is the set of probability measures on A, and 1 G (x, y) = , x, y ∈ d, d d−2 R κd|x − y| is the Green function. Think of the above integral as the electrostatic energy when the set A is a conductor with a unit charge.
A different formula is needed for the case d = 2, which we will not consider. The following strong LLN holds: 1 d ≥ 5: limt→∞ t Ct = cd P − a.s. Asselah, Schapira, Sousi 2017
The existence of the limit follows from the subaddivity property
d Cap(A ∪ B) ≤ Cap(A) + Cap(B) ∀A, B ⊂ R Borel.
The fact that the limit is non-degenerate requires some estimates. No explicit formula is known for cd. The case d = 4 is critical: log t 2 limt→∞ t Ct = c4 = 8π P − a.s. Asselah, Schapira, Sousi 2016
The CLT is expected (!) to hold with t2/ log4 t, d = 4, Var(Ct) t log t, d = 5, t, d ≥ 6, Asselah, Schapira, Sousi 2016 – discrete setting
The limit law is expected (!) to be Gaussian for d ≥ 5 and non-Gaussian for d = 4.
The cases d = 2, 3 are interesting too, but are less well understood. We will not consider them. § LARGE DEVIATIONS FOR THE VOLUME
THEOREM 1 Van den Berg, Bolthausen, dH, 2001 + 2004
Let d ≥ 3. For every b > 0,
1 ↓ lim log P(Vt ≤ bt) = −I (b), t→∞ t(d−2)/d d where Z I↓(b) = inf |∇φ|2(x) dx d d φ∈Φd(b) R with Z 1 d 2 Φd(b) = φ ∈ H (R ): φ (x) dx = 1, Rd Z 2 1 − e−κdφ (x) dx ≤ b , Rd HEURISTICS:
The idea is that the optimal strategy for β to realise the event {Vt ≤ bt} is to behave like a Brownian motion in a drift field 1/d t x 7→ (∇φ/φ)(x) for some φ ∈ Φd(b). The cost of adopting this drift during a time t is (d−2)/d 2 exp[t k∇φk2], to leading order. Conditional on adopting this drift, the path spends time tφ2(x)dx in the volume element t1/ddx and the Wiener sausage covers a fraction 2 1 − e−κdφ (x) of this volume element. CONCLUSION:
The optimal strategy for the Wiener Sausage to achieve a downward large deviation of its volume is to look like a Swiss cheese! THEOREM 2 Van den Berg, Bolthausen, dH, 2001 + 2004
Let d = 2. For every b > 0,
1 ↓ lim log P(Vt ≤ bt/ log t) = −I (b), t→∞ log t 2 where Z I↓(b) = inf |∇φ|2(x) dx 2 2 φ∈Φ2(b) R with Z Φ (b) = φ ∈ H1( 4): φ(x)2 dx = 1, 2 R 2 R Z 2 1 − e−κ2φ(x) dx ≤ b . 2 R NOTE:
The cost of a downward large deviation in the critical dimension d = 2 is polynomial rather than stretched exponential. § RATE FUNCTIONS
By Brownian scaling, the variational formula for the rate function can be standardised, namely, 1 I↓(b) = χ (b/κ ), b > 0, d 2/d d d κd where
2 1 d χd(u) = inf k∇ψk2 : ψ ∈ H (R ), Z −ψ2 kψk2 = 1, (1 − e ) ≤ u . Rd
The function χd is continuous on (0, ∞), strictly decreasing on (0, 1), and equal to zero on [1, ∞). If a minimiser exists, then it is unique modulo translations, radially symmetric and radially strictly decreasing. Moreover, 2/d lim u χd(u) = −λd, u↓0 with λd ∈ (0, ∞) the principal Dirichlet eigenvalue of −∆ on B1(0).
The latter corresponds to the regime where Vt = o(E(Vt)) and the Swiss cheese squeezes out its holes to becomes a ball. This regime was studied earlier and is much easier to handle. Donsker, Varadhan 1975 Bolthausen 1990 + 1994 Sznitman 1990 Qualitative picture of u 7→ χd(u):
d = 2 d = 3, 4 d ≥ 5
•
0 1 0 1 0 ud 1
For d ≥ 5, there is a critical value ud ∈ (0, 1) above which the variational formula has no minimiser: mass leaks a way to infinity.
The optimal strategy is time-inhomogeneous: √ partly on scale t1/d, partly on scale t. § WHAT ABOUT UPWARD LARGE DEVIATIONS?
Upward large deviations are more costly. For d ≥ 3, 1 ↑ limt→∞ t log P(Vt ≥ bt) = −Id(b), b > 0, Hamana, Kesten 2001 with ↑ Id(b) > 0 ∀ b > κd. Upward large deviations are much harder to capture than downward large deviations: they require a local dilation of the Wiener sausage essentially everywhere.
↑ No variational formula is known for Id. Only bounds are available via estimates on exponential moments. van den Berg, T´oth1991 van den Berg, Bolthausen 1994 § LARGE DEVIATIONS FOR THE CAPACITY
THEOREM 3 Van den Berg, Bolthausen, dH, in progress
Let d ≥ 5. For every b > 0,
1 ↓ lim log P(Ct ≤ bt) = −I (b), t→∞ t(d−4)/(d−2) d where Z I↓(b) = inf |∇φ|2(x) dx d d φ∈Φd(b) R with Z 1 d 2 Φd(b) = φ ∈ H (R ): φ (x) dx = 1, Rd Z 2 φ (x)[ cduφ(x)] dx ≤ b , Rd where uφ is the unique solution of the equation
2 ! 1 (d−2)/d ∆u = Adcd φ u, u(∞) ≡ 1,Ad = d ωd , 2κd with πd/2 ω = Vol(B (0)) = . d 1 d+2 Γ( 2 )
Note that the function uφ is the solution of the 2 Schr¨odingerequation with potential −Adcd φ and with boundary condition1 at infinity. HEURISTICS:
The idea is that the optimal strategy for β to realise the event {Ct ≤ bt} is to behave like a Brownian motion in a drift field 1/(d−2) t x 7→ (∇φ/φ)(x) for some φ ∈ Φd(b). The cost of adopting this drift during a time t is (d−4)/(d−2) 2 exp[t k∇φk2], to leading order. Conditional on adopting this drift, the path spends time tφ2(x)dx in the volume element t1/(d−2)dx and the capacity associated with the Wiener sausage in this volume element is 2 uφ(x)cd [tφ (x)dx]. In the latter expression:
¯ – cd is the probability that β escapes locally from Wt, i.e., moves out of the volume element t1/(d−2)dx without hitting Wt.
– uφ(x) is the probability that β escapes globally from Wt, i.e., moves to infinity without hitting the part of 1/(d−2) Wt that lies outside the volume element t dx.
For capacity both local and global properties control the downward large deviations. CONCLUSION:
The optimal strategy for the Wiener Sausage to achieve a downward large deviation of its capacity is to look like thin Italian spaghetti! THEOREM 4 Van den Berg, Bolthausen, dH, in progress
Let d = 4. For every b > 0,
1 ↓ lim log P(Ct ≤ bt/ log t) = −I (b), t→∞ log t 4 where Z I↓(b) = inf |∇φ|2(x) dx 4 4 φ∈Φ4(b) R with Z Φ (b) = φ ∈ H1( 4): φ(x)2 dx = 1, 4 R 4 R Z φ(x)2 [c u (x)] dx ≤ b , 4 4 φ R where uφ is the unique solution of the equation
2 2 1/2 ∆u = A4c4 φ u, u(∞) ≡ 1,A4 = ω4 . κ4
The cost of a downward large deviation in the critical dimension d = 4 is polynomial rather than stretched exponential. § RATE FUNCTIONS By Brownian scaling, we get 1 I↓(b) = χ (b/c ), b > 0, d 2/(d−2) d d (Adcd) where
2 1 d χd(u) = inf k∇ψk2 : ψ ∈ H (R ), kψk = 1, R ψ2u ≤ u 2 Rd ψ 2 with uψ the solution of∆ u = ψ u, u(∞) ≡ 1.
Qualitatively, the behaviour of χd should be similar to that for the volume, but we have nothing (!) to offer yet.
We expect that upward large deviations are exponentially costly, but no results are yet available. LECTURE 2
Sketch of the proof of the LDP: skeleton approach § UPPER BOUND
We prove the upper bound for the volume:
1 ↓ lim sup log P(Vt ≤ bt) ≤ −Id(b), b > 0, d ≥ 3. t→∞ t(d−2)/d
The lower bound follows after an easy modification, as we explain later. Afterwards we consider the capacity.
The proof comes in 6 steps:
(1) Compactification (2) Excursion decomposition (3) Concentration (4) Donsker-Varadhan LDP (5) Short excursion limit (6) Decompactification (1) COMPACTIFICATION
N N d Pick N > 0, and letΛ N = [− 2 , 2 ] . Clearly,
Vol(WN,t) ≤ Vol(Wt) with WN,t the Wiener sausage of the Brownian motion 1/d wrapped around t ΛN .
By Brownian scaling,
−1 law τ t Vol(WN,t) = Vol(WN,τ ) with τ WN,τ = ∪s∈[0,τ]Brτ (βN,s), where (d−2)/d −1/(d−2) τ = t , rτ = τ , and βN is Brownian motion wrapped aroundΛ N . Hence, τ putting VN,τ = Vol(WN,τ ), we have
P(Vt ≤ bt) ≤ PN (VN,τ ≤ b)
where PN is the law of βN starting from0.
The right-hand side involves the Wiener sausage onΛ N with a radius rτ that shrinks with τ. The desired upper bound follows from the following two propositions.
PROPOSITION 1
Let d ≥ 3. For every b > 0,
1 ↓ lim log PN (VN,τ ≤ b) = −I (b) τ→∞ τ d,N ↓ ↓ with Id,N (b) given by the same variational formula as Id(b), d except that R is replaced by ΛN with a periodic boundary.
PROPOSITION 2
↓ ↓ limN→∞ Id,N (b) = Id(b) for every b > 0. (2) EXCURSION DECOMPOSITION
To prove Proposition 1, we fix N > 0 and suppress it from the notation. Decompose β into excursions of length > 0. Let
τ/ Xτ, = {βi}i=1
denote the endpoints of the excursions, which we refer to as the skeleton. In what follows we will write
Eτ,(Vτ ) = E(Vτ | Xτ,) to denote the average of Vτ conditional on Xτ,. Let τ/ 1 X Lτ, = δ (β(i−1),βi) τ/ i=1 denote the empirical pair distribution of the skeleton. Let Wi denote the Wiener sausage associated with the i-th excursion. We have τ/ Z [ Eτ,(Vτ ) = dx Pτ, x ∈ Wi Λ N i=1 τ/ ! Z Y h i = dx 1 − 1 − Pτ,(x ∈ Wi) Λ N i=1 Z τ Z = dx 1 − exp ΛN ΛN ×ΛN log[1 − qτ,(y − x, z − x)] Lτ,(dy, dz) ,
(∗) where q (y, z) = τ ≤ τ, Py,z Brτ (0) with Py,z the law of the Brownian Bridge of length starting at y and ending at z.
Through( ∗), we have written Eτ,(Vτ ) as an exponential functional of the random variable Lτ,. This functional also involves the Brownian Bridge function qτ,, which for large τ is small.
y- - z
Brτ (0)
0 LEMMA 1 For every δ > 0, 1 lim lim sup log P |Vτ − Eτ,(Vτ )| ≥ δ = −∞. ↓0 τ→∞ τ PROOF: The proof uses concentration of measure. The τ/ key observation is that Vt = Vol(∪i=1Wi) is a Lipschitz function of the Wi’s, which are independent under the law Pτ,. Talagrand 1995 (4) DONSKER-VARADHAN LDP By the DV-LDP for the empirical pair distribution of a uniformly ergodic Markov chain,( Lτ,)τ>0 satisfies the LDP (2) on P(ΛN × ΛN ) with rate τ and with rate function I / given by ( (2) h(µ | µ ⊗ π ), if µ = µ , I (µ) = 1 1 2 ∞, otherwise. Here, h(· | ·) denotes relative entropy, µ1 and µ2 are the left- and right-marginal of µ, and π(x, dy) = p(y − x)dy is the time-τ Brownian transition kernel onΛ N . Donsker, Varadhan 1975 We can apply this LDP to( ∗) to obtain the following. LEMMA 2 (Eτ,(Vτ ))τ>0 satisfies the LDP on [0, ∞) with rate τ and with rate function J given by 1 (2) J(b) = inf I (µ): µ ∈ P(Λ × Λ ), Φ(µ) = b , N N where Z κd Z Φ(µ) = dx 1 − exp − ΛN ΛN ×ΛN ϕ(y − x, z − x) µ(dy, dz) with R 0 ds ps(−y)p−s(z) ϕ(y, z) = . p(z − y) PROOF: The key observation is the following fact: lim |τq (y, z) − κ ϕ (y, z)| = 0 τ→∞ τ, d ‘uniformly’ in y, z ∈ ΛN . Inserting this into( ∗), and using that lim q (y, z) = 0 τ→∞ τ, ‘uniformly’ in y, z ∈ ΛN to linearise the logarithm, we deduce the claim with the help of the contraction principle. Some technicalities arise in order to deal with y, z that are either close to or far away from the origin on scale rτ . (5) SHORT EXCURSION LIMIT Let (1) (2) I (ν) = inf{I (µ): µ1 = ν} (2) be the projection of I onto P(ΛN ). LEMMA 3 For every ν ∈ P(ΛN ), 1 (1) lim I (ν) = I(ν), ↓0 where ( R |∇φ|2(x) dx, if dν = φ2 with φ ∈ H1(Λ ), I(ν) = ΛN dx N ∞, otherwise. PROOF: The representation formula (1) Z (πu)(x) I (ν) = − inf ν(dx) log 0 The limit can be identified by looking at the behaviour of π, the time-τ Brownian transition kernel onΛ N , as ↓ 0. Donsker, Varadhan 1975 I is the large deviation rate function for the empirical distribution of Brownian motion onΛ N . For ν ∈ P(ΛN ), define Z κd Z Z Ψ(ν) = dx 1 − exp − ds ps(x − y)ν(dy) . ΛN 0 ΛN LEMMA 4 lim sup |Φ(µ) − Ψ(µ1)| = 0. ↓0 1 (2) ! µ∈P(ΛN ×ΛN ): I (µ)< ∞ PROOF: Estimate for µ ∈ P(ΛN × ΛN ), |Φ(µ) − Ψ(µ1)| = |Φ(µ) − Φ(µ1 ⊗ π)| ≤ κakµ − µ1 ⊗ πkTV withTV the total variation norm. The claim follows from Pinsker’s inequality: q (2) kµ − µ1 ⊗ πkTV ≤ 8 I (µ). LEMMA 5 Define Z −κ f(x) Γ(f) = dx 1 − e d , f ∈ L1(ΛN ). ΛN Then dν lim sup Γ − Ψ(ν) = 0, ↓0 1 (1) ! dx ν∈P(ΛN ): I (ν)< ∞ PROOF: Estimate for ν ∈ P(ΛN ), Z Z dν κd d(ν ⊗ πs) dν Γ − Ψ(ν) ≤ dx ds − (x) dx ΛN 0 dx dx κd Z = ds kν ⊗ πs − νkTV. 0 The claim again follows from Pinsker’s inequality. We are now ready to prove Proposition 1. PROOF: For any test function f : [0, ∞) → R that is bounded and continuous, we have, by Lemmas 1-5, lim 1 log exp [τf(V )] τ→∞ τ E τ L1 1 h i = lim lim log E exp τf Eτ,(Vτ ) ↓0 τ→∞ τ L2 1 (2) = lim sup f(Φ(µ)) − I (µ) ↓0 µ∈P(ΛN ×ΛN ) L4 1 (2) = lim sup f(Ψ(µ1)) − I (µ) ↓0 µ∈P(ΛN ×ΛN ) L5 1 (1) = lim sup f(Ψ(ν)) − I (ν) ↓0 ν∈P(ΛN ) L3 n dν o = sup f Γ dx − I(ν) ν∈P(ΛN ) L3 n 2 2o = sup f(Γ(φ )) − k∇φk2 . 1 2 φ∈H (ΛN ): kφk2=1 Since f is arbitrary, and Z 2 Γ(φ2) = dx 1 − e−κdφ(x) ΛN is the key integral for the constraint in our variational formula, the claim in Proposition 1 follows from Bryc’s Lemma: Moment-Generating Function ⇐⇒ Rate Function Bryc 1990 (6) DECOMPACTIFICATION It remains to prove Proposition 2. PROOF: With the help of convexity properties of the three integrals appearing in the variational formula for the rate d function, it is not hard to prove that, asΛ N → R , the one d onΛ N converges to the one on R . § LOWER BOUND PROOF: Let CN,τ denote the event that βN does not hit τ ∂ΛN−1. In that case WN,τ does not hit ∂ΛN . Clearly, PN (VN,τ ≤ b) ≥ PN (VN,τ ≤ b, CN (τ)). We can now simply repeat the argument for the upper bound, using the same skeleton approach, the result being that 1 ¯↓ lim log PN VN,τ ≤ b | CN (τ) = −I (b), τ→∞ τ d,N ¯↓ with Id,N (b) given by the same variational formula, except d that R is replaced byΛ N with killing boundary. We have 1 lim log PN (CN,τ ) = −λN , τ→∞ τ with λN the principal Dirichlet eigenvalue of∆ onΛ N . It therefore follows that 1 ¯↓ lim inf log PN (VN,τ ≤ b) ≥ −I (b) − λN . τ→∞ τ d,N But limN→∞ λN = 0. Moreover, lim I¯↓ (b) = I↓(b) N→∞ d,N d by the same type of argument as sketched in the proof of Proposition 2. OUFF, that was a long and intricate proof! § SKELETON APPROACH FOR CAPACITY The skeleton approach can also be used for the capacity. However, there are several important changes. – For the volume the scale of the Brownian motion is t1/d, while for the capacity it is t1/(d−2). – The variational formula for the capacity has a more involved constraint than for the volume. In what follows we sketch the main differences. (1) COMPACTIFICATION d Note that Cap(A) ≥ Cap(TA) for any Borel set A ⊂ R and any map T that is a reflection with respect to a( d − 1)- dimensional hyperplane. By successive reflections of β with respect to the faces of 1/(d−2) t ΛN , we get Cap(Wt) ≥ Cap(WN,t), where WN,t is the Wiener sausage at time t of a reflected 1/(d−2) Brownian motion on t ΛN . By Brownian scaling, −1 law τ t Cap(WN,t) = Cap(WN,τ ) with τ WN,τ = ∪s∈[0,τ]Brτ (βN,s), where (d−4)/(d−2) −1/(d−4) τ = t , rτ = τ , and βN is a reflected Brownian motion onΛ N . Putting τ CN,τ = Cap(WN,τ ), we have P(Ct ≤ bt) ≤ PN (CN,τ ≤ b), where PN is the law of βN starting from0. Henceforth we suppress N from the notation. (2) EXCURSION DECOMPOSITION The same as before. The key players are τ/ Xτ, = {βi}i=1, τ/ 1 X Lτ, = δ . (β(i−1),βi) τ/ i=1 Below we write Eτ,(Cτ ) in terms of Lτ,. (3) CONCENTRATION LEMMA 1 For every δ > 0, 1 lim lim sup log P |Cτ − Eτ,(Cτ )| ≥ δ = −∞. ↓0 τ→∞ τ τ/ Again use that Cτ = Cap(∪i=1Wi) is Lipschitz in the Wi’s. (4) DONSKER-VARADHAN LDP τ/ Note that Wτ = ∪i=1Wi. The radius of Wi is0 < rτ 1, which tends to zero as τ → ∞. Hence the union is close to being disjoint. Write d−2 ¯ Cτ = lim κd|y| Py(¯τW < ∞), (∗) |y|→∞ τ where bar refers to an auxiliary Brownian motion. Write ¯ Py(¯τWτ < ∞) τ/ Z ! X ¯ ∼ ¯y τ¯W < ∞, τ¯ > τ¯W , βτ¯ ∈ dx P i Wτ \Wi i Wi i=1 ∂Wi τ/ = X ¯ ¯ Py τWi < ∞ i=1 Z ¯ τ¯ > τ¯ , β¯ ∈ dx τ¯ < ∞ . Py Wτ \Wi Wi τ¯W Wi ∂Wi i Wi−1 Wi Wi+1 x • ←→ 6 √ •? y time reversal But d−2 ¯ Cap(Wi) = lim κd|y| Py(¯τW < ∞), |y|→∞ i while, by time reversal, Z lim ¯ τ¯ > τ¯ , β¯ ∈ dx τ¯ < ∞ Py Wτ \Wi Wi τ¯W Wi |y|→∞ ∂Wi i Z =! p (dx) ¯ τ¯ = ∞ τ¯ = ∞ i,∂Wτ Px Wτ \Wi Wi ∂Wi def = ui(Wτ ), where h∂Wτ (dx) pi,∂Wτ (dx) = ,x ∈ ∂Wi, h∂Wτ (∂Wi) with h∂Wτ the harmonic measure on ∂Wτ . Combining this with( ∗), we obtain τ/ ! X Cτ ∼ Cap(Wi) ui(Wτ ). i=1 By Brownian scaling, we have Cap(Wi) = Cap ∪s∈[(i−1),i]Brτ (βs) law = Cap ∪s∈[0,]Brτ (βs) law d−2 = r Cap ∪ −2 B1(βs) τ s∈[0, rτ ] ! d−2 −2 cd ∼ rτ cd rτ = , τ → ∞, P − a.s. ∀ i, τ/ where in the last line we use the LLN. It follows that τ/ 1 X Eτ,(Cτ ) ∼ cd Eτ,(ui(Wτ )) τ/ i=1 ! Z ∼ cd Lτ,(dx, dy) ΛN ×ΛN ¯ × Eτ, Px+y τ¯ x+y = ∞ , 2 Wτ \Bδ()( 2 ) ↓ 0, P − a.s. In the last line we use that if( β(i−1)s, βis) = (x, y), then x+y √ |β¯τ¯ − | = O( ) as ↓ 0, τ,-a.s., while the condition Wi 2 P ¯ = pushes ¯ a distance τWi ∞ β √ δ() 1 away from Wi. Subsequently, β¯ must avoid Wτ . But from distance δ() it is unlikely to hit Wτ ∩ Bδ()(x). y • Wi • x - √ - δ() x+y Wi is unlikely to exit the ball of radius δ() around 2 , while the constraint¯ = pushes ¯ out of this ball. τWi ∞ β Our next observation is that, conditional on 2 Lτ, ∼ φ λ ⊗ π, τ → ∞, ↓ 0, λ = Lebesgue, we have ¯ Eτ, Px+y τ¯ x+y = ∞ 2 Wτ \Bδ()( 2 ) Z ∞ ! ¯ ¯ 2 ∼ Ex exp −Adcd dsφ (βs) , τ → ∞, ↓ 0. 0 Indeed, β spends time τφ(x)2dx inside the volume element 2 ¯ dx. Hence Cap(Wτ ∩ dx) = cdφ(x) dx. When β is inside dx, it has a probability AdCap(Wτ ∩ dx) ds to hit Wτ during the time interval ds. The right-hand side equals uφ(x), which is the solution of 2 the Schr¨odingerequation with potential −Adcd φ and with boundary condition u(∞) ≡ 1. (5) SHORT EXCURSION LIMIT As we saw before, 2 h 2 i P Lτ, ∼ φ λ ⊗ π ∼ exp − τI(φ λ) , τ → ∞, ↓ 0, with Z I(φ2 λ) = |∇φ|2(x) dx. ΛN Combining the above observations, we get the analogue of Proposition 1: the LDP upper bound onΛ N . (6) DECOMPACTIFICATION The variational formula for the rate function onΛ N with d reflecting boundary converges to the one on R as N → ∞. Hence we also get the analogue of Proposition 2. For the lower bound we can again use the killing boundary. This completes the sketch of the proof of the downward LDP for the capacity. Many technical details were left out and are still under construction.