Large deviation for non-symmetric random walks on ...

Large deviation for non-symmetric random walks on crystal lattices Toshikazu Sunada Department of Mathematics, Meiji University

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The topic covered here is Asymptotic behavior of random walks on crystal lattices Our view point is quite a bit geometric.

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Notations and conventions ◦ A graph is denoted as X = (V, E), where V = the set of vertices, E = the set of all oriented edges o(e) = the origin of e ∈ E, t(e) = the terminus of e ∈ E, e = the inversion of e ∈ E. Ex = {e ∈ E; o(e) = x}

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Homology groups and cohomology groups Let A be an abelian group (for insatnce, A = Z, R). The the group of 0-chains X C0(X, A) = { axx; ax ∈ A} x

The the group of 1-chains X C1(X, A) = { aee; ae ∈ A}/he + ei e

The boundary map ∂ : C1(X, A) −→ C0(X, A) is defined by ∂e = t(e) − o(e)

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The homology groups are defined as H0(X, A) = C0(X, A)/Image ∂, H1(X, A) = Ker ∂ (⊂ C1(X, A)) Put C 0(X, R) = C(V ), C 1(X, R) = {ω ∈ C(E); ω(e) = −ω(e)} The coboundary operator d : C 0(X, R) −→ C 1(X, R) is defined by df (e) = f (te) − f (oe). The cohomology groups are defined as H 0(X, R) = Ker d(= R), H 1(X, R) = C 1(X, R)/Image d 5

Random walks A random walk (RW) on a (finite or infinite) graph X = (V, E) is a stochastic process with values in V characterized by a transition probability p : E −→ R, where

X

p(e) ≥ 0,

p(e) = 1,

e∈Ex

p(e) + p(e) > 0 We think of p(e) to be the probability that a “particle” at o(e) moves to t(e) along e in unit time. 6

Put E +(p) = supp p ⊂ E. Our assumption =⇒ E +(p) ∪ E +(p) = E. “two-way traffic” : E +(p) = E, “one-way traffic” : E +(p) ∩ E +(p) = ∅ “mixed traffic” : E +(p) ∩ E +(p) 6= ∅

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Transition operators and discrete Laplacians for random walks The transition operator L : C(V ) −→ C(V ) is defined by X (Lf )(x) = p(e)f (te) e∈Ex

The RW Laplacian is defined by ∆ = L − I, that is, X £ ¤ (∆f )(x) = p(e) f (te) − f (oe) . e∈Ex

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A stochastic process (with discrete time) in general is a probability space (Ω, P ) together with a family of maps into a set S ξn : Ω → S

(n = 0, 1, 2, . . .)

For the random walk, Ω = Ωx = {c = (e1, e2, . . .); one-sided infinite paths with o(c) = x}, S = V, ξn(c) = o(en+1). The probability measure P = Px is defined in such a way that, for a given path (e1, . . . , en) of length n, ¡ ¢ Px {c = (e1, , . . . , en, ∗, ∗, . . .)} = p(e1) · · · p(en) 9

n-step transition probability Define the n-step transition probability by p(n, x, y) = Px(c; ξn(c) = y) Then p(n, x, y) = (Lnδy )(x). p(n + 1, x, y) − p(n, x, y) = ∆p(n, ·, y) Namely u(x) = p(n, x, y) satisfies a discrete analogue of the heat equation ∂u ∂t

= ∆u

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A RW on the Z-lattice

µ p(n, 0, x) =

¶ n n + x p(n+x)/2q (n−x)/2

2 (n ≡ x (mod. 2))

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The central limit theorem p(n, 0, x) ¡ ¢2 ´ ³ ¡ ¢ 1 x − (p − q)n =√ exp − 1 + rn(x) 2πnpq 8npq where lim rn(x) = 0 uniformly in x with n→∞ √ |x − (p − q)n| ≤ A n. Large deviation asymptotic Let |ξ| < 1, and Suppose that xn − nξ is bounded. Then 1 lim log p(n, 0, xn) = −H(ξ), n→∞ n

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where 1 1 H(ξ) = − (1 + ξ) log p − (1 − ξ) log q − log 2 2 2 1 1 + (1 + ξ) log(1 + ξ) + (1 − ξ) log(1 − ξ) 2 2

Both central limit theorem and large deviation asymptotic are proved by using the Stirling formula for n!. Question What about p(n, x, y) in general cases ? 13

Irreducibility A RW is said to be irruducible if, for any x, y ∈ V , there exists n with p(n, x, y) > 0 k For any x, y, one can find a path c = (e1, . . . , en) such that o(c) = x, t(c) = y and ei ∈ E +(p) = supp p for every i, i.e. the walk along c obeys the “traffic rule”. ◦ In general, a subset E + ⊂ E with E = E + ∪ E + is said to be irreducible if it has this property. ◦ A path c = (e1, . . . , en) is said to be admissible with respect to E + if ei ∈ E + for every i.

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Perron-Frobenius theorem For a while, X is supposed to be finite. Given a function ϕ ∈ C(E), consider the operator Aϕ : C(V ) −→ C(V ) defined by X (Aϕf )(x) = ϕ(e)f (te). e∈Ex

Theorem Let ϕ be non-negative valued, and suppose E +(ϕ) = supp ϕ is irreducible. 1. Aϕ has a positive eigenvalue. The maximal one α among all positive eigenvalues is simple and has a positive-valued eigenfunction. 2. If Aϕf = λf , f ≥ 0, then λ = α and f > 0. 15

3. If Aϕf ≥ αf , f ≥ 0, f 6≡ 0, then Aϕf = αf 1 4. lim log Aϕn1 = log α, where 1 denotes the n→∞ n function identically equal to 1. 5. Define X t ( Aϕf )(x) = ϕ(e)f (te). e∈Ex

Then α is the maximal positive eigenvalue of tAϕ. In fact, tAϕL is the adoint operator (transpose) of Aϕ with respect to the inner product: X hf1, f2i0 = f1(x)f2(x). x∈V

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Invariant measures ◦ 1 is the maximal positive eigenvalue of L with the eigenfunction 1 (the constant function with value 1). ◦ If the RW is irreducible, then there exists a positive valued function m ∈ C(V ) such that tLm = m (apply (4) in the Perron-Frobenius theorem). m is unique up to a constant multiple. P P t ◦ Lm = m ⇐⇒ x∈V Lf (x)m(x) = x∈V f (x)m(x) for every f ∈ C(V ). m is called an invariant measure. From now on, we consider an irreducible RW on a finite graphs. 17

Symmetric random walks ◦ When there exists a positive-valued function mV on V such that p(e)mV (oe) = p(e)mV (te), the RW is said to be symmetric. A symmetric RW is two-way traffic, and hence irreducible. mV turns out to be an invariant measure. ◦ The RW Laplacian for a symmetric RW is a SRW Laplacian. Exercise Show p(n, x, y)m(y)−1 = p(n, y, x)m(x)−1 .

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Simple random walks The simple random walk on X is a RW with the transition probability defined by 1 p(e) = . deg o(e) The invariant measure is m(x) = deg x. Clearly the simple RW is symmetric, and mE ≡ 1. The RW Laplacian for the simple RW coincides with the canonical Laplacian X £ ¤ 1 (∆f )(x) = f (te) − f (x) deg x e∈E x

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Law of large numbers Let f be a (vector-valued) function on E. Consider ¤ 1£ f (e1) + · · · + f (en) n for c = (e1, e2, . . .) ∈ Ωx. Let m be the normalized P invariant measure ( x∈V m(x) = 1). lim

n→∞

1£ n

¤

f (e1) + · · · + f (en) =

X

p(e)m(oe)f (e)

e∈E

in probability one. This is a consequence of the ergodic theorem. 20

Special cases ◦ f (e) = o(e) ∈ C0(X, R). ¤ 1£ o(e1) + · · · + o(en) lim n→∞ n X X = p(e)m(oe)o(e) = m(x)x. e∈E

x∈V

◦ f (e) = e ∈ C1(X, R). X ¤ 1£ lim e1 + · · · + en = p(e)m(oe)e n→∞ n e∈E P The 1-chain γp = e∈E p(e)m(oe)e is in H1 (X, R), called the homological position (direction). 21

Modified discrete Hodge-Kodaira theorem Define δ : C 1(X, R) −→ C 0(X, R) by X p(e)ω(e) (δω)(x) = e∈Ex

This plays a role of the adjoint (transpose) of d. Indeed, ∆ = −δd. In symmetric case, δ is actually the adjoint of d. Lemma (1) dim Ker δ = b1(X) if and only if the RW is symmetric. In non-symmetric case, dim Ker δ = b1(X) − 1. (2) The RW is symmetric if and only if γp = 0.

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Definition ω ∈ C 1(X, R) is said to be a (modified) harmonic 1-form if δω + hγp, ωi = 0. b 1(X) the space of harmonic 1-forms. Denote by H b 1(X) ⊂ C 1(X, R) induces Theorem The inclusion H b 1(X) ∼ an isomorphism H = H 1(X, R). Suppose that p is one-way traffic (so that E +(p) is an orientation). Define θp ∈ C 1(X, R) by ( 1 (e ∈ E +(p)) θp(e) = −1 (e ∈ E +(p)) b 1(X). Then θp ∈ H 23

“Universal” free energy Let ω ∈ C 1(X, R). Let eFuni(ω) be the maximal positive eigenvalue of the operator Lω defined by X p(e)eω(e)f (te) (Lω f )(x) = e∈Ex

Funi(ω) = Funi(p, ω) is said to be the universal free energy. ◦ Lω+du = e−uLω eu so that Funi(p, ω +du) = Funi(p, ω), and hence Funi(p, ·) induces a function on H 1(X, R), which we denote by the same symbol Funi. ◦ Let fp,ω be a positive-valued eigenfunction for the maximal positive eigenvalue eFuni(p,ω) of Lω , that is, 24

X

eω(e)p(e)fp,ω (te) = eFuni(p,ω)fp,ω (x).

e∈Ex

If we put fp,ω (te) pω (e) = e−Funi(p,ω)+ω(e)p(e) , fp,ω (oe) X + then supp pω = E (p), pω (e) = 1, so that pω is e∈Ex

an irreducible transition probability. ◦ It is checked that Funi(pω , u) = Funi(p, ω + u) − Funi(p, ω) ◦ When p is one-way traffic, Lω+tθp = etLω so that Funi(p, ω + tθp) = Funi(p, ω) + t 25

Gradient and Hessian of Funi Define, for ω, u ∈ H 1(X, R), d ¯¯ (∇uFuni)(ω) = ¯ Funi(ω + tu), dt t=0 d2 ¯¯ (Hessω Funi)(u, u) = 2 ¯ Funi(ω + tu), dt t=0 Theorem (1) (∇uFuni)(0) = hγp, ui b 1(X), (2) For u ∈ H X (Hess0Funi)(u, u) = p(e)m(oe)u(e)2 e∈E

³X ´2 − p(e)m(oe)u(e) e∈E 26

We employ a perturbation technique to prove the theorem, say, by putting α(t) = eFuni(p,tu), ft = fp,tu, differentiate both sides of the following equation X etu(e)p(e)ft(te) = α(t)ft(x). e∈Ex

(ft being unique up to positive multiple constants). In the discussion, we need to adjust ft by multiplying a suitable scalar ct. (Hess0Funi) is non-negative as a quadratic form since 1 X (Hess0Funi)(u, u) = p(e1)m(oe1)p(e2)m(oe2) 2 e ,e ∈E 1 2 ¡ ¢2 × u(e1) − u(e2) . 27

Theorem (1) Hess0Funi is positive definite if and only if E +(p) ∩ E +(p) 6= ∅ (mixed traffic). (2)¡ If E +(p) ¢∩ E +(p) = ∅ (one-way traffic), then b 1(X)) Null Hess0Funi = Rθp(⊂ H In the case of a simple RW, ¡ ¢ 1 X Hess0Funi (u, u) = u(e)2, m(V ) e∈E P where m(V ) = x∈V deg x.

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Gradient maps The correspondence ω 7→ (∇·Funi)(ω) is a map from ¡ ¢ ∗ H 1(X, R) into H 1(X, R) = H1(X, R). Theorem (1) In the case E +(p) ∩ E +(p) 6= ∅, the map ω 7→ (∇·Funi)(ω) is a diffeomorphism onto the interior DE +(p) of a convex polyhedron ¡ ¢∗ in H1(X, R) = H 1(X, R) (2) In the case E +(p) ∩ E +(p) = ∅, the map ω 7→ (∇·Funi)(ω) induces a diffeomorphism from H 1(X, R)/Rθp onto the interior DE +(p) of a convex polyhedron in the affine space {ξ ∈ H1(X, R); hξ, [θp]i = 1}. Note ∇Funi(0) = γp. 29

Characterization of Image ∇Funi DE +(p) = Image ∇Funi depends only on E +(p) (namely, if E +(p1) = E +(p2), then DE +(p1) = DE +(p2)). Indeed X DE + = { Q(e)e ∈ H1(X, R); e∈E

Q(e) ≥ 0,

X

Q(e) = 1, supp Q = E +}

e∈E

Exercise Let E + be a subset of E with E + ∪E + = E, and suppose that there exists Q : E −→ R satisfying X X Q(e)e ∈ H1 (X, R), Q(e) ≥ 0, Q(e) = 1, supp Q = E + . e∈E

e∈E

Show that E + is irreducible. 30

The case of two-way traffic RW Define the norm k · k1 on C1(X, R) by ° ° X X ° ° aee° = |ae|, ° e∈E o

1

e∈E o

where E o is an orientation. Note that this definition does not depend on the choice of E o. Theorem DE = {α ∈ H1(X, R); kαk1 < 1} Theorem x ∈ H1(X, R) is an extreme point of DE if and only if x = c/kck1 for a simple closed path c in X. Here a closed path is said to be simple if it constitutes a circuit subgraph.

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The case of one-way traffic RW Theorem If E + is one-way traffic, then DE + is a facet (a face of the maximal dimension) of DE .

The case of mixed traffic RW Theorem If E + ∩ E + 6= ∅, then DE + is the intersection of DE and a convex cone with the center 0. In particular, 0 ∈ DE + .

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Example Consider the 2-bouquet graph X. Then H 1(X, R) = Re1 + Re2 = R2 and kx1e1 + x2e2k1 = |x1| + |x2| so that DE is the square.

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RW on a crystal lattice

Crystal lattices A graph X = (V, E) is said to be a d-dimensional crystal lattice if Aut(X) has a free abelian subgroup Γ of rank d such that (1) Γ acts freely both on V and the set of nonoriented edges, and (2) the quotient graph X0 = Γ\X is finite. The group Γ is said to be a lattice (group) of X, and X0 = (V0, E0) is said to be the fundamental finite graph, where V0 = Γ\V , E0 = Γ\E.

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Γ

This definition coincides with the previous one (X → X0 is an abelian covering graph). =⇒ surjective homomorphism µ : H1(X0, Z) −→ Γ =⇒ sujective linear map µR : H1(X0, R) −→ Γ ⊗ R =⇒ injective linear map t

µR : Hom(Γ, R) −→ H 1(X0, R),

where Hom(Γ, R) denotes the linear space of homomorphisms of Γ into R, which is identified with (Γ ⊗ R)∗. Later, we shall think of Hom(Γ, R) as a subspace of H 1(X0, R).

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Explicit description of µ Represent α ∈ H 1(X0, Z) by a closed path c0 in X0, and let c be a lift of c0 in X. Since o(c) and t(c) project down to the same vertex o(c0) = t(c0), one can find g ∈ Γ with t(c) = go(c). Then µ(α) = g.

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Periodic realizations Let X = (V, E) be a d-dimensional crystal lattice. A map Φ : V −→ Rd is said to be a periodic realization of X if there exist a lattice group Γ and an injective homomorphism ρ : Γ −→ Rd such that (1) Φ(gx) = Φ(x) + ρ(g) for x ∈ V and g ∈ Γ. (2) ρ(Γ) is a lattice in Rd. Here a lattice means a discrete subgroup of Rd of maximal rank.

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“Normalized” expression of periodic realizations The inclusion ρ : Γ −→ Rd extends to a linear isomorphism of Γ ⊗ R onto Rd. Identify Rd with Γ ⊗ R. Then Φ is a map of X into Γ ⊗ R satisfying Φ(gx) = Φ(x) + g

(∗),

where g ∈ Γ ⊂ Γ ⊗ R. From now on, a periodic realization mean a map Φ : V −→ Γ ⊗ R satisfying (∗). Put v(e) = Φ(te) − Φ(oe) (e ∈ E). Then v(e) = −v(e) and v(ge) = v(e), so that v is regarded as an element of C 1(X0, Γ ⊗ R). The 1-cochain v is a ‘‘building block” of Φ. 39

RW with a periodic transition probability A transition probability p on a crystal lattice X is said to be periodic if there exists a lattice group Γ such that p(ge) = p(e) (g ∈ Γ, e ∈ E). A periodic transition probability yields a transition probability p0 on X0. Conversely, a transition probability on X0 induces a periodic transition probability p on X. Similarly, we have the following correspondence Γ-invariant subsets E + ⊂ E ⇐⇒ subsets E0+ ⊂ E0 ◦ E + ∪ E + = E ⇐⇒ E0+ ∪ E0+ = E0 ◦ E +(p) corresponds to E0+(p0). 40

Irreducibility ◦ E + is irreducible =⇒ E0+ is irreducible. Warning: Converse is, in general, not true.

◦ If X is the maximal abelian covering of X0, then E + is irreducible if and only if E + = E (two-way traffic). 41

Probability spaces Let π : V −→ V0 be the covering map. When π(x) = x0, (Ωx, Px) is identified with (Ωx0 , Px0 ).

Thus a stochastic process with the probability space (Ωx, Px) is regarded as a stochastic process with the probability space (Ωx0 , Px0 ). 42

Law of large numbers From now on, p0 is supposed to be irreducible. Given a periodic realization Φ : V −→ Γ⊗R, we have the stochastic process {ξn}∞ n=0 with values in Γ ⊗ R defined by ¡ ¢ ¡ ¢ ξn(c) = Φ c(n) c(n) = o(en+1), c = (e1, e2, · · · ) Theorem lim

n→∞

1 n

ξn(c) = µR(γp0 ) in probability one.

Proof We may assume Φ(x) = 0. Then ξn(c) = v(e1) + · · · + v(en)

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Applying the general law of large numbers, we have X 1 p0(e)m0(e)v(e) = hv, γp0 i lim ξn(c) = n→∞ n e∈E What remains to do is to show is that, for any α ∈ H1(X0, Z), hv, αi = µ(α) Represent α by a closed path c0 = (e1, · · · , en). Then ¡ ¢ ¡ ¢ hv, αi = v(e1) + · · · + v(en) = Φ t(c) − Φ o(c) , where c is a lift of c0. Since t(c) = µ(α)o(c), the RHS is µ(α). Exercise Let x ∈ Hom(Γ, R), and define ω ∈ C 1 (X0 , R) by ω(e) = hv(e), xi. Show that t µR (x) = [ω] (Note (Γ ⊗ R)∗ = Hom(Γ, R)). 44

Martingale realizations There are many ways to realize periodically a given crystal lattice in space. Which is the most natural one when we take account of RW ? Definition A periodic realization Φ is said to be a martingale realization (or harmonic realization) if v ∈ C 1(X0, Γ ⊗ R) is a harmonic 1-from. The following conditions are equivalent: (1) Φ is a martingale realization (2) ∆Φ = µR(γp0 ) (3) {ξn − nµR(γp0 )}∞ n=0 is martingale A martingale realization is unique up to additive constant vectors. 45

Large deviation theory Let {ηn}∞ n=0 be a general stochastic process with values in a finite dimensional vector space S. Assumption (1) For all x ∈ S ∗, the limit ¡ ¢ 1 F (x) = lim log E ehηn,xi n→∞ n exists (E denotes the expectation). F is said to be the free energy (F is a convex function). (2) F is smooth on S ∗.

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Define the function H(ξ) on S by ¡ ¢ H(ξ) = sup hξ, xi − F (x) x∈S ∗

(Legendre-Fenchel transform). This is a convex function possibly taking ∞, and called the entropy. For a subset A ⊂ S, put H(A) := inf {H(ξ) | ξ ∈ A}. Large deviation principle ³1 ´ 1 −H(intA) ≤ lim inf log P ηn ∈ intA n→∞ n n ³1 ´ 1 ≤ lim sup log P ηn ∈ A ≤ −H(A), n n→∞ n

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The free energy for {ξn}∞ n=0 Let x ∈ Hom(Γ, R). Then X hξn ,xi E(e ) = p(c)ehΦ(tc),xi c;|c|=n o(c)=x

=

X

p0(c)eω(c) = Lω n1

c;|c|=n o(c)=x

where ω(e) = hv(e), xi, and ω(c) = ω(e1)+· · ·+ω(en). Therefore 1 lim log E(ehξn,xi) = Funi(ω), n→∞ n and F (x) = Funi(tµR(x)), where Funi is the universal free energy. 48

◦ Let ∇F : Hom(Γ, R) −→ Γ⊗R be the gradient map. Then (∇F )(0) = µR(γp0 ), Image ∇F = µR(DE +(p0)) 0

Image ∇F is the interior of a convex polyhedron in Γ ⊗ R or in an affine hyperplane of Γ ⊗ R. ◦ The entropy H assumes finite values on µR(DE +(p0)). 0 When ξ = (∇F )(x), H(ξ) = hξ, xi − F (x). ◦ H(µR(γp0 )) = 0 ◦ RW on X with the transition probability p is irreducible if and only if 0 ∈ µR(DE +(p0)). 0

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Large deviation asymptotic Let H be the entropy for {ξn}. Theorem Suppose that RW on X is irreducible. Let ξ ∈ µR(DE +(p0)), and let {yn}∞ n=1 be a sequence 0 in V such that {Φ(yn) − nξ} is bounded. Then lim

n→∞

1 n

log p(n, x, yn) = −H(ξ)

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Central limit theorem Theorem Suppose that RW on X is irreducible. As n ↑ ∞, ν ¡ ¢ p(n, x, y)m(y)−1 ∼ (4πn)d/2vol J (Γ) ³ ´ 1 2 × exp − kΦ(y) − Φ(x) − nγp)k 4n √ as far as kΦ(x) − Φ(y) − nγp)k ≤ A n. ◦ Φ : martingale realization, ◦ ν : the period of E +(p), the greatest common divisor of length of closed admissible paths. ◦ γp = µR(γp0 ), 51

◦ k · k : the dual norm on Γ ⊗ R of the norm on Hom(Γ, R) defined by ¡ ¢ 1 2 kxk = (Hess0Funi) µR(x), µR(x) 2 ◦ J (Γ) = Hom(Γ, R)/Hom(Γ, Z) with the flat metric induced from kxk.

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References [1] M. Kotani and T. Sunada, Large deviation and the tangent cone at infinity of a crystal lattice, to appear in Math. Z., (2006). [2] M. Kotani and T. Sunada, Albanese maps and an off diagonal long time asymptotic for the heat kernel, (with M.Kotani), Comm. Math. Phys. 209(2000), 633-670.

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Appendix

RW on the triangular lattice

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One-way traffic RW on the triangular lattice

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Scaling limit of accessible parts Fix x ∈ V , and suppose Φ(x) = 0. Denote Cn = {y ∈ V ; p(n, x, y) > 0}, Dp = µR(DE +(p0)). If Dp has non-empty interior, then lim

n→∞

1 n

Φ(Cn) = Dp

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