An example on the discrete circle

4.7 Loop clusters

4.7.1 An example on the discrete circle

Consider a discrete circle G with n vertices 1, . . . , nand 2n oriented edges

E ={(1,2),(2,3), . . . ,(n−1, n),(n,1),(2,1),(3,2), . . . ,(n, n−1),(1, n)}

Define the clockwise edges set E+ = {(1,2),(2,3), . . . ,(n − 1, n),(n,1)} and the counter clockwise edges E− =E−E₊. Consider a Markovian generatorL such that for anye∈E₊, Lê−_e+ =p, Lê+_e− = 1−p, Lê−_e−=−(1 +c)and Lis null elsewhere. Then, we have a loop measure and Poissonian ensembles associated with L. The rest of this subsection is devoted to study the loop cluster C_α in this example.

Lemma 4.7.3. Let T3,n be a n×n tri-diagonal Toeplitz matrix of the following form:







a b 0 · · · 0 c a b . .. ...

0 . .. ... ... 0 ... . .. c a b 0 · · · 0 c a







n×n

Let S_n be the following n×n matrix:







a b 0 c

c a b . ..

0 . .. ... ... 0 . .. c a b

b 0 c a







n×n

Let x₁, x₂ be the roots of x²−ax+bc= 0. Then,

• det(T_3,n) = xⁿ⁺¹₁ −xⁿ⁺¹₂ x₁−x₂ ,

• det(S_n) =xⁿ₁ +xⁿ₂ + (−1)ⁿ⁺¹(bⁿ+cⁿ).

Proposition 4.7.4.

Set x₁ =1

2(1 +c+p

(1 +c)²−4p(1−p)), x₂ =1

2(1 +c−p

(1 +c)² −4p(1−p)).

Then,

P[{1, n} is closed.] =

(xⁿ₁ −xⁿ₂)²

(x₁−x₂)(xⁿ⁻¹₁ −xⁿ⁻¹₂ )(xⁿ₁ +xⁿ₂ −(pⁿ+ (1−p)ⁿ)) −α

Proof. By Proposition 3.2.1 and Proposition 3.5.8

P[{1, n} is closed] =e^−αµ(Nⁿ¹^(l)+N¹ⁿ^(l)>0) =e^−αµ(l^visits¹^andⁿ⁾

det(V_{1,n}) V₁¹V_nⁿ

det(−L|{2,...,n}×{2,...,n}) det(L|{1,...,n−1}×{1,...,n−1}) det(−L|{2,...,n−1}×{2,...,n−1}) det(−L)

^−α

(xⁿ₁ −xⁿ₂)²

(x₁−x₂)(xⁿ⁻¹₁ −xⁿ⁻¹₂ )(xⁿ₁ +xⁿ₂ −(pⁿ+ (1−p)ⁿ)) −α

where x₁ = ^1+c+

√

(1+c)²−4p(1−p)

2 and x₂ = ^1+c−

√

(1+c)²−4p(1−p)

2 .

Proposition 4.7.5. Conditionally on{1, n} being closed,Cαis a renewal process conditioned to jump at time n. To be more precise, by deleting edges {1, n} and adding{0,1},{n, n+ 1}, we get a discrete segment with vertices {0,1, . . . , n, n+ 1}and edges {{0,1}, . . . ,{n, n+ 1}}.

Conditionally to {1, n} being closed,Cα induces a partition on{1, . . . , n}. The clusters ofCα

are the intervals between the edges closed at time α (namely the edges which are not crossed by any loop of L_α). Then the left points of these closed edges, together with the left points of {0,1} and {n, n+ 1}, form a renewal process conditioned to jump at n.

Proof. Among the Poissonian loop ensembles, the ensemble of loops crossing {1, n} and the rest are independent. Therefore, the conditional law Q of the loops not crossing {1, n} conditioned on the event that no loop is crossing {1, n}is exactly the same as the unconditioned law. Consider another Poissonian loop ensembles on Z driven by the following generator:

L^m_m =−(1 +c), L^m_m+1 =p, L^m_m−1 = 1−p for all m∈Z, and L is null elsewhere.

Then, Q is the same as the conditional law of the loop ensembles contained in {1, . . . , n}

given the condition that{0,1}or{n, n+1}are closed. By Proposition 3.1.2, after a harmonic transform, L is modified as follows:

L^m_m =−(1 +c), L^m_m+1 =L^m_m−1 =p

p(1−p)for all m∈Z, and Lis null elsewhere.

According to Proposition 3.1 in [LJL12], in the case of Z, conditionally to the event that {0,1} is closed, the left points of the closed edges form a renewal process. There is an obvious one-to-one correspondence between the jumps of the renewal process and the closed edges. Finally, in the case of the circle, conditioning on {1, n} being closed, we can identify C_α to a renewal process conditioned to jump at time n. It is not hard to see the parameter κ in [LJL12] equals ^1+c−2

√p(1−p)

√

p(1−p) .

We can go back to the symmetric model conditionally on {1, n}being closed. Hence, we use the following modified model. Consider a pure-jump Markov process on {1, . . . , n, . . .} with generator L: L^m_m+1 = L^m+1_m = 1/2, L^m_m = −(1 +κ/2) for m ∈ N+, L¹₂ = L²₁ = 1/2 and L is null elsewhere. Then, associated with this L, we have a loop measure µand a Poisson point process of loops of intensity αµ. Let us treat the caseα ∈]0,1[.

Hypothesis 4.7.1. Supposeα ∈]0,1[.

The corresponding loop probability depends on κ and we will denote it by P^(κ). It has been showed in [LJL12] that the left points of the closed edges form a renewal process (Sm^(κ), m ≥ 0)(S₀^κ = 0), see Proposition 3.1 in [LJL12]. Moreover, in Proposition 3.1 of [LJL12], it has been proved that (S_b^(κα−1²⁾tc, t ≥ 0) converges to a subordinator (X_t^(κ), t ≥ 0) with potential density U(x, y) = 1{y>x}

2√ κ 1−e⁻²^√^κ(y−x)

in the sense of finite marginal distribution. Once we show the tightness in the sense of Skorokhod, we could replace the finite marginal convergence by the convergence in law in the sense of Skorokhod.

Lemma 4.7.6 (Tightness of (S_b^(κα−1²⁾tc, t ≥0)). The distribution (S_b^(κα−1²⁾tc, t≥ 0) is tight in the Skorokhod space. Therefore, (S_b^(κα−1²⁾tc, t≥0)converges to a subordinator (X_t^(κ), t≥0)in the sense of Skorokhod.

Proof. Define F_n = σ(S₁^(κ²⁾, . . . , Sn^(κ²⁾) for n ∈ N and F⁽⁾_t = F_b^α−1_tc for t ≥ 0. Then, (F⁽⁾_t , t≥0)is a right-continuous filtration. As usual, by adding the negligible sets, we get a the complete filtration which are denoted by the same notation.

Let T be a (F⁽⁾_t , t ≥0)stopping time. Then, b^α−1Tc is a (F_n, n∈N) stopping time.

In order to show the tightness, it is enough to verify the following Aldous’ criteria, see [JS03].

For each M > 0, δ >0,

K→∞lim lim

→0P⁽⁾[S_b^(κα−1²⁾Mc > K] = 0 (4.1)

limθ↓0 lim

→0 sup

T1,T2∈F⁽⁾_M,T1≤T2≤T1+θ

P⁽⁾[|S_b^(κα−1²⁾T2c−S_b^(κα−1²⁾T1c|> δ] = 0 (4.2) Since we already know the finite marginal convergence, condition (4.1) reduces to P[X_M^(κ) =

∞] = 0. Since S^(κ²⁾ is a renewal process, for T₁, T₂ ∈ F⁽⁾_M such that T₁ ≤ T₂ ≤ T₁ +θ,

limθ↓0 lim

→0 sup

T1,T2∈F⁽⁾_M,T1≤T2≤T1+θ

P⁽⁾[|S_b^(κα−1²⁾T2c−S_b^(κα−1²⁾T1c|> δ]

≤lim

θ↓0 lim

→0P⁽⁾[|S_d^(κα−1²⁾θe|> δ]

≤lim

θ↓0 P[|X_2θ^(κ)|> δ] = 0 The proof is complete.

Immediately, we get the convergence in the sense of the Skorokhod. Using this result, we will the convergence of the corresponding bridge processes in the following proposition.

Proposition 4.7.7. Define (Zm^(κ/n²⁾, m ≥ 0) by Zm^(κ/n²⁾ = Sm^(κ/n²⁾∧n. The law of Z^(κ/n²⁾ depends on n, κ and is are denoted by Q^n,κ. Conditioned on {n, n+ 1} being closed, the left points of the closed edges together with the left point of {0,1} form a renewal process conditioned to jump at n. Define a conditioned loop probability as follows: P˜^(κ/n

2)[·] = P^(κ/n

2)[·|{n, n + 1} is closed]. Let Q˜^n,κ be the law of Z^n,κ under P˜^(κ/n

2). As n tends to infinity, under P˜^(κ/n²⁾, (Z_bn^(κ/n1−α²⁾tc/n, t ≥ 0) converges in law to (X_t^(κ)∧1, t ≥ 0) conditioned on {X_T^(κ)

]1,∞[− = 1}, in the sense of finite marginal convergence.

Before proving this, let us precise the law of (X_t^(κ), t < T]1,∞[) conditioned on the event {X_T^(κ)

]1,∞[− = 1} in the following lemma. Recall that the potential density U(x, y) of X^(κ) equals 1{y>x}

2√ κ 1−e⁻²^√^κ(y−x)

. Set u(x) = U(0, x)and h(x) =U(x,1) = u(1−x).

Lemma 4.7.8.

1. For all positive functions f, we have E⁰[f(X_s^(κ), s ∈[0, t])1{t<T_]1,∞[}, X_T^(κ)

]1,∞[− ∈db]

=E⁰[X_T^(κ)

]1,∞[− ∈db]E⁰[f(X_s^(κ), s∈[0, t])1{t<T_]1,∞[}

u(b−X_t^(κ)) u(b) ].

2. The conditioned process³ is a h-transform of the original subordinator with respect to the excessive function x →u(1−x). To be more precise, for y∈[x,1[, its semi-group is given by

Q^(κ)_t (x, dy) = u(1−y)

u(1−x)P_t^(κ)(x, dy).

The process is right-continuous on [0, ζ[.

3. Denote by ζ the lifetime of the conditioned process Y^(κ). Then, Y_ζ−^(κ)= 1.

4. The semi-group Q is a Feller semi-group.

3More precisely, the process defined by the probabilityE⁰[f(Xs^(κ), s∈[0, t])1_{t<T_]1,∞[_}^u(1−X

(κ) t ) u(1) ].

5. The time reversal from the lifetime of the processY^(κ)is the left-continuous modification of 1−Y^(κ) under Q⁰.

Proof.

1. The subordinator (X_t^(κ), t ≥ 0) has the potential density U(x, y) = ( ²

√κ

1−e⁻²^√^κ(y−x))^α for y≥x. When y tends to x, U(x, y) tends to∞. As a consequence, the drift coefficient d = 0. It is proved by H. Kesten [Kes69] that for a fixed x > 0, x does not belong to the range of the subordinator with probability 1, see Proposition 1.9 in [Ber99]. By using the strong Markov property at stopping time S,

E⁰[f(X_s^κ, s∈[0, S])1{S<T_]1,∞[}, X_T^(κ)

]1,∞[− ∈db]

=E⁰ h

f(X_S^κ, S ∈[0, t])1{S<T_]1,∞[}E^X

(κ) S [X_T^(κ)

]1,∞[− ∈db]i . Then, we use Lemma 1.10 in [Ber99]:

• E^X

(κ) S [X_T^(κ)

]1,∞[− ∈db] = ¯Π(1−b)u(b−X_S^(κ))db,⁴

• E⁰[X_T^(κ)

]1,∞[− ∈db] = ¯Π(1−b)u(b)db.

Immediately, we have E⁰

f(X_s^κ, s∈[0, S])1{S<T_]1,∞[}E^X

(κ) S [X_T^(κ)

]1,∞[− ∈db]

=E⁰[X_T^(κ)

]1,∞[ ∈db]E⁰[f(X_s^κ, s∈[0, S])1{S<T_]1,∞[}

u(b−X_S^(κ)) u(b) ].

In particular, we take fixed time t, E⁰[f(X_s^κ, s∈[0, t])1_{t<T_]1,∞[_}|X_T^(κ)

]1,∞[− = 1]

=E⁰[f(X_s^κ, s∈[0, t])1{t<T_]1,∞[}

u(1−X_t^(κ)) u(1) ]. (*) 2. It is enough to show thatx→u(1−x)is excessive. The rest follows from the classical results on the h-transform, see Chapter 11 of [CW05]. Take a positive function g, we have P_tU g =

∞

P_sg ds and U g =

∞

P_sg ds. Then, for all positive function g, we have P_tU g ≤ U g and lim

t→0P_tU g = U g. As a consequence, except on a set N(x) of zero Lebesgue measure, y →u(y, z) is an excessive function for all z, i.e.

• R

P_t(x, dy)u(y, z)≤u(x, z),

4Here, Π¯ represents the tail of the characteristic measure of the subordinator.

• lim

t→0Pt(x, dy)u(y, z) =u(x, z).

Take a decreasing sequence z₁ >· · · with limit1. As the increasing limit of a sequence of excessive functions y→u(y, z_n), y→u(y,1)is excessive.

3. It is enough to show that Q^x[T[1−δ,∞[< ζ] = 1 for any δ >0. In fact, by Theorem 11.9 of [CW05],

Q^x[T[1−δ,∞[< ζ] =P^x

T[1−δ,∞[< T]1,∞[,u(1−XT_[1−δ,∞[) u(1−x)

. P⁰[X_T_[1−δ,∞[−∈da, X_T_[1−δ,∞[−−X_T_[1−δ,∞[− ∈db] =u(a)daΠ(db).

Consequently,

Q^x[T[1−δ,∞[< ζ] =P⁰

T[1−x−δ,∞[< T]1−x,∞[,u(1−x−X_T_{[1−x−δ,∞[}) u(1−x)

0<a<1−x−δ<a+b<1−x

u(1−x−a−b)

u(1−x) u(a)daΠ(db) Set c= 1−x−a−b.

Q^x[T[1−δ,∞[< ζ] =

0<c<δ<c+b<1−x

u(c)

u(1−x)u(1−x−c−b)dcΠ(db)

=P⁰

T_[δ,∞[ < T_]1−x,∞[,u(1−x−X_T_[δ,∞[) u(1−x)

=Q^x[T[x+δ,∞[< ζ] By the right-continuity of the path,

δ→0limQ^x[T_{[1−x−δ,∞[}< ζ] = lim

δ→0Q^x[T_[x+δ,∞[< ζ] = 1.

But Q^x[T[x+a,∞[< ζ]decreases as a increases. Then, we must have Q^x[T[y,∞[< ζ] = 1 for y∈[x,1[.

4. We know that P_t^(κ) is a Feller semi-group. For f ∈ C_K([0,1[)⁵, x → Q_tf(x) belongs to C₀([0,1[). By the Markov property of the semi-group Q_t, i.e. ||Q_tf||∞ ≤ ||f||∞, we have Q_tf = lim

n→∞Q_t(f|[0,1−1/n[)∈C₀([0,1[). For fixed x∈[0,1[, limt→0Q_tf(x) = lim

t→0P^x[1_{t<T_]1,∞[_}f(X_t^(κ))u(1−X_t^(κ))

u(1−x) ] =f(x).

Then we see that the semi-group (Q_t, t ≥0)is Feller.

5The collection of compact supported continuous function over [0,1[.

5. By a classical result about time reversal, the reversed process is a moderate Markov process, its semi-group Qˆ_t(x, dy)is given by the following formula:

hg, QtfiG =hQˆtg, fiG. where Q_t(x, dy) = ^U(y,1)_U(x,1)P_t(x, dy) and G(dx) =

∞

Q_t(0, dx)dt = U(0,x)U(x,1)

U(0,1) dx. Denote by( ˆP_t, t≥0)the dual semi-group of(P_t, t≥0)or the semi-group of−X^(κ)equivalently.

Denote by u(x)the function U(0, x)and by h(x)the function U(x,1).

hg, Q_tfi_G=P_t(hf)(x)

h(x) g(x)u(x)h(x) u(1) dx

=f(x)

Pˆt(ug) u(x)

u(x)h(x) u(1) dx

=hf,Pˆ_t(ug) u iG.

This implies that the semi-group ( ˆQ_t, t ≥0)associated with the reversed process of Y is given by

Qˆ_t(x, dy) = ˆP_t(x, dy)U(0, y)

U(0, x) = ˆP_t(x, dy)U(1−y,1) U(1−x,1).

Then, by a change of variable, we find it equals the semi-group of 1−Y^(κ). By 3, the reversed process starts from 1. Then, it is exactly the left-continuous modification of 1−Y^(κ) for Y^(κ) starting from 0.

The above proposition gives the Radon-Nikodym derivative between the subordinator and its bridge on a sub-σ-field. We will prove Proposition 4.7.7 by showing the convergence of the Radon-Nikodym derivative from the discrete model to the continuous case.

Proof of Proposition 4.7.7. Define f^n,κ(k) = P^κ/n

2[S₁^(κ/n²⁾ = k], g^n,κ(k) = P^κ/n

2[S₁^(κ/n²⁾ ≥ k]

and C^κ/n²(k) = P^κ/n

2[{k, k + 1}is closed]. Let W_i^(κ) = S_i^(κ/n²⁾ − S_i−1^(κ/n²⁾ for i ∈ N+. As mentioned above, W_i, i∈N+ is a sequence of i.i.d. variables. Then,

dQ˜^n,κ

dQ^n,κ = f^n,κ(n−S_T^(κ/n²⁾

n−1 ) C^κ/n²(n)g^n,κ(n−S_T^(κ/n²⁾

n−1 )

whereT_p = inf{l ∈N:S_l^(κ/n²⁾ =p}. LetF_m =σ(S₀^(κ/n²⁾, . . . , Sm^(κ/n²⁾)andG_m =σ(Z₀, . . . , Z_m).

Then, G_m∩ {m < T_n}=F_m∩ {m < T_n} and G_m∩ {T_n ≤m}=F_T_n−1∩ {T_n≤m}.⁶ dQ˜^n,κ

dQ^n,κ _G

=E^(κ/n

dQ˜^n,κ dQ^n,κ

G_m

6Tn−1 is a(Fm+1, m≥0)stopping time.

=1{T_n≤m}

f^n,κ(n−S_T^(κ/n_n₋₁²⁾) C^κ/n²(n)g^n,κ(n−S_T^(κ/n_n₋₁²⁾) +E^(κ/n

1_{T_n_≥m+1} f^n,κ(n−S_T^(κ/n²⁾

n−1 ) C^κ/n²(n)g^n,κ(n−S_T^(κ/n_n₋₁²⁾)

G_m

=1{Tn≤m}

f^n,κ(n−S_T^(κ/n²⁾

n−1 ) C^κ/n²(n)g^n,κ(n−S_T^(κ/n²⁾

n−1 ) +E^(κ/n

1{Tn≥m+1}

f^n,κ(n−Sm^(κ/n²⁾−(S_T^(κ/n²⁾

n−1 −Sm^(κ/n²⁾)) C^κ/n²(n)g^n,κ(n−Sm^(κ/n²⁾−(S_T^(κ/n²⁾

n−1 −Sm^(κ/n²⁾))

G_m

=1{T_n≤m}

f^n,κ(n−S_T^(κ/n_n₋₁²⁾) C^κ/n²(n)g^n,κ(n−S_T^(κ/n²⁾

n−1 )+ 1{T_n≥m+1}

C^κ/n²(n−Sm^(κ/n²⁾) C^κ/n²(n) In the proof of Proposition 3.1 of [LJL12], it is been showed that

C^κ/n²(m) =





1−(1 + _2n^κ2 + qκ

n² +_4n^κ²4)⁻² 1−(1 + _2n^κ2 +

qκ

n² + _4n^κ²4)^−2(m+1)





(In fact, C^κ/n²(m) is denoted q^κ/n²(m) there). We deduce the following estimation for C^κ/n²(bbnc)(b >0)as n tends to ∞:

C^κ/n²(bbnc)∼





 ( ²

√κ

1−e^−2b^√^κ)^αn^−α κ >0 (bn)^−α κ= 0

Moreover, for any compact subset K ⊂]0,∞[, (C^κ/n²(bbnc)n^α, b ∈ K) converge uniformly.

Define G_t⁽ⁿ⁾ =G_bn^1−α_tc for t≥0. Then, dQ˜^n,κ

dQ^n,κ _G(n)

1{Z^(κ/n2)

bn1−αtc/n<1} = dQ˜^n,κ dQ^n,κ _G(n)

1{S^(κ/n2)

bn1−αtc/n<1} = 1

{S^(κ/n2)

bn1−αtc/n<1}

C^κ/n²(n−S_bn^(κ/n1−α²⁾tc) C^κ/n²(n) . It is proved in [Ber96] that for any fixed x >0, X_T^(κ)

]x,∞[ > x > X_T^(κ)

]x,∞[− holds with probability 1 if d= 0.

From Lemma 4.7.6, we know that the sequence of renewal process S^(κ/n²⁾ converges towards the subordinatorX^(κ) in the sense of Skorokhod. By the coupling theorem of Skorokhod and Dudley, we can suppose thatS^(κ/n²⁾ converges toX^(κ)almost surely as long as our result only depends on the law. Since for any x > 0, X_T^(κ)

]x,∞[ > x > X_T^(κ)

]x,∞[− holds with probability 1,

S_Tn−1

n converges toX_T^(κ)

]1,∞[−. Moreover, if we fixt ≥0,1

{S^(κ/n2)

bn1−αtc/n<1}

C^κ/n²(n−S^(κ/n2)

bn1−αtc)

C^κ/n²(n) converges to 1{T_]1,∞[>t}

u(1−X_t^(κ))

u(1) since X^(κ) is continuous at time t and X_T^(κ)

]1,∞[ > 1 > X_T^(κ)

]1,∞[− almost surely. Consequently, ^d_dQ^Q^˜^n,κn,κ|_G_t1

{Z^(κ/n2)

bn1−αtc/n<1} converges to 1_{T_]1,∞[_>t}^u(1−X

(κ) t )

u(1) almost surely.

The Proposition 3.1 in [LJL12] gives the density of the renewal measure of the subordinator (X_t^(κ), t≥0):

u(s) = U(0, s) = ( 2√ κ

1−e⁻²^√^κs)^α for s >0.

Let Q^x stand for the law of the Markov process with sub-Markovian semi-group Qt(x, dy) =

u(1−y)

u(1−x)P_t(x, dy) and initial state x. By Lemma 4.7.8, Q^x[1_A, t < ζ] =P^x

1_Au(1−X_t^(κ)) u(1−x) 1{t<ζ}

for A∈ F_t

where P^x is the law of the processX^(κ) starting from x. By dominated convergence, for any δ >0,

n→∞lim Q˜^n,κ

nZ_bn^(κ/n1−α²⁾tc<1−δ

=Q⁰[X_t^(κ) <1−δ]

It is clear that Q˜^n,κ[_n¹Z_bn^(κ/n1−α²⁾tc ≤1] = 1 =Q⁰[X_t^(κ)≤ 1]. Therefore, for any fixed t, ¹_nZ_bn^(κ/n1−α²⁾tc

converges in law towards X_t^(κ)(under the lawQ˜^n,κandQ⁰ respectively) asntends to infinity.

In particular, we have

n→∞lim Q^n,κ[1

{S^(κ/n2)

bn1−αtc/n<1}

C^κ/n²(n−S_bn^(κ/n1−α²⁾tc)

C^κ/n²(n) ] = lim

n→∞

Q˜^n,κ[1

nZ_bn^(κ/n1−α²⁾tc <1]

=Q⁰[X_t^(κ) <1]

=P⁰[X_t^(κ) <1,u(1−X_t) u(1) ].

Taking any bounded continuousf, by the coupling assumption and dominated convergence⁷,

n→∞lim Q^n,κ



f 1

nS_bn^κ/n1−α² sc, s∈[0, t]

1{S^(κ/n2)

bn1−αtc/n<1}

C^κ/n²(n−S_bn^(κ/n1−α²⁾tc) C^κ/n²(n)





=P⁰

f(X_s^(κ), s∈[0, t])u(1−X_t^(κ))

u(1) , X_t^(κ) <1

Equivalently,

n→∞lim Q˜^n,κ

nZ_bn^(κ/n1−α²⁾sc, s∈[0, t]

nZ_bn^(κ/n1−α²⁾tc<1

=Q⁰[f(X_s^(κ), s∈[0, t]), X_t^(κ)<1]

Therefore, we have the finite marginals convergence.

7The dominating sequence is



1{S^{(κ/n2 )}

bn1−α tc/n<1}

C^κ/n²(n−S_bn^(κ/n1−α²⁾tc) C^κ/n²(n) , n >0



.

Chapter 5 Loop erasure and spanning tree

In this section we will show that Poisson processes of loops appear naturally in the construc- tion of random spanning trees.

5.1 Loop erasure

Suppose ω is the path of a minimal transient canonical Markov process, then its path can be expressed as a sequence (x0, t0, x1, t1, . . .). The corresponding discrete path(x0, x1, . . .) is the embedded Markov chain. From the transience assumption, P

n∈N

1{xn=x} <∞ a.s..

Definition 5.1.1 (Loop erasure). The loop erasure operation which maps a path ω to its loop erased path ω_BE is defined as: ω_BE = (y₀, . . .) with y₀ = x₀. Define T₀ = inf{n ∈ N :

∀m ≥ n, x_m 6=y₀}, then set y₁ = x_T₀. Similarly define T₁ = inf{n ∈N :∀m ≥ n, x_m 6=y₁}, sety₂ =x_T₁ and so on. LetP^νBE be the image measure ofP^ν whereν is the initial distribution of the Markov process.

Recall that ∂ is the cemetery point, that Q^x_∂ = 1− P

y6=∂

Q^x_y for x 6= ∂ and Q^∂_x = δ^∂_x. Set L^x_∂ =−P

y6=∂

L^x_y forx6=∂, L^∂_∂ =−1and L^∂_x = 0 for x6=∂.

Proposition 5.1.1. We have the following finite marginal distribution for the loop-erased random walk:

P^νBE[ω_BE = (x₀, x₁, . . . , x_n, . . .)]

=νx0det(V{x0,...,xn−1})L^x_x⁰₁· · ·L^x_xⁿ⁻¹_n P^xⁿ[T{x0,...,xn−1} =∞]

=νx0L^x_x⁰₁· · ·L^x_xⁿ⁻¹_n

V_x^x₀⁰ · · · V_x^x_n−1⁰ 1 ... . .. ... ... V_x^x₀ⁿ⁻¹ · · · V_x^x_n−1ⁿ⁻¹ 1

V_x^xⁿ

0 · · · V_x^x_n−1ⁿ 1 .

Proof. Starting from x_n, the probability that the Markov process never reaches x₀, . . . , xn−1, P^xⁿ[T{x0,...,xn−1} = ∞] equals the same probability for the trace of the Markov process on x₀, . . . , x_n, P^x{xⁿ0,...,xn}[T{x₀,...,xn−1} = ∞]. It equals the one step transition probability from x_n to ∂ for the trace of the process. Let L{x₀,...,xn} be the generator of the trace of the Markov process on {x0, . . . , xn}. Then, the one step transition probability from xn

to ∂ equals (L_{x₀_,...,x_n_})^x_∂ⁿ

−(L_{x₀_,...,x_n_})^x_xⁿ

. Since (L{x₀,...,xn})^x_∂ⁿ = −(L{x₀,...,xn})^x_xⁿ_n −

n−1

i=0

(L{x₀,...,xn})^x_xⁿ

i and

−(L{x₀,...,xn})^x_xⁿ_i = (−1)^i+1+n+1det(V|{x₀,...,xn}\{x_i}×{x₀,...,xn−1})

det(V_{x₀_,...,x_n_}) , we have

(L_{x₀_,...,x_n_})^x_∂ⁿ = 1 det(V{x0,...,xn})

V_x^x₀⁰ · · · V_x^x_n−1⁰ 1 ... . .. ... ... V_x^x₀ⁿ⁻¹ · · · V_x^x_n−1ⁿ⁻¹ 1 V_x^x₀ⁿ · · · V_x^x_n−1ⁿ 1

and −(L_{x₀_,...,x_n_})^x_xⁿ

n =

V_x^x₀⁰ · · · V_x^x_n−1⁰ ... . .. ... V_x^x₀ⁿ⁻¹ · · · V_x^x_n−1ⁿ⁻¹

Therefore, P^xⁿ[T{x₀,...,xn−1} =∞] =

V_x^x₀⁰ · · · V_x^x_n−1⁰ 1 ... . .. ... ... V_x^x₀ⁿ⁻¹ · · · V_x^x_n−1ⁿ⁻¹ 1 V_x^x₀ⁿ · · · V_x^x_n−1ⁿ 1 /

V_x^x₀⁰ · · · V_x^x_n−1⁰ ... . .. ... V_x^x₀ⁿ⁻¹ · · · V_x^x_n−1ⁿ⁻¹

Set D₀ = φ and D_k = {x₀, . . . , xk−1} for k ∈ N+. Note that Q|_D^c

k×D_k^c is the transition probability for the process restricted in D^c_k. In order for the loop-erased path ω_BE to be (x₀, x₁, . . . , x_n, . . .), the random walk must start from x₀. After some excursions back to x₀, it should jump to x₁ and never return to x₀. Next, after some excursions from x₁ to x₁, it jumps to x₂ and never returns to x₀, x₁, etc. Accordingly,

P^νBE[ω_BE = (x₀, x₁, . . . , x_n, . . .)]

=ν_x₀

n−1

k=0

n≥0

((Q|_D^c

k×D_k^c)ⁿ)^x_x^k

k)Q^x_x^k

k+1P^xⁿ[T{x₀,...,xn−1} =∞]

=νx0

n−1

k=0

(V^D^c^k)^x_x^k_kL^x_x^k_k+1P^xⁿ[T{x0,...,xn−1} =∞]

where L^D^c^k =L|_D^c

k×D^c_k is the generator of the Markov process restricted in D_k^c, and V^D^c^k be the corresponding potential, see Definition 2.3.1.

Let V_F stands for the sub-matrix of V restricted to F ×F. It is also the potential of the trace of the Markov process on F and let PF stand for its law. Then, for all D⊂F, we have (V^D^c)_F = (V_F)^D^c. In particular, for k < n, we have (V^D^c^k)^x_x^k

k = ((V^D^c^k)_D_n)^x_x^k

k = ((V_D_n)^D^c^k)^x_x^k

k. One can apply Jacobi’s formula

det(A|B×B) det(A⁻¹) = det(A⁻¹|B^c×B^c)

for A = (V_D_n)^D^c^k and B ={x_k}. To be more precise, since ((V_D_n)^D^k^c)⁻¹ = (−L_D_n)|_D^c

k×D_k^c = (−L_D_n)|_(D_n−D_k)×(Dn−D_k), we have

(V^D^c^k)^x_x^k

k = ((V_D_n)^D^c^k)^x_x^k

k = det(−L_D_n|_(D_n−D_k+1)×(Dn−D_k+1)) det(−L_D_n|_(D_n_−D_k_)×(D_n_−D_k₎) with the convention that det(−LDn|φ) = 1. Then,

n−1

k=0

(V^D^c^k)^x_x^k

k =

n−1

k=0

det(−L_D_n|_(D_n−D_k+1)×(Dn−D_k+1))

det(−L_D_n|_(D_n_−D_k_)×(D_n_−D_k₎) = det(−L_D_n|_(D_n−D_n)×(D_n−D_n)) det(−L_D_n|_(D_n_−D₀_)×(D_n_−D₀₎)

= 1

det(−L_D_n) = det((VDn)Dn) = det(V{x0,...,xn−1}).

Finally, by combining the results above, we conclude that P^νBE[ω_BE = (x₀, x₁, . . . , x_n, . . .)]

=ν_x₀det(V_{x₀_,...,x_n−1_})L^x_x⁰

1· · ·L^x_xⁿ⁻¹

n P^xⁿ[T_{x₀_,...,x_n−1_} =∞]

=ν_x₀L^x_x⁰

1· · ·L^x_xⁿ⁻¹

V_x^x₀⁰ · · · V_x^x_n−1⁰ 1 ... . .. ... ... V_x^x₀ⁿ⁻¹ · · · V_x^x_n−1ⁿ⁻¹ 1

V_x^xⁿ

0 · · · V_x^x_n−1ⁿ 1 .

Remark 14. Since a Markov chain in a countable space could be viewed as a pure-jump sub-Markov process with jumping rate 1, the above result holds for a sub-Markov chain if we replace Lby the transition matrix Q−Id and V = (Id−Q)⁻¹.

The following property was discovered by Omer Angel and Gady Kozma, see Lemma 1.2 in [Koz07]. Here, we give a different proof as an application of Proposition 5.1.1.

Proposition 5.1.2. Let (Xm, m ∈[0, ζ[) be a discrete Markov chain in a countable space S with time lifeζ and initial pointx₀. Fix some w∈S\ {x₀}, defineT₁ = inf{n >0 :X_n=w}

and T_N = inf{m > TN−1 : X_m = w} with the convention that infφ = ∞. We can perform loop-erasure for the path (X0, . . . , XT_N), and let LE[0, TN] stand for the loop-erased self- avoiding path obtained in that way. If T_N <∞ with positive probability, then the conditional law of LE[0, T_N] given that {T_N <∞} is the same as the conditional law of LE[0, T₁] given that {T1 <∞}.

Proof. We suppose T₁ <∞ with positive probability. By adding a small killing rate at all states and taking ↓0, we could suppose that we have a positive probability to jump to the cemetery point from any state. In particular, the Markov chain is transient.

Let∂ be the cemetery point. Letτ(p)be a geometric variable with mean 1/p, independent of the Markov chain. Let(Xm^(p), m∈[0,(ζ−1)∧T_τ(p)]be the sub-Markov chainX stopped after T_τ(p) which is again sub-Markov. Let P^xp⁰ stand for the law of X^(p) and let P^xp,BE⁰ stand for the law of the loop-erased random walk associated to (Xm^(p), m∈[0,(ζ−1)∧T_τ(p)]). Let Q^(p) be the transition matrix of X^(p) and use the notation Qfor Q⁽⁰⁾. Then, (Q^(p))^w_i = (1−p)Q^w_i for i∈S and (Q^(p))ⁱ_j =Qⁱ_j for i∈S\ {w} and j ∈S. Accordingly, (Q^(p))^w_∂ =p+Q^w_∂ −pQ^w_∂. Define V = (I −Q)⁻¹, V_qδ_w = (M_qδ_w +I −Q)⁻¹ for q ≥ 0 and V^(p) = (I − Q^(p))⁻¹ = (M(1−p)δ_w(I+_1−p^p −Q))⁻¹ =V ^p

1−pδwM ¹

1−pδw. SetC_ω ={the loop-erased random walk stopped at w} Then,

C_Ω ={the random walk stopped at w}

n≥1

{the random walk stopped at w at timeT_k}

k≥1

{τ(p) =k, Tk < ζ}[ [

k≥1

{Tk=ζ−1, τ(p)> k}.

For xn=w,

P^xp,BE⁰ [ω_BE = (x₀, x₁, . . . , x_n=w)]

= (Q^(p))^x_x⁰

1· · ·(Q^(p))^x_xⁿ⁻¹_n (Q^(p))^x_∂ⁿ

(V^(p))^x_x⁰₀ · · · (V^(p))^x_x⁰_n ... . .. ... (V^(p))^x_xⁿ₀ · · · (V^(p))^x_xⁿ_n

= p+Q^w_∂ −pQ^w_∂ (1−p)Q^w_∂ Q^x_x⁰

1· · ·Q^x_xⁿ⁻¹_n Q^x_∂ⁿ

(V ^p

1−pδw)^x_x⁰

0 · · · (V ^p

1−pδw)^x_x⁰_n ... . .. ... (V ^p

1−pδw)^x_xⁿ₀ · · · (V ^p

1−pδw)^x_xⁿ_n .

By the resolvent equation, V_jⁱ = (V ^p

1−pδw)ⁱ_j+ _1−p^p (V ^p

1−pδw)ⁱ_wV_j^w = (V ^p

1−pδw)ⁱ_j +_1−p^p (V ^p

1−pδw)^w_jV_wⁱ. Therefore,

V_x^x⁰

0 · · · V_x^x⁰ .. n

. . .. ... V_x^x₀ⁿ · · · V_x^x_nⁿ

1 −_1−p^p V_x^x₀ⁿ · · · −_1−p^p V_x^x_nⁿ (V ^p

1−pδw)^x_x⁰_n (V ^p

1−pδw)^x_x⁰₀ · · · (V ^p

1−pδw)^x_x⁰_n ... ... . .. ... (V ^p

1−pδw)^x_x⁰_n (V ^p

1−pδw)^x_xⁿ₀ · · · (V ^p

1−pδw)^x_xⁿ_n

1 + _1−p^p V_x^x_nⁿ −_1−p^p V_x^x₀ⁿ · · · −_1−p^p V_x^x_nⁿ

0 (V ^p

1−pδw)^x_x⁰₀ · · · (V ^p

1−pδw)^x_x⁰_n

... ... . .. ...

0 (V ^p

1−pδw)^x_xⁿ₀ · · · (V ^p

1−pδw)^x_xⁿ_n

=(1 + p 1−pV_x^xⁿ

(V ^p

1−pδw)^x_x⁰

0 · · · (V ^p

1−pδw)^x_x⁰

... . .. ... (V ^p

1−pδw)^x_xⁿ₀ · · · (V ^p

1−pδw)^x_xⁿ_n .

Accordingly,(1−p+pV_w^w)Q^w_∂

p+Q^w_∂ −pQ^w_∂ P^xp,BE⁰ [ω_BE = (x₀, x₁, . . . , x_n=w)]does not depend on p. Con- sequently, it must be equal to P^x0,BE⁰ [ω_BE = (x₀, x₁, . . . , x_n =w)]. Equivalently,

(1−p+pV_w^w)Q^w_∂

p+Q^w_∂ −pQ^w_∂ P^xp,BE⁰ [·, Cw] =P^x0,BE⁰ [·, Cw] Therefore,

P^x0,BE⁰ [C_w] = (1−p+pV_w^w)Q^w_∂

p+Q^w_∂ −pQ^w_∂ P^xp,BE⁰ [C_w].

Immediately, it implies that conditionally on C_w, the law of the loop-erased random walk does not depend on p:

P^xp,BE⁰ [·|C_w] =P^x0,BE⁰ [·|C_w].

Since

P^xp,BE⁰ [ω_BE ∈ ·, C_w] =X

k≥1

P^x⁰[τ(p) = k, T_k< ζ, LE[0, T_k]∈ ·]

k≥1

P^x⁰[τ(p)> k, Tk =ζ−1, LE[0, Tk]∈ ·]

k≥1

(1−p)^k−1pP^x⁰[T_k <∞, LE[0, T_k]∈ ·]

k≥1

(1−p)^kP^x⁰[T_k <∞, LE[0, T_k]∈ ·]Q^w_∂

k≥1

(1−p)^k−1(p+Q^w_∂ −pQ^w_∂)P^x⁰[Tk<∞]P^x⁰[LE[0, Tk]∈ ·|Tk <∞],

we have

P^xp,BE⁰ [ω_BE ∈ ·|C_w] = P

k≥1

(1−p)^k−1P^x⁰[Tk <∞]P^x⁰[LE[0, Tk]∈ ·|Tk <∞]

k≥1

(1−p)^k−1P^x⁰[T_k<∞] . (*) Since P^xp,BE⁰ [ωBE ∈ ·|Cw] does not depend on p ∈ [0,1], we will denote it by Q. Then the equation (*) can be written as follows:

Q[·]X

k≥1

(1−p)^k−1P^x⁰[T_k <∞]

k≥1

(1−p)^k−1P^x⁰[T_k <∞]P^x⁰[LE[0, T_k]∈ ·|T_k<∞].

Finally, by identifying the coefficients, we conclude that P^x⁰[LE[0, T_k]∈ ·|T_k<∞] =Q[·] as long as P^x⁰[T_k <∞] for k ≥1and we are done.

Consider (et, t ≥ 0), a Poisson point process of excursions of finite lifetime at x with the intensity Leb⊗(−L^x_x− _V¹x

x)ν_{x},ex^x→x . (Recall that ν_{x},ex^x→x is the normalized excursion measure at x, see Definition 3.3.2.) Let (γ(t), t ≥ 0) be an independent Gamma subordinator¹ with the Laplace exponent

Φ(λ) =

∞

(1−e^−λs)s⁻¹e^−s/V^x^xds.

LetRαbe the closure of the image of the subordinatorγup to timeα, i.e. Rα={γ(t) :t∈[0, α]}.

Then, [0, γ(α)]\ R_α is the union of countable disjoint open intervals, {]γ(t−), γ(t)[: t ∈ [0, α], γ(t−) < γ(t)}. To such an open interval ]g, d[, one can associate a based loop l as follows: During the time interval ]g, d[, the Poisson point process(et, t≥0)has finitely many excursions, namely e_t₁,· · · , e_t_n, g < t₁ < · · · < t_n < d. Each excursion e_t_i is viewed as a càdlàg path of lifetime ζ_t_i: (e_t_i(s), s∈[0, ζ_t_i[). Define l : [0, d−g+P

ζ_t_i]→S as follows:

l(s) =







e_t_i(s−(P

j<i

ζ_t_j +t_i)) if s∈[P

j<i

ζ_t_j +t_i,P

j≤i

ζ_t_j+t_i[

x otherwise.

This mapping between an open interval ]g, d[ and a based loop l depends on ]g, d[ and (et, t∈]g, d[)and we denote is by Ψ^]g,d[ (l= Ψ^]g,d[(e)). By mapping a based loop into a loop, we get a countable collection of loops for α≥0, namelyO_α.

Proposition 5.1.3. (Oα, α ≥ 0) has the same law as the Poisson point process of loops intersecting {x}, i.e. ({l ∈ L_α :l^x>0}, α >0).

1See Chapter III of [Ber96].

Proof. As both sides have independent stationary increment, it is enough to show O1 = {l ∈ L₁ : l^x > 0}. It is well-known that (γ(t)−γ(t−), t ∈ R) is a Poisson point process with characteristic measure ¹_se^−s/V^x^xds. Therefore, P

l∈Oα

δ_l^x is Poisson random measure with intensity ¹_se^−s/V^x^xds. On the other hand, for the Poisson ensemble of loopsL_α, by taking the trace of the loops on x and dropping the empty ones, as a consequence of Proposition 3.2.1, we get a Poisson ensemble of trivial Markovian loops with intensity measure ¹_se^s(L^{x}⁾^x^xds where (−L_{x})^x_x = 1/V_x^x. Consequently, we have

{l^x :l∈ O₁} has the same law as{l^x :l∈ L₁, l^x >0}

In other words, by disregarding the excursions attached to each loop, the set of trivial loops in x obtained from O1 and {l∈ L1 :l^x >0}is the same. In order to restore the loops, we need to insert the excursions into the trivial loops. Then, it remains to show that the excursions are inserted into the trivial loops in the same way. Finally, by using the independence between (et, t ≥ 0) and (γ(t), t ≥ 0) and the stationary independent increments property with respect to time t, it ends up in proving the following affirmation: Ψ^]0,T[(e) induces the same probability on the loops with l^x = T as the loop measure conditioned by {l^x = T}.

By Proposition 3.1.7, we have l^xµ(dl) = µ^x,x(dl). Hence, µ(dl|l^x = T) = µ^x,x(dl|l^x = T) where µ^x,x is considered to be a loop measure. Let P^x be the law of the Markov process (X_t, t∈ [0, ζ[) associated with the Markovian loop measure µ. Let (L(x, t), t ∈[0, ζ[)be the local time process at xand L⁻¹(x, t)be its right-continuous inverse. Letτ be an independent exponential variable with parameter 1. Define the processX^L⁻¹^(x,τ⁾with lifetimeL⁻¹(x, τ)∧ζ as follows: X^L⁻¹^(x,τ⁾(T) = X(T), T ∈[0, L⁻¹(x, τ)∧ζ[. Denote byQ[dl]the law of X^L⁻¹^(x,τ⁾. Then,e^−l^xµ^x,x(dl) = Q[dl]whereµ^x,x(dl)is considered to be a based loop measure. Therefore,

µ^x,x(dl|l^x =T) = Q[dl|l^x =T] =Q[dl|τ =T] =the law of Ψ^]0,T^[(e)

in the sense of based loop measures. Then, the equality stills holds for loop measures and we are done.

Suppose (X_t, t ∈[0, ζ[)is a transient Markov process onS. (Assume the process stays at the cemetery point after lifetime ζ.) Define the local time at x L(x, t) =

1{Xs=x}ds. Denote by L⁻¹(x, t) its right-continuous inverse and by L⁻¹(x, t−) its left-continuous inverse. The excursion process(et, t≥0)is defined by et(s) = X_s+L⁻¹(x,t−), s∈[0, L⁻¹(x, t)−L⁻¹(x, t−)[.

Define a measure on the excursion which never returns to x by

ν^x→[dl] =X

y∈S

Q^x_yP^y[dl,the process never hits x].

We can calculate the total mass of ν˜^x→ as follows:

ν^x→[1] =X

y∈S

Q^x_yP^y[the process never hitsx]

= 1−E^x[{after leaving x, the process returns to x}]

= (1−(R^{x})^x) = (L{x})^x_x

L^x_x =− 1 V_x^xL^x_x. After normalization, we get a probability measure ν^x→. The law of the first excursion is

−_Vx¹

xL^x_xν^x→+

1 + _Vx¹ xL^x_x

ν_{x},ex^x→x . In particular, the first excursion is not an excursion from x back to x with probability−_Lx¹

xV_x^x. According to the excursion theory, the excursion process is a Poisson point process stopped at the appearing of an excursion of infinite lifetime or an excursion that ends up at the cemetery. The characteristic measure is proportional to the law of the first excursion. By taking the trace of the process on x, we know that the total occupation time is an exponential variable with parameter (−L{x})^x_x = _V¹x

x . According to the excursion theory, it is an exponential variable with parameter _V^−dx

xL^x_x, d being the mass of the characteristic measure. Immediately, we get d = −L^x_x. If we focus on the process of excursions from x back to x, it is a Poisson point process with characteristic measure (−L^x_x − _V¹x

x)ν_{x},ex^x→x stopped at an independent exponential time with parameter _V¹x x . Let (γ(t), t≥0) be a Gamma subordinator with Laplace exponent

Φ(λ) =

∞

(1−e^−λs)s⁻¹e^−s/V^x^xds.

Then,γ(t)follows theΓ(t,_V¹x

x )distribution with densityρ(y) = _Γ(t)(V¹ x

x)^ty^t−1e^−y/V^x^x. In particular, γ(1) is an exponential variable of the parameter 1/V_x^x. It is known that (_γ(1)^γ(t), t∈[0,1]) is independent of γ(1), and that it is a Dirichlet process. (One can prove this by a di- rect calculation on the finite marginal distribution.) Moreover, the jumps of the process (_γ(1)^γ(t), t∈[0,1])rearranged in decreasing order follow the Poisson-Dirichlet(0,1)distribution.

For x ∈ S, let Z_x be the last passage time in x: Z_x = sup{t ∈ [0, ζ[: X(t) = x}. Suppose the loop erased path ωBE equals (x1, . . .). DefineSn =Txn for n ≥1 and S0 = 0. LetOi be (X_s, s ∈ [S_i, S_i+1[) i.e. the i-th loop erased from the process X. Then O₁ can be viewed as a Poisson point process (e⁽¹⁾_t , t ∈ [0, L(x₁, ζ)[) of excursions at x₁ killed at the arrival of an excursion with infinite lifetime or an excursion ending up at the cemetery. Conditionally on ω_BE = (x₁, x₂, . . .), the shifted process(X(s+T₁), s∈[0, ζ[)is the Markov process restricted in S \ {x₁} starting from x₂ = X(T₁) . Moreover, it is conditionally independent of the Poisson point process e⁽¹⁾. Once again, we can view O2 as an killed Poisson point process

No documento Contribution à l’étude des lacets markoviens (páginas 77-94)