Weakly pinned random walk on the wall: pathwise descriptions of the phase transition^{1}

Yasuki Isozaki^{2} and Nobuo Yoshida^{3}

Key words and phrases: random walk, weak pinning, wall condition, entropic repulsion, wetting transition, limit theorems.

Abstract

We consider a one-dimensional random walk which is conditioned to stay non-negative and is “weakly pinned” to zero. This model is known to exhibit a phase transition as the strength of the weak pinning varies. We prove path space limit theorems which describe the macroscopic shape of the path for all values of the pinning strength. If the pinning is less than (resp. equal to) the critical strength, then the limit process is the Brownian meander (resp. reflecting Brownian motion). If the pinning strength is supercritical, then the limit process is a positively recurrent Markov chain with a strong mixing property.

1 Introduction

In this paper, we discuss a localization-delocalization phase transition of a discrete interface above the attractive wall. Large number of works have been done on this subject, especially for one-dimensional interfaces, e.g. [BDZ00, Bur81, Cha82, Fis84, Upt99, vLHi81] among all (See also [BDZ00, CaVe00, DeVe00, DMRR92, IoVe00] for extensions in higher dimensions). The model considered in this paper is one-dimensional and is based on a symmetric, nearest neighbor random walk ({S_{n}}_{n≥0}, {P^{x}}_{x∈ℤ}) on ℤ. To be precise, {S_{n}}_{n≥0} are integer valued random variables and {P^{x}}_{x∈ℤ} are probability measures such that the following hold for all n ≥ 1 and x ∈ ℤ; S_{1} − S_{0}, …, S_{n} − S_{n−1} are P^{x}-independent and

P^{x}{S_{0} = x} = 1, P^{x}{S_{n} − S_{n−1} = 0} = r < 1, P^{x}{S_{n} − S_{n−1} = ±1} = v/2,  (1.1)

where r + v = 1. For n ≥ 1 and x ∈ ℤ, we denote the number of 0 ≤ j ≤ n − 1 such that S_{j} = x by L_{n}^{x}. We define a probability measure μ_{n}^{x} for x, n ∈ ℕ = {0, 1, …} by μ_{0}^{x} = P^{x} and

μ_{n}^{x}(dω) = \frac{1}{Z_{n}^{x}} P^{x} [exp (βL_{n}^{0}) : dω ∩ \{S_{1} ≥ 0, …, S_{n} ≥ 0\}] , n ≥ 1  (1.2)

where β ∈ ℝ and Z_{n}^{x} is the normalizing constant.

In the case β > 0 which we are mainly interested in, the basic feature of the measure μ_{n}^{x} can be explained as follows. We refer to the exponential factor in (1.2) as the weak pinning which tries to localize the path (S_{k})_{k=1}^{n} near the origin. The effect of the weak pinning, however, has to compete with the entropic fluctuation of the random walk, which together with the restriction to {S_{1} ≥ 0, …, S_{n} ≥ 0}, has a tendency to push the path away from the origin. We call the latter tendency entropic repulsion. The competition between weak pinning and entropic repulsion has been well studied at the level of the free energy. We set

1to appear in Stoch. Proc. Appl. (2001)

2Department of Mathematics, Graduate School of Science, Osaka University,Machikaneyama-cho, Toyonaka 560-0043, Japan. email: yasuki@math.sci.osaka-u.ac.jp

3Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan.

email:nobuo@kusm.kyoto-u.ac.jp

β_{c} = ln \frac{2}{1 + r}, .  (1.3)

For β > 0, an equation

q(λ) =^{def.} 4(e^{β} − 1)λ^{2} − 4e^{β}(e^{β} − 1)rλ − e^{2β}v^{2} = 0  (1.4)

has real solutions λ_{±} such that

λ_{−} < 0 < λ_{+}, and λ_{+} = |λ_{−}| + re^{β}.  (1.5)

Since q(1) = − ((1 + r)e^{β} − 2)^{2}, we see that

λ_{+} ≥ 1 with the equality if and only if β = β_{c}.  (1.6)

It is known [Bol99, Cha82, Fis84] and will also be seen later in this paper that the free energy for this model can explicitly be given as follows;

ψ(β) =^{def.} lim_{n↗∞} n^{−1} ln Z_{n}^{x} = \{ \begin{matrix}0 & if β ≤ β_{c}, \\ ln(λ_{+}) > 0 & if β > β_{c}. \end{matrix}  (1.7)

for all x ∈ ℕ. In particular, there is a phase transition at β = β_{c} in the sense that ψ(β) = ψ(0) for β ≤ β_{c} and ψ(β) > ψ(0) for β > β_{c}. This phase transition is often called the wetting transition. The phenomenalistic explanation of the wetting transition is the following. For β < β_{c}, the effect of the weak pinning is smaller than that of the entropic repulsion and the pinning effect does not change the diffusive nature of the original random walk in a qualitative way. On the other hand, if β > β_{c}, the effect of the weak pinning overwhelms that of the entropic repulsion and it confines the path near the origin. The purpose of this paper is to justify the above expected phenomena as limit theorems.

We start by describing the (scaling) limit of a random variable S_{n} under the measure μ_{n}^{x}. To do so, we introduce the following definitions.

Let p ≥ 0, m ≥ 1 and f : I^{m} → ℝ, where I is either [0, ∞) or ℕ. We say f is p-bounded if

sup_{(x_{j})_{j=1}^{m}∈I^{m}}(1 + |x_{1}| + … + |x_{m}|)^{−p}|f(x_{1}, …, x_{m})| < ∞.  (1.8)

We say f is polynomially bounded if it is p-bounded for some p ≥ 0. We introduce a bilinear form

⟨ f_{1}, f_{2} ⟩_{β} = e^{−β}f_{1}(0)f_{2}(0) + ∑_{x≥1} f_{1}(x)f_{2}(x)  (1.9)

for f_{i} : ℕ → ℝ (i = 1, 2) when the summation on the right-hand-side converges absolutely. For β > β_{c}, we set

ρ_{+}(1) = \frac{−(e^{β} − 1)r + \sqrt{(e^{β} − 1)^{2}r^{2} + (e^{β} − 1)v^{2}}}{(e^{β} − 1)v} ∈ (0, 1)  (1.10)

and define functions ρ_{±} : ℕ → [0, 1] by

ρ_{+}(x) = ρ_{+}(1)^{x},
ρ_{−}(x) = (−ρ_{+}(1))^{x} if r = 0,
ρ_{−}(x) ≡ 0 if r > 0.  (1.11)

Finally, for β > β_{c}, we define probability measures m_{β,k} on ℕ (k ∈ ℤ) by

m_{β,k}[f] = \frac{⟨ f, ρ_{+} + (−1)^{k}ρ_{−} ⟩_{β}}{⟨ 1, ρ_{+} + (−1)^{k}ρ_{−} ⟩_{β}} , f : ℕ → [0, ∞).  (1.12)

If r > 0, m_{β,k} is independent of the choice of k, while, if r = 0, (1.12) defines exactly two different measures: m_{β,0} and m_{β,1}.

Theorem 1.1 a) If β ≤ β_{c}, then

lim_{n↗∞} μ_{n}^{x} [f(S_{n}/\sqrt{vn})] = ∫_{[0,∞)} fdμ  (1.13)

for all x ∈ ℕ and for any polynomially bounded continuous function f : [0, ∞) → ℝ, where

μ(dt) = \{ \begin{matrix}e^{−t^{2}/2}tdt & if β < β_{c}, \\ \sqrt{2/π}e^{−t^{2}/2}dt & if β = β_{c}. \end{matrix}  (1.14)

b) For β > β_{c}, there are constants C_{1.15} = C_{1.15}(β, v) > 0 and ε_{1.15} = ε_{1.15}(β, v) ∈ (0, 1) as follows; if n ≥ (1 + x)C_{1.15} and f : ℕ → ℝ is polynomially bounded, then

|μ_{n}^{x}[f(S_{n})] − m_{β,x+n}[f]| ≤ B_{1.15}(f)ε_{1.15}^{n},  (1.15)

where the measure m_{β,x+n} is defined by (1.12) and B_{1.15}(f) is a constant which depends only on β, v and f.

Remark 1.1 The situation described by (1.15) is different for r > 0 and r = 0. For r > 0, (1.15) in particular says that, as n ↗ ∞, the measures μ_{n}^{x}{S_{n} ∈ ·} converge to m_{β,0} for all x ∈ ℕ. For r = 0, the measure μ_{n}^{x}{S_{n} ∈ ·} is concentrated on the set {x ∈ ℕ ; (−1)^{x+n} = 1}, so that the limit obtained from even n’s and odd n’s are different, depending on the parity of x.

We have the following process level limit theorem for β ≤ β_{c};

Theorem 1.2 For β ≤ β_{c}, consider probability measures μ̃_{n}^{x}, n ≥ 1, x ∈ ℕ on C([0, 1]) defined by

μ̃_{n}^{x} = μ_{n}^{x} \{(S_{nt}/\sqrt{vn})_{0≤t≤1} ∈ ·\} ,  (1.16)

where (S_{t})_{t≥0} is the linear interpolation of the random walk (S_{n})_{n∈ℕ}. Then, for any fixed x ∈ ℕ, as n ↗ ∞, μ̃_{n}^{x} converges weakly to the law of a process W = (W_{t})_{0≤t≤1}. The process W is described as follows;

W_{t} = \{ \begin{matrix}(1 − τ)^{−1/2}B_{τ+t(1−τ)} & if β < β_{c}, \\ |B_{t}| & if β = β_{c}, \end{matrix}  (1.17)

where B = (B_{t})_{0≤t≤1} is a one-dimensional Brownian motion starting from 0 and τ = sup{t ∈ [0, 1] ; B_{t} = 0}.

Remark 1.2 Similar limit theorems are discussed in the literatures. Bolthausen et al consider the case of one-dimensional weakly pinned Gaussian field in the delocalized phase [BDZ00, Section4]. In [Upt99], Upton discusses one-dimensional SOS interfaces and the phase boundaries of two-dimensional Ising model, where the thermodynamic parameters are assumed to be sufficiently inside delocalized region.

We have a process level limit theorem also for β > β_{c}. We will have a positively recurrent Markov chain as the limit process. To describe the Markov chain obtained as the limit, we introduce matrices Q = (Q_{xy})_{x,y∈ℕ} and R = (R_{xy})_{x,y∈ℕ} by

Q_{xy} = ((e^{β} − 1)δ^{0,x} + 1) P^{x}{S_{1} = y},  (1.18)
R_{xy} = λ_{+}^{−1}ρ_{+}(x)^{−1}Q_{xy}ρ_{+}(y).  (1.19)

We see by direct computation that

λ_{+} = e^{β} (r + \frac{v}{2}ρ_{+}(1)) = r + \frac{v}{2} (ρ_{+}(1) + ρ_{+}(1)^{−1}) .  (1.20)

This implies that ∑_{y∈ℕ} Q_{xy}ρ_{+}(y) = λ_{+}ρ_{+}(x) for all x ∈ ℕ and hence that R is a transition probability for an ℕ-valued Markov chain. We denote this Markov chain by ({S_{n}}_{n∈ℕ}, {ν^{x}}_{x∈ℕ}). It is easy to check that the Markov chain {ν^{x}}_{x∈ℕ} is reversible with respect to the probability measure m on ℕ defined by

m[f] = \frac{⟨ f, ρ_{+}^{2} ⟩_{β}}{⟨ 1, ρ_{+}^{2} ⟩_{β}} , f : ℕ → [0, ∞).  (1.21)

In particular, {ν^{x}}_{x∈ℕ} is positively recurrent.

Theorem 1.3 For β > β_{c}, there are constants C_{1.22} = C_{1.22}(β, v) > 0 and ε_{1.22} = ε_{1.22}(β, v) ∈ (0, 1) as follows; if n ≥ k + (1 + x)C_{1.22}, f : ℕ^{k} → ℝ is polynomialy bounded and F = f(S_{1}, …, S_{k}), then

|μ_{n}^{x}[F] − ν^{x}[F]| ≤ B_{1.22}(f)ε_{1.22}^{n−k},  (1.22)

where B_{1.22}(f) is a constant which depends only on β, v and f.

By Theorem 1.3, we get the following strong mixing properties for both μ_{n}^{x} and ν^{x} for β > β_{c};

Theorem 1.4 For β > β_{c}, there are constants C_{1.23} = C_{1.23}(β, v) > 0 and ε_{1.23} = ε_{1.23}(β, v) ∈ (0, 1) as follows; if n ≥ l + m, l ≥ k + (1 + x)C_{1.23}, f : ℕ^{k} → ℝ and g : ℕ^{m} → ℝ are polynomially bounded, F = f(S_{1}, …, S_{k}) and G = g(S_{l+1}, …, S_{l+m}) then

|μ_{n}^{x}(F G) − μ_{n}^{x}(F)μ_{n}^{x}(G)| ≤ B_{1.23}(f, g)ε_{1.23}^{l−k},  (1.23)
|ν^{x}(F G) − ν^{x}(F)ν^{x}(G)| ≤ B_{1.23}(f, g)ε_{1.23}^{l−k},  (1.24)

where B_{1.23}(f, g) is a constant which depends only on β, v, f and g.

Remark 1.3 In [BDZ00, Section4], it is mentioned that a similar statement to (1.23) is true for the one-dimensional weakly pinned Gaussian field in the localized phase. Localization of two-dimensional weakly pinned surfaces are discussed in [DeVe00, DMRR92, IoVe00], however without the positivity constraint.

2 Proof of Theorem 1.1 (a)

2.1 Preliminaries for the proof

For x, n ∈ ℕ, we introduce the following random variables;

h_{n} = 1{S_{1} ≥ 0, …, S_{n} ≥ 0}, n ≥ 1,  (2.1)
e_{n} = exp (βL_{n}^{0}) , β ∈ ℝ, n ≥ 1,  (2.2)
z_{n} = e_{n}h_{n}, n ≥ 1,  (2.3)
z_{0} = e_{0} = h_{0} = 1,  (2.4)

where 1{…} denotes the indicator function. Then, the measure μ_{n}^{x} can be rewritten as the following concise form;

μ_{n}^{x}(dω) = \frac{1}{Z_{n}^{x}} P^{x}[z_{n} : dω], x, n ∈ ℕ,  (2.5)

We need to introduce some more notations. For a function φ : ℕ → ℝ, a formal power series

φ̂(s) = ∑_{n∈ℕ}s^{n}φ(n), −1 ≤ s ≤ 1,  (2.6)

is called the generating function of φ. If φ is non-negative and the convergence radius is one, then the following important relation between φ and φ̂ is known as a Tauberian theorem [Fel70, page 447]; for α ∈ [0, ∞) and C > 0,

φ̂(s) ∼^{s↗1} C(1 − s)^{−α} if and only if φ(0) + ⋯ + φ(n − 1) ∼^{n↗∞} \frac{Cn^{α}}{Γ(α + 1)}.  (2.7)

Moreover, if α ≠ 0 and φ is monotone, then (2.7) is equivalent to

φ(n) ∼^{n↗∞} \frac{Cn^{α−1}}{Γ(α)} .  (2.8)

We set

f^{x}(n) = \{ \begin{matrix}0 & if n = 0, \\ P^{x}\{T_{1} = n\}, & if n ≥ 1, \end{matrix}  (2.9)

where T_{m} denotes the m^{th}-hitting time to the origin;

T_{m} = inf {n ≥ 1 | ∑_{k=1}^{n} 1{S_{k} = 0} = m} .  (2.10)

It is then well known that the generating function is given by

f̂^{x}(s) = \{ \begin{matrix}(\frac{1−rs−γ(s)^{1/2}}{vs} )|x| & if x ≠ 0, |s| ≤ 1, \\ 1 − γ(s)^{1/2} & if x = 0, |s| ≤ 1, \end{matrix}  (2.11)

where γ(s) = (1 − rs)^{2} − s^{2}v^{2}. 2.2 Lemmas We define

θ_{±}(s) = 1 − \frac{e^{β}}{2} (rs + 1 ± \sqrt{γ(s)}) for s ∈ [−1, 1].  (2.12)

Note that θ_{−}(s) is strictly concave on [−1, +1] and is strictly decreasing on [0, 1]. Since, θ_{−}(0) = 1 and θ_{−}(1) = 1 − \frac{e^{β}}{2} (1 + r), we have that

\{s ∈ [0, 1] ; θ_{−}(s) = 0\} = \{ \begin{matrix}∅ & if β < β_{c}, \\ \{1\} & if β = β_{c}. \end{matrix}  (2.13)

Lemma 2.1 For x, y, n ∈ ℕ, define

z_{n}^{y} = z_{n}1{S_{n} = y},  (2.14)
Z_{n}^{x,y} = P^{x}[z_{n}^{y}].  (2.15)

Then, for 0 ≤ s < 1,

Ẑ^{x,0}(s) = \frac{1}{θ_{−}(s) ∨ 0} (δ^{x,0} + (1 − δ^{x,0})f̂^{x}(s)) .  (2.16)

In particular,

Ẑ^{x,0}(1) = 1/θ_{−}(1) if β < β_{c},  (2.17)

and

Ẑ^{x,0}(s) ∼^{s↗1} C_{2.18}/(1 − s)^{1/2}, if β = β_{c}.  (2.18)

where C_{2.18} = \frac{1+r}{\sqrt{2v}}.

Proof: The equality (2.16) is essentially the same as the representation of Green function of the random walk in terms of its hitting times. We follow [Bol99]. Note first that

Ẑ^{x,0}(s) = δ^{x,0} + ∑_{n≥1} s^{n}P^{x}[z_{n}^{0}].  (2.19)

To compute the right-hand-side, let us introduce

g^{x}(n) = \{ \begin{matrix}0 & if n = 0, \\ P^{x}\{h_{n} : T_{1} = n\} & if n ≥ 1. \end{matrix}  (2.20)

It is then easy to see that

g^{x}(n) = \{ \begin{matrix}f^{x}(n) & if x ≥ 1, \\ f^{x}(n) = r & if x = 0 and n=1, \\ f^{x}(n)/2 & if x = 0 and n ≥ 2. \end{matrix}  (2.21)

and consequently that for 0 ≤ s ≤ 1,

ĝ^{x}(s) = \{ \begin{matrix}(rs + f̂^{0}(s))/2 & if x = 0, \\ f̂^{x}(s) & if x ≥ 1. \end{matrix}  (2.22)

We compute the right-hand-side of (2.19), first for the case x ≥ 1;

∑_{n≥1} s^{n}P^{x}[z_{n}^{0}]  (2.23)
= ∑_{n≥1} s^{n} ∑_{m≥1} e^{(m−1)β} ∑_{0<i_{1}<...<i_{m}=n} P^{x}[h_{n} : T_{1} = i_{1}, …, T_{m} = i_{m}]
= ∑_{n≥1} s^{n} ∑_{m≥1} e^{(m−1)β} ∑_{{j_{1}≥1,...,j_{m}≥1 \atop j_{1}+...+j_{m}=n}} g^{x}(j_{1})g^{0}(j_{2}) ⋯ g^{0}(j_{m})

= ∑_{m≥1} ∑_{j_{1}≥1} … ∑_{j_{m}≥1} s^{j_{1}}g^{x}(j_{1}) (e^{β}s^{j_{2}}g^{0}(j_{2})) ⋯ (e^{β}s^{j_{m}}g^{0}(j_{m}))

= ĝ^{x}(s) ∑_{m≥1} (e^{β}ĝ^{0}(s))^{m−1}
= ĝ^{x}(s) ∑_{m≥1} (1 − θ_{−}(s))^{m−1} ,  (2.24)

which, implies (2.16) for the case x ≥ 1. Equality (2.16) for the case x = 0 can be seen similarly as above just by replacing e^{(m−1)β} and g^{x}(j_{1}) respectively by e^{mβ} and g^{0}(j_{1}). □

Lemma 2.2 For 0 ≤ s < 1,

Ẑ^{x}(s) = (1 − δ^{x,0}) \frac{1 − f̂^{x}(s)}{1 − s} + (\frac{1}{2}e^{β} \frac{1 − f̂^{0}(s)}{1 − s} − \frac{1}{2}e^{β} + 1) Ẑ^{x,0}(s).  (2.25)

In particular,

Ẑ^{x}(s) ∼^{s↗1} C_{2.26}/(1 − s)^{1/2} if β < β_{c},  (2.26)

where C_{2.26} = \sqrt{\frac{2}{v}} (x + \frac{ve^{β}}{2θ_{−}(1)}).

Proof: We begin by showing that

Z_{n}^{x} = \frac{1}{2}e^{β} ∑_{m=0}^{n−1} Z_{m}^{x,0}F_{n−m}^{0} + Z_{n}^{x,0} + (1 − δ^{x,0})F_{n}^{x},  (2.27)

where F_{n}^{x} = P^{x}{T_{1} > n}. To prove this, we set

k_{m,n} = 1{S_{m+1} > 0, …, S_{n} > 0}, 0 ≤ m ≤ n,
d_{n} = max D_{n} if D_{n} =^{def.} {0 ≤ j ≤ n ; S_{j} = 0} ≠ ∅.

We have

P^{x}[k_{0,n}] = \{ \begin{matrix}F_{n}^{x}/2 & if x = 0, \\ F_{n}^{x} & if x ≥ 1. \end{matrix}  (2.28)

Note also that D_{n} = ∅ if and only if x ≥ 1 and T_{1} > n. Since we have either d_{n} ∈ {0, …, n} or D_{n} = ∅, we see that z_{n} can be decomposed as follows;

z_{n} = ∑_{m=0}^{n−1} z_{n}1{d_{n} = m} + z_{n}1{d_{n} = n} + (1 − δ^{0,x})z_{n}1{T_{1} > n}

= ∑_{m=0}^{n−1} e^{β}z_{m}^{0}k_{m,n} + z_{n}^{0} + (1 − δ^{0,x})k_{0,n}.

We therefore have

Z_{n}^{x} = e^{β} ∑_{m=0}^{n−1} P^{x}[z_{m}^{0}k_{m,n}] + Z_{n}^{x,0} + (1 − δ^{x,0})P^{x}[k_{0,n}]

= \frac{1}{2}e^{β} ∑_{m=0}^{n−1} Z_{m}^{x,0}F_{n−m}^{0} + Z_{n}^{x,0} + (1 − δ^{x,0})F_{n}^{x},

where we have used (2.28) in the last line. This proves (2.27).

It is easy to see that

F̂^{x}(s) = \frac{1 − f̂^{x}(s)}{1 − s}.  (2.29)

Therefore, (2.25) is an easy consequence of (2.16), (2.29) and (2.27). We see (2.26) from (2.25) and the following asymptotic property for f̂^{x}(s);

1 − f̂^{x}(s) ∼ \{ \begin{matrix}\sqrt{2v(1 − s)} & if x = 0, \\ x\sqrt{2(1 − s)/v} & if x ≠ 0, \end{matrix} as s ↗ 1.  (2.30)

□

To state and prove the next lemma, we will use the following notation. For a sequence (a_{n})_{n≥0} (random or non-random), we set Δa_{n} = a_{n} − a_{n−1} for n ≥ 1.

Lemma 2.3 Let Φ_{0} be a constant and Φ_{n} = φ_{n}(S_{1}, …, S_{n}) for n ≥ 1, where each φ_{n} is a function on ℕ^{n}. Then,

P^{x}[Φ_{n}z_{n}] = Φ_{0} + ∑_{k=1}^{n} P^{x}[z_{k−1}ΔΦ_{k}] + ∑_{k=1}^{n} P^{x}[z_{k−1}^{0}(e^{β}1{ΔS_{k} ≥ 0} − 1)ΔΦ_{k}]

−θ_{−}(1) ∑_{k=1}^{n} P^{x}[z_{k−1}^{0}Φ_{k−1}].  (2.31)

In particular,

Z_{n}^{x} = 1 − θ_{−}(1) ∑_{k=0}^{n−1} Z_{k}^{x,0},  (2.32)

P^{x}[S_{n}^{p}z_{n}] = x^{p} + v ∑_{1≤j≤p/2} \frac{p!}{(2j)!(p − 2j)!} ∑_{k=0}^{n−1} P^{x}[S_{k}^{p−2j}z_{k}]

+\frac{v}{2}\{e^{β} − 1 − (−1)^{p}\} ∑_{k=0}^{n−1} Z_{k}^{x,0}.  (2.33)

Proof: We observe that

Δz_{n} = (e^{β}1{ΔS_{n} ≥ 0} − 1) z_{n−1}^{0}.

We therefore have

Δ(z_{n}Φ_{n}) = z_{n}ΔΦ_{n} + Φ_{n−1}Δz_{n}
= (z_{n−1} + Δz_{n})ΔΦ_{n} + Φ_{n−1}Δz_{n}
= z_{n−1}ΔΦ_{n} + (e^{β}1{ΔS_{n} ≥ 0} − 1) z_{n−1}^{0}ΔΦ_{n} + (e^{β}1{ΔS_{n} ≥ 0} − 1) z_{n−1}^{0}Φ_{n−1}.

This implies (2.31), since P^{x}[Φ_{n}z_{n}] = Φ_{0} + ∑_{k=1}^{n} P^{x}[Δ(z_{k}Φ_{k})] and P^{x}[e^{β}1{ΔS_{n} ≥ 0} − 1] = −θ_{−}(1). The other two equalities are obtained easily from (2.31). □

2.3 Proof of Theorem 1.1(a)

It is enough to prove that for all x, p ∈ ℕ,

lim_{n↗∞} μ_{n}^{x} [(S_{n}/\sqrt{vn})^{p}] = ∫_{0}^{∞} x^{p}μ(dx) = \{ \begin{matrix}2^{p/2}Γ(\frac{p}{2} + 1) & if β < β_{c}, \\ \sqrt{2^{p}/π}Γ(\frac{p+1}{2}) & if β = β_{c}. \end{matrix}  (2.34)

We begin with the case β < β_{c}. Since Z_{n}^{x} is decreasing in n by (2.32), we see from (2.26) and (2.8) that

Z_{n}^{x}∼C_{2.26}/(πn)^{1/2} as n ↗ ∞.  (2.35)

Therefore, if we set M_{n}(p) = P^{x}[S_{n}^{p}z_{n}], then, (2.34) for β < β_{c} is equivalent to that;

lim_{n↗∞} n^{− \frac{p−1}{2}} M_{n}(p) = (C_{2.26}/\sqrt{π})(2v)^{p/2}Γ(\frac{p}{2} + 1).  (2.36)

We prove (2.36) by induction. For p = 0, (2.36) can immediately be seen from (2.35). For p = 1, (2.36) can be seen from (2.17), (2.33) and (2.35). Let us consider the case p ≥ 2. We may now assume (2.36) with p replaced by p−2j (1 ≤ j ≤ p/2). We denote the right-hand-side of (2.36) by m(p). It is then not difficult to see that

n^{− \frac{p−1}{2}} ∑_{k=0}^{n−1} M_{k}(p − 2j)

= n^{− \frac{p−1}{2}} ∑_{k=0}^{n−1} k^{− \frac{p−2j−1}{2}} M_{k}(p − 2j)k^{\frac{p−2j−1}{2}} ⟶^{n↗∞} \{ \begin{matrix}\frac{m(p−2)}{(p−1)/2}, & if j = 1, \\ 0 & if otherwise. \end{matrix}  (2.37)

We have

M_{n}(p) = v ∑_{1≤j≤p/2} \frac{p!}{(2j)!(p − 2j)!} ∑_{k=0}^{n−1} M_{k}(p − 2j) + O(1) as n ↗ ∞

by (2.17) and (2.33). We see from this and (2.37) that

lim_{n↗∞} n^{− \frac{p−1}{2}} M_{n}(p) = v \frac{p!}{2!(p − 2)!} \frac{m(p − 2)}{(p − 1)/2}
= vpm(p − 2) = m(p),

which completes the proof of (2.36).

Proof of (2.34) for β = β_{c} is similar as above. What we have to show in this case is that;

lim_{n↗∞} n^{− \frac{p}{2}} M_{n}(p) = \sqrt{(2v)^{p}/π}Γ(\frac{p+1}{2}).  (2.38)

We prove (2.38) by induction. For p = 0, (2.38) can immediately be seen from (2.32). For p = 1, we see from (2.18) and (2.7) that

∑_{k=0}^{n−1} Z_{k}^{x,0}∼2C_{2.18}\sqrt{n/π} as n ↗ ∞.  (2.39)

This and (2.33) imply (2.38) for p = 1. Let us consider the case p ≥ 2. We may now assume (2.38) with p replaced by p − 2j (1 ≤ j ≤ p/2). We denote the right-hand-side of (2.38) by m(p). It is then not difficult to see that

n^{− \frac{p}{2}} ∑_{k=0}^{n−1} M_{k}(p − 2j)

= n^{− \frac{p}{2}} ∑_{k=0}^{n−1} k^{− \frac{p−2j}{2}} M_{k}(p − 2j)k^{\frac{p−2j}{2}} ⟶^{n↗∞} \{ \begin{matrix}\frac{m(p−2)}{p/2} & if j = 1, \\ 0 & if otherwise. \end{matrix}  (2.40)

We have

M_{n}(p) = v ∑_{1≤j≤p/2} \frac{p!}{(2j)!(p − 2j)!} ∑_{k=0}^{n−1} M_{k}(p − 2j) + O(n^{1/2}) as n ↗ ∞

by (2.33) and (2.39). We see from this and (2.40) that

lim_{n↗∞} n^{− \frac{p}{2}} M_{n}(p) = v \frac{p!}{2!(p − 2)!} \frac{m(p − 2)}{p/2}
= v(p − 1)m(p − 2) = m(p),

which completes the proof of (2.38). □

3 Proof of Theorem 1.2

We consider the case x = 0 for simplicity. The modifications if otherwise are straightforward.

3.1 The critical case

In this case, the paths of {S_{n}} can be embedded in the paths of W (t) ≡ |B(t)| without changing the law under μ_{n}. Note first that for all k < n,

If S_{k−1} > 0, μ_{n}{ΔS_{k} = 0|S_{k−1}} = r; μ_{n}{ΔS_{k} = ±1|S_{k−1}} = v/2.  (3.1)
If S_{k−1} = 0, μ_{n}\{ΔS_{k} = 0|S_{k−1}\} = \frac{2r}{1 + r}; μ_{n}\{ΔS_{k} = 1|S_{k−1}\} = \frac{v}{1 + r}.  (3.2)

By modifying the Skrokhod embedding, we can construct a series of stopping times {σ_{k}}_{k≥0} so that {W (σ_{k})}_{k=1}^{n} is a realization of ({S_{k}}_{k=1}^{n}, μ_{n}^{0}). In fact, for constants K_{0} and K_{1} we define {σ_{k}}_{k≥0} as follows; σ_{0} := 0 and if W (σ_{k−1}) ∈ ℕ \ {0} then

σ_{k} := inf{t ≥ σ_{k−1}; |W (t) − W (σ_{k−1})| = 1 or (t > K_{1} + σ_{k−1} and W (t) = W (σ_{k−1}))}

if W (σ_{k−1}) = 0 then

σ_{k} := inf{t ≥ σ_{k−1}; W (t) = 1 or (t > K_{0} + σ_{k−1} and W (t) = 0)}.

We can then choose K_{0} and K_{1} (depending upon β and v) so that S̃_{k} =^{def.} W (σ_{k}) satisfies

If S̃_{k−1} > 0, P^{W} {ΔS̃_{k} = 0|S̃_{k−1}} = r; P^{W} {ΔS̃_{k} = ±1|S̃_{k−1}} = v/2.  (3.3)
If S̃_{k−1} = 0, P^{W} \{ΔS̃_{k} = 0|S̃_{k−1}\} = \frac{2r}{1 + r}; P^{W} \{ΔS̃_{k} = 1|S̃_{k−1}\} = \frac{v}{1 + r}.  (3.4)

By discarding k’s such that W (σ_{k−1}) = 0, we obtain a series of i.i.d. {τ_{k}; k ≥ 0} := {σ_{k} − σ_{k−1}; k ≥ 0, W (σ_{k−1}) ≠ 0}. It is elementary to prove E^{W} [τ_{k}] = v. By well-known Kolmogorov’s inequality, we have

P^{W} [∃k ≤ n, | ∑_{j=1}^{k}(τ_{j} − v)| > δn] ≤ V ar(τ_{1})/δ^{2}n.  (3.5)

Since the number of downcrossings from 1/\sqrt{n} to 0 approximate the local time, it holds, after rescaled by \sqrt{n}, for any ε > 0, there is K_{2} > 0 such that for all large n,

P^{W} [#\{k ≤ n; W (σ_{k−1}) = 0\} > K_{2}\sqrt{n}/v] < ε.  (3.6)

The following is its natural consequence: there is K_{3} > 0 such that for all large n,

P^{W} [ ∑_{k≤n;W (σ_{k−1})=0}(σ_{k} − σ_{k−1}) > K_{3}\sqrt{n}] < ε.  (3.7)

Choose n so large that 1/n < δ and (K_{2} ∨ K_{3})\sqrt{n} < δn, where δ if beforehand chosen so that

P^{W} [ sup_{0<t,s<1;|s−t|<δ} |W (vns) − W (vnt)| > ε\sqrt{n}] < ε.  (3.8)

Now it remains to put these together:

sup_{0<t<1} |\frac{W (vnt)}{\sqrt{n}} − \frac{S̃_{nt}}{\sqrt{n}}| < ε + sup_{0≤k≤n} |\frac{W (vk)}{\sqrt{n}} − \frac{S_{k}}{\sqrt{n}}|

≡ ε + sup_{0≤k≤n} |\frac{W (vk)}{\sqrt{n}} − \frac{W (σ_{k})}{\sqrt{n}}|

< 2ε + sup_{0≤k≤n} |\frac{W (vk)}{\sqrt{n}} − \frac{W (∑_{j≤k;W (σ_{j−1})≠0}(σ_{j} − σ_{j−1}))}{\sqrt{n}}|

< 3ε + sup_{0≤k≤n} |\frac{W (vk)}{\sqrt{n}} − \frac{W (∑_{j≤k;W (σ_{j−1})≠0} v)}{\sqrt{n}}|
< 4ε

on the intersection of the events that appeared in (3.5), (3.6), (3.7) and (3.8).

We have thus proven μ̃_{n} converges weakly to the law of \{W (vnt)/\sqrt{n}; 0 ≤ t ≤ 1\} which in turn is identical to that of \{\sqrt{v}W (t); 0 ≤ t ≤ 1\}. □

3.2 Convergence of finite dimensional distributions for β < β_{c}

The purpose of this subsection is to prove the following

Lemma 3.1 Let β < β_{c}, k ≥ 1 and 0 < t_{1} < …, < t_{k} ≤ 1. Suppose that {t_{j,n} > 0 ; 1 ≤ j ≤ k, n ≥ 1} are such that 0 < t_{1,n} < …, < t_{k,n} for all n ≥ 1 and that sup_{n≥1} |t_{j,n} − nt_{j}| < ∞ for all 1 ≤ j ≤ k. Then, probability measures on ℝ^{k} defined by

μ_{n}^{0} \{(S_{t_{j,n}}/\sqrt{vn})_{j=1}^{k} ∈ ·\} , ; n = 1, 2, …  (3.9)

converge weakly to the law of (W_{t_{j}})_{j=1}^{k}.

We introduce the following functions;

g_{t}(x) = (2πt)^{−1/2} exp (−\frac{x^{2}}{2t} ) , t > 0, x ∈ ℝ,  (3.10)
h(x) = 2 ∫_{0}^{x} g_{1}(y)dy, x ≥ 0,  (3.11)
g_{t}(x, y) = g_{t}(x − y) − g_{t}(x + y), t > 0, x, y ∈ ℝ,  (3.12)
h(t, x) = h (\frac{x}{\sqrt{1 − t}}) , (t, x) ∈ [0, 1] × [0, ∞)\\{(1, 0)\}  (3.13)

According to Belkin [Bel72], the process W for β < β_{c} is a (time-inhomogeneous) diffusion process on [0, ∞) with the following transition probability; for 0 ≤ s < t ≤ 1 and x, y ≥ 0

p(s, x|t, y) = \{ \begin{matrix}t^{−3/2}y exp (−\frac{y^{2}}{2t} ) h(t, y) & if s = 0 and x = 0 \\ h(s, x)^{−1}g_{t−s}(x, y)h(t, y) & if s > 0 and x > 0, \\ 0 & if otherwise \end{matrix}  (3.14)

We set

S_{n}^{min} = min_{0≤j≤n} S_{j}.  (3.15)

Lemma 3.2 Let β < β_{c}. Then, for 0 ≤ s < t < 1, x, y ≥ 0,

lim_{n↗∞} P^{y\sqrt{vn}} [z_{n−⌊ns⌋} : \frac{S_{⌊nt⌋−⌊ns⌋}}{\sqrt{vn}} ≤ x] = P \{ min_{0≤u≤1−s} B_{u} ≥ −y, B_{t−s} ≤ x − y\} (3.16)
= h(s, y) ∫_{0}^{x} p(s, y|t, z)dz.

The convergence is uniform in (x, y) ∈ [0, ∞) × [0, b] for any b > 0. In particular,

lim_{n↗∞} Z_{n−⌊ns⌋}^{y\sqrt{vn}} = h(s, y).  (3.17)

uniformly in y ∈ [0, b] for any b > 0.

Proof: We write ⟨ u ⟩ = ⌊u⌋ − ⌊ns⌋ (u ≥ 0) in this proof. We define

τ = τ(y, n) = inf\{j ≥ 1 ; S_{j} ≤ S_{0} − y\sqrt{vn}\}

and divide the expectation on the left-hand-side of (3.16) as follows;

P^{y\sqrt{vn}} [z_{⟨ n ⟩} : S_{⟨ nt ⟩} ≤ x\sqrt{vn}]
= P^{0} [exp (βL_{⟨ n ⟩}^{−y\sqrt{vn}}) : \frac{S_{⟨ n ⟩}^{min}}{\sqrt{vn}} ≥ −y, \frac{S_{⟨ nt ⟩}}{\sqrt{vn}} ≤ x − y]

= P^{0} [\frac{S_{⟨ n ⟩}^{min}}{\sqrt{vn}} > −y, \frac{S_{⟨ nt ⟩}}{\sqrt{vn}} ≤ x − y]  (3.18)

+ ∑_{k=0}^{⟨ n ⟩−1} P^{0} [exp (βL_{⟨ n ⟩}^{−y\sqrt{vn}}) : \frac{S_{⟨ n ⟩}^{min}}{\sqrt{vn}} ≥ −y, \frac{S_{⟨ nt ⟩}}{\sqrt{vn}} ≤ x − y, τ = k]  (3.19)

The first term (3.18) converges to the right-hand-side of (3.16) by the invariance principle of Donsker. Moreover, the convergence is uniform in (x, y) ∈ [0, ∞)^{2} by a result of Billingsley and Topsϕe [BiTo67, Theorem 2]. On the other hand, the second term (3.19) is bounded from above by

∑_{k=0}^{⟨ n ⟩−1} P^{0}{τ = k}Z_{⟨ n ⟩−k}^{0}.  (3.20)

By using (2.35) and the fact that P^{0}\{τ = k\} is not greater than a constant times y\sqrt{n}k^{−3/2}, it is not difficult to bound (3.20) by a constant times yn^{−1/3}. This completes the proof. □

Proof of Lemma 3.1: Note first that it is enough to prove (3.9) for a special case: t_{j,n} = ⌊nt_{j}⌋ (1 ≤ j ≤ k). To see this, let {t_{j,n}} be an arbitrary sequence which satisfies the assumptions in Lemma 3.1. Since S_{t_{j,n}} and S_{⌊nt_{j}⌋} differ at most by |t_{j,n} − ⌊nt_{j}⌋|, the L^{1}(μ_{n}^{0})-difference of two vectors (S_{t_{j,n}}/\sqrt{vn})_{j=1}^{k} and (S_{⌊nt_{j}⌋}/\sqrt{vn})_{j=1}^{k} vanishes as n ↗ ∞. Therefore, their limits in law coincide.

Following Iglehart [Igl74, proof of (2.23)], we prove (3.9) by induction on k. We will prove that for all x_{1}, …, x_{k} > 0,

lim_{n↗∞} μ_{n}^{0}(A_{k}) = ∫_{0}^{x_{1}} dy_{1} … ∫_{0}^{x_{k}} dy_{k}p(0, 0|t_{1}, y_{1})p(t_{1}, y_{1}|t_{2}, y_{2}) ⋯ p(t_{k−1}, y_{k−1}|t_{k}, y_{k}), (3.21)

where A_{k} = \{\frac{S_{⌊nt_{1}⌋}}{\sqrt{vn}} ≤ x_{1}, …, \frac{S_{⌊nt_{k}⌋}}{\sqrt{vn}} ≤ x_{k}\}. We set (s, t, x) = (t_{k−1}, t_{k}, x_{k}) to simplify the expressions. We start with the case k = 1. If t = 1, then, (3.21) follows from (1.13). Let us therefore assume t < 1. We then have

μ_{n}^{0} \{\frac{S_{⌊nt⌋}}{\sqrt{vn}} ≤ x\} = \frac{Z_{⌊nt⌋}^{0}}{Z_{n}^{0}} ∫_{y≤x} μ_{⌊nt⌋}^{0} \{\frac{S_{⌊nt⌋}}{\sqrt{vn}} ∈ dy\} Z_{n−⌊nt⌋}^{y\sqrt{vn}}  (3.22)

We see from Lemma 3.2 that Z_{n−⌊nt⌋}^{y\sqrt{vn}} converges to h (t, y) locally uniformly in y. We therefore have by (1.13) and (2.35) that

lim_{n↗∞} μ_{n}^{0} \{\frac{S_{⌊nt⌋}}{\sqrt{vn}} ≤ x\} = t^{−1/2} ∫_{0}^{x} μ(t^{−1/2}dy)h (t, y) dy
= ∫_{0}^{x} p(0, 0|t, y)dy

which proves (3.21) for k = 1. We now consider the case k ≥ 2. Since

\frac{S_{⌊nt_{j}⌋}}{\sqrt{vn}} = \sqrt{\frac{m}{n}} \frac{S_{m·⌊nt_{j}⌋/m}}{\sqrt{vm}} , m = ⌊ns⌋,

we have by the induction hypothesis and the observation at the beginning of the proof that

lim_{n↗∞} μ_{⌊ns⌋}^{0} \{(S_{⌊nt_{j}⌋}/\sqrt{vn})_{j=1}^{k−1} ∈ ·\} = the law of (\sqrt{s}W_{t_{j}/s})_{j=1}^{k−1}.

Note that the above limit has the density

p_{s} (0, 0|t_{1}, y_{1}) p_{s} (t_{1}, y_{1}|t_{2}, y_{2}) ⋯ p_{s} (t_{k−2}, y_{k−2}|s, y_{k−1}) ,  (3.23)

where

p_{s} (t, x|t′, y) = s^{−1/2}p (t/s, x/\sqrt{s}|t′/s, y/\sqrt{s}) .

If we set h_{s}(t, x) = h(t/s, x/\sqrt{s}), then we have obviously that

p_{s} (t, x|t′, y) = \{ \begin{matrix}s^{1/2}p (0, 0|t′/s, y/\sqrt{s}) \frac{h_{s}(t′,y)}{h(t′,y)}, & if t = 0 and x = 0 \\ \frac{h_{s}(t,x)}{h(t,x)}p (t/s, x/\sqrt{s}|t′/s, y/\sqrt{s}) \frac{h_{s}(t′,y)}{h(t′,y)}, & if t > 0 and x > 0. \end{matrix}  (3.24)

We rewrite the probability on the left-hand-side of (3.21) as follows;

μ_{n}^{0}(A_{k}) = \frac{Z_{⌊ns⌋}^{0}}{Z_{n}^{0}} ∫_{0}^{x_{k−1}} μ_{⌊ns⌋}^{0} \{A_{k−2}, \frac{S_{⌊ns⌋}}{\sqrt{vn}} ∈ dy\} P^{y\sqrt{vn}} [z_{n−⌊ns⌋} : \frac{S_{⌊nt⌋−⌊ns⌋}}{\sqrt{vn}} ≤ x] (3.25)

By (2.35), (3.25), (3.16) and (3.24) we conclude that

lim_{n↗∞} μ_{n}^{0}(A_{k})

= s^{−1/2} ∫_{0}^{x_{1}} dy_{1} … ∫_{0}^{x_{k−2}} dy_{k−2} ∫_{0}^{x_{k−1}} dyp_{s} (0, 0|t_{1}, y_{1}) p_{s} (t_{1}, y_{1}|t_{2}, y_{2}) ⋯ p_{s} (t_{k−2}, y_{k−2}|s, y)
×h(s, y) ∫_{0}^{x} p(s, y|t, z)dz
= ∫_{0}^{x_{1}} dy_{1} … ∫_{0}^{x_{k}} dy_{k}p(0, 0|t_{1}, y_{1})p(t_{1}, y_{1}|t_{2}, y_{2}) ⋯ p(t_{k−1}, y_{k−1}|t_{k}, y_{k}).  (3.26)

This completes the proof of Lemma 3.1. □

3.3 Tightness for β < β_{c}

We continue to consider the distribution of (S_{nt}/\sqrt{vn})_{0≤t≤1} under μ_{n}^{0} with β < β_{c}. We have shown in Lemma 3.1 that any finite dimensional distribution converges. The purpose of this subsection is to prove the tightness of the distribution, which, together with Lemma 3.1, implies Theorem 1.2 for β < β_{c}. By the well-known Kolmogorov’s tightness criterion, it is enough to prove that for ε > 0,

lim_{δ↘0} sup_{n≥1} μ_{n}^{0} \{ max_{{t,u≤1 \atop |t−u|≤δ}} |S_{nt} − S_{nu}| ≥ ε\sqrt{n}\} = 0,  (3.27)

It is easy to see that (3.27) follows from the following estimate.

Lemma 3.3 For β < β_{c}, there is a constant C_{3.28} = C_{3.28}(β, v) ∈ (0, ∞) such that for all l ≥ 1 and λ ≥ 1;

μ_{n}^{0} \{ max_{{0≤j≤l \atop 0≤k≤n−j}} |S_{k+j} − S_{k}| ≥ λ\sqrt{vl}\} ≤ C_{3.28} (n/l)^{3/2} exp (−λ/C_{3.28}) .  (3.28)

We first prove the following exponential integrability property.

Lemma 3.4 Let β < β_{c} and ε > 0 be such that

\frac{v}{2}(e^{ε} − 1)(e^{β} + e^{−ε} − 1) ≤ θ_{−}(1).  (3.29)

Then,

sup_{n≥ε^{2}} P^{x} [z_{n} exp (ε |\frac{S_{n} − x}{\sqrt{n}}|)] ≤ 2 exp (\frac{cosh(1)vε^{2}}{2} ) .  (3.30)

Proof: We will show that

P^{x} [z_{n} exp (αS_{n} − αx)] ≤ exp ((cosh(α) − 1)vn) , if α ∈ (−∞, ε]  (3.31)

Since e^{|t|} ≤ e^{t} + e^{−t} (t ∈ ℝ) and cosh(t) − 1 ≤ \frac{1}{2} cosh(1)t^{2} (|t| ≤ 1), the desired inequality (3.30) follows from (3.31) by plugging ±ε/\sqrt{n} in place of α. To prove (3.31), we set Φ_{n} = exp (αS_{n} − αx) and a_{n} = P^{x}[z_{n}Φ_{n}]. Then, it is easy to see that

P^{x} [z_{k−1}ΔΦ_{k}] = (cosh(α) − 1)va_{k−1},
P^{x}[z_{k−1}^{0}(e^{β}1\{ΔS_{k} ≥ 0\} − 1)ΔΦ_{k}] = e^{−αx} \frac{v}{2}(e^{α} − 1)(e^{β} + e^{−α} − 1)Z_{k−1}^{x,0},
P^{x}[z_{k−1}^{0}Φ_{k−1}] = e^{−αx}Z_{k−1}^{x,0}.

Hence we see from (2.31) that

a_{n} ≤ 1 + (cosh(α) − 1)v ∑_{k=1}^{n} a_{k−1},

which implies (3.31). □

Proof of Lemma 3.3: We introduce the following notation;

M_{k,n} = max_{k≤j≤n} |S_{j} − S_{k}|.  (3.32)

We divide the proof into two steps;

Step1: We first prove that there is a constant C_{3.33} = C_{3.33}(β, v) ∈ (0, ∞) such that for all y > 0 and λ ≥ 1;

sup_{n∈ℕ}P^{y\sqrt{vn}} [z_{n} : \frac{M_{0,n}}{\sqrt{vn}} ≥ λ] ≤ C_{3.33} exp (−λ/C_{3.33}) ,  (3.33)
sup_{n∈ℕ} \sqrt{n}P^{0} [z_{n} : \frac{M_{0,n}}{\sqrt{vn}} ≥ λ] ≤ C_{3.33} exp (−λ/C_{3.33}) .  (3.34)

We follow Belkin [Bel72], cf. proof of (2.9) in that paper. We set A_{n} = \{\frac{M_{0,n}}{\sqrt{vn}} ≥ λ\} and

τ = inf \{j ≥ 0 ; \frac{|S_{j}−S_{0}|}{\sqrt{vn}} ≥ λ\}. Then, we have for any m ≥ 1 that

P^{y\sqrt{vn}} [z_{n} : A_{n}]
= P^{y\sqrt{vn}} [z_{n} : τ ≤ n − m] + P^{y\sqrt{vn}} [z_{n} : A_{n} ∩ \{τ > n − m\}] .  (3.35)

We divide the first term of (3.35) into

I_{1} = P^{y\sqrt{vn}} [z_{n} : τ ≤ n − m, \frac{S_{τ}}{\sqrt{vn}} < y − λ] ,
I_{2} = P^{y\sqrt{vn}} [z_{n} : τ ≤ n − m, \frac{S_{τ}}{\sqrt{vn}} ≥ y + λ] .

Note that I_{1} ≠ 0 only if y > λ, in which case we have z_{τ} = 1. Hence

I_{1} = P^{y\sqrt{vn}} [Z_{n−τ}^{S_{τ}} : τ ≤ n − m, \frac{S_{τ}}{\sqrt{vn}} < y − λ]
≤ P^{y\sqrt{vn}} \{τ ≤ n − m, \frac{S_{τ}}{\sqrt{vn}} < y − λ\}
≤ P^{y\sqrt{vn}} \{\frac{M_{0,n}}{\sqrt{vn}} > λ\}
≤ C_{3.36} exp(−λ^{2}/2).  (3.36)

To give an upper bound of I_{2}, we observe that there is C_{3.37} = C_{3.37}(v) ∈ (1, ∞) such that

P^{x\sqrt{vn}} [z_{n−k} : \frac{S_{n−k}}{\sqrt{vn}} ≥ x] ≥ 1/C_{3.37} if x ≥ 1 and n − k ≥ C_{3.37}.  (3.37)

In fact,

P^{x\sqrt{vn}} [z_{n−k} : \frac{S_{n−k}}{\sqrt{vn}} ≥ x] ≥ P^{0} [\frac{S_{n−k}^{min}}{\sqrt{nv}} > −x, \frac{S_{n−k}}{\sqrt{vn}} ≥ 0]

≥ P^{0} [\frac{S_{n−k}^{min}}{\sqrt{v(n − k)}} > −1, \frac{S_{n−k}}{\sqrt{v(n − k)}} ≥ 0]

The last displayed probability converges, as n − k ↗ ∞, to

P { min_{0≤t≤1} B_{t} ≥ −1, B_{1} ≥ 0} > 0.

This proves (3.37). We now take m = ⌊C_{3.37}⌋ + 1. We then have by (2.32) and (3.37) that

Z_{n−k}^{x\sqrt{vn}} ≤ 1 ≤ C_{3.37}P^{x\sqrt{vn}} [z_{n−k} : \frac{S_{n−k}}{\sqrt{vn}} ≥ x] for x ≥ 1,

and therefore that

I_{2} = ∑_{k≤n−m} ∫_{x≥y+λ} P^{y\sqrt{vn}} [z_{k} : τ = k, \frac{S_{k}}{\sqrt{vn}} ∈ dx] Z_{n−k}^{x\sqrt{vn}}

≤ C_{3.37} ∑_{k≤n−m} ∫_{x≥y+λ} P^{y\sqrt{vn}} [z_{k} : τ = k, \frac{S_{k}}{\sqrt{vn}} ∈ dx] P^{x\sqrt{vn}} [z_{n−k} : \frac{S_{n−k}}{\sqrt{vn}} ≥ x]

= C_{3.37}P^{y\sqrt{vn}} [z_{n} : \frac{S_{n}}{\sqrt{vn}} ≥ y + λ] .  (3.38)

We divide the second term of (3.35) into

J_{1} = P^{y\sqrt{vn}} [z_{n} : A_{n} ∩ \{τ > n − m, \frac{|S_{n} − S_{0}|}{\sqrt{vn}} > \frac{λ}{2}\}] ,
J_{2} = P^{y\sqrt{vn}} [z_{n} : A_{n} ∩ \{τ > n − m, \frac{|S_{n} − S_{0}|}{\sqrt{vn}} ≤ \frac{λ}{2}\}] .

We have

J_{2} = ∫_{x:|x−y|≤λ} P^{y\sqrt{vn}} [z_{n−m} : \frac{S_{n−m}}{\sqrt{vn}} ∈ dx] P^{x\sqrt{vn}} [z_{m} : max_{j≤n} |\frac{S_{j}}{\sqrt{vn}} − y| ≥ λ, |\frac{S_{m}}{\sqrt{vn}} − y| ≤ \frac{λ}{2}]
≤ e^{βm} ∫_{x:|x−y|≤λ} P^{y\sqrt{vn}} [z_{n−m} : \frac{S_{n−m}}{\sqrt{vn}} ∈ dx] P^{0} [\frac{M_{0,m}}{\sqrt{vn}} ≥ \frac{λ}{2}]
≤ C_{3.39} exp (−λ^{2}n/C_{3.39}) ,  (3.39)

where C_{3.39} = C_{3.39}(β). Combining (3.36), (3.38) and (3.39), we get

P^{y\sqrt{vn}} [z_{n} : A_{n}] ≤ (1 + C_{3.37})P^{y\sqrt{vn}} [z_{n} : \frac{|S_{n} − S_{0}|}{\sqrt{vn}} ≥ \frac{λ}{2}] + C_{3.39} exp (−λ^{2}n/C_{3.39}) . (3.40)

This and (3.30) imply (3.33). On the other hand, we have by (1.13) that

lim_{n↗∞} μ_{n}^{0} \{\frac{S_{n}}{\sqrt{vn}} ≥ \frac{λ}{2}\} = exp(−λ^{2}/8).

This, together with (2.35) and (3.40), implies (3.34).

Step2: Proof of (3.28). It is easy to see that the probability on the left-hand-side of (3.28) is bounded from above by

∑_{p=0}^{⌊n/l⌋} μ_{n}^{0}(A_{p}),  (3.41)

where A_{p} = \{M_{pl,(p+1)l} ≥ λ\sqrt{vl}/3\}, cf. [KaSh91, proof of (4.19)]. For p ≥ 1, we have by (2.32), (3.33) and (2.35) that

P^{0} [z_{n} : A_{p}] ≤ P^{0} [z_{(p+1)l} : A_{p}]
= ∫ P^{0} [z_{pl} : \frac{S_{pl}}{\sqrt{vpl}} ∈ dx] P^{x\sqrt{vpl}} [z_{l} : M_{l} ≥ λ\sqrt{vl}/3]
≤ C_{3.42}Z_{(p+1)l}^{0} exp (−λ/C_{3.42})  (3.42)
≤ C_{3.43}l^{−1/2} exp (−λ/C_{3.42})  (3.43)

For p = 0, we use (3.34) to see that

P^{0} [z_{n} : A_{0}] ≤ P^{0} [z_{l} : A_{0}] ≤ C_{3.44}l^{−1/2} exp (−λ/C_{3.44}) .  (3.44)

Therefore,

∑_{p=0}^{⌊n/l⌋} μ_{n}^{0}(A_{p}) ≤ C_{3.45}(n/l)n^{1/2}l^{−1/2} exp (−λ/C_{3.43})
= C_{3.45}(n/l)^{−3/2} exp (−λ/C_{3.43}) .  (3.45)

This proves (3.28). □

4 Supercritical phase

4.1 Lemmas

The argument we present here is similar to the one developed by van Leeuwen and Hilhorst [vLHi81, Section2]. Namely, we look at the matrix Q and reduce the original problem to the spectral properties of the matrix. Before the next lemma, let us recall that we have defined functions θ_{±}(s) by (2.12). For β > 0, we see by direct computations that

{s ∈ [−1, 1] ; θ_{−}(s) = 0} ⊂ {1/λ_{−}, 1/λ_{+}}.  (4.1)

Since θ_{−}(s) is strictly decreasing on [0, 1], θ_{−}(0) = 1 and θ_{−}(1) < 0 for β > β_{c}, we see that

{s ∈ [0, 1] ; θ_{−}(s) = 0} = {1/λ_{+}} for β > β_{c}.  (4.2)

Lemma 4.1 Let s = 1/λ ∈ (−1, +1). Then, for β > β_{c}, an equation

Qu = λu,  (4.3)

given the value of u(0), can uniquely be solved as

u(x) = \frac{u(0)}{e^{β}\sqrt{γ(s)}} (θ_{−}(s)f̂^{x}(s)^{−1} − θ_{+}(s)f̂^{x}(s)) , x ≥ 1.  (4.4)

In particular, the equation (4.3) for |λ| > 1 has a bounded solution only if λ = λ_{±}.

Proof: The equation (4.3) reads

e^{β}ru(0) + \frac{1}{2}e^{β}vu(1) = λu(0),  (4.5)
ru(x) + \frac{1}{2}v (u(x − 1) + u(x + 1)) = λu(x) x ≥ 1.  (4.6)

If s = 1/λ ∈ (−1, +1), then the characteristic equation

rt + \frac{1}{2}v (1 + t^{2}) = λt

for (4.6) has f̂^{±1}(s) as its two distinct real solutions (|f̂^{1}(s)| < 1 < |f̂^{−1}(s)|). It is therefore elementary to get (4.4). □

Lemma 4.2 For β > β_{c}, there is a constant ε_{4.7} = ε_{4.7}(β, v) ∈ (0, 1) such that

|λ_{+}^{−n}P^{x}[f(S_{n})z_{n}] − \frac{⟨ f, ρ_{+} + (−1)^{x+n}ρ_{−} ⟩_{β}}{⟨ ρ_{+}, ρ_{+} ⟩_{β}} ρ_{+}(x)| ≤ (1 + |x|)^{p}B_{4.7}(f)ε_{4.7}^{n}  (4.7)

for all x, n ∈ ℕ and a p-bounded function f : ℕ → ℝ, where B_{4.7}(f) is a constant which depends only on β, v and f.

Proof: Since Q_{xy} = P^{x}[z_{1}^{y}], we have

P^{x}[f(S_{n})z_{n}] = ∑_{y∈ℕ}P^{x}[z_{n}^{y}]f(y)
= ∑_{x_{1},x_{2},...,x_{n−1},y∈ℕ}Q_{xx_{1}}Q_{x_{1}x_{2}} ⋯ Q_{x_{n−1}y}f(y)
= Q^{n}f(x).  (4.8)

We will prove (4.7) by studying the matrix Q. To do so, let us introduce a Hilbert space

ℓ_{β}^{2} = {f : ℕ → ℝ ; ⟨ f, f ⟩_{β} < ∞},

equipped with the inner product defined by (1.9). For f : ℕ → ℝ, we set

Πf(x) = \frac{⟨ f, ρ_{+} ⟩_{β}ρ_{+}(x) + ⟨ f, ρ_{−} ⟩_{β}ρ_{−}(x)}{⟨ ρ_{+}, ρ_{+} ⟩_{β}}, x ∈ ℕ.

Then, Π : ℓ_{β}^{2} → ℓ_{β}^{2} is nothing but the orthogonal projection to the space spanned by ρ_{±}. Note also that

λ_{+}^{−n}Q^{n}Πf(x) = \frac{⟨ f, ρ_{+} + (−1)^{x+n}ρ_{−} ⟩_{β}}{⟨ ρ_{+}, ρ_{+} ⟩_{β}} ρ_{+}(x)  (4.9)

for all f : ℕ → ℝ and x ∈ ℕ. Let us first prove that

∥λ_{+}^{−n}Q^{n}(f − Πf)∥_{β} ≤ ∥f − Πf∥_{β}ε_{4.10}^{n} for f ∈ ℓ_{β}^{2},  (4.10)

where ∥ · ∥_{β} denots the ℓ_{β}^{2}-norm. Consider Q as a bounded operator on ℓ_{β}^{2}. Then, Q is symmetric and bounded. Therefore, σ(Q): the set of ℓ_{β}^{2}-spectra of Q, is a compact set in ℝ. We claim that

σ(Q) ⊂ [−1, 1] ∪ {λ_{−}, λ_{+}},  (4.11)
λ_{+} > sup{|λ| ; λ ∈ σ(Q), |λ| ≠ λ_{+}}.  (4.12)

We let σ^{disc.}(Q) and σ^{ess.}(Q) denote respectively, the discrete and the essential spectra of Q. Now, consider a matrix P = (P^{x}{S_{1} = y})_{x,y∈ℕ}. Then, P is ℓ_{0}^{2}-symmetric and has the operator norm one on ℓ_{0}^{2}. This implies σ(P) ⊂ [−1, 1], since the set of ℓ_{β}^{2}-spectra of a bounded operator is independent of the choice of β. Note also that Q−P is of finite rank and hence is a compact operator, which implies that σ^{ess.}(Q) = σ^{ess.}(P) ⊂ [−1, 1], cf. [ReSi78, page 111, Lemma 3].

On the other hand, we see from Lemma 4.1 that σ^{disc.}(Q)\[−1, +1] ⊂ {λ_{−}, λ_{+}}. This, together with (1.5), implies (4.11) and (4.12). The claim (4.10) can now easily be seen from (4.8),(4.11), (4.12) and the spectral decomposition. The desired estimate (4.7) for polynomially bounded f can be seen by applying (4.10) to a cut-off function f_{x,n}(y) = f(y)1{|x − y| ≤ n} as follows. By (4.9) and the equality Q^{n}f_{x,n}(x) = Q^{n}f(x), the left-hand-side of (4.7) is bounded by

|λ_{+}^{−n}Q^{n}(f_{x,n} − Πf_{x,n})(x)| + |λ_{+}^{−n}Q^{n}Π(f_{x,n} − f)(x)|  (4.13)

Since f_{x,n} ∈ ℓ_{β}^{2}, we see from (4.10) that the first term of (4.13) is not greater than ε_{4.10}^{n}∥f_{x,n}∥_{β} which has a bound in the form of the right-hand-side of (4.7). On the other hand, it is easy to bound the second term of (4.13) by B(f)ρ_{+}(n), where B(f) is a constant which depends only on β, v and f. This completes the proof. □

Lemma 4.3 Suppose that f : ℕ^{k} → ℝ is p-bounded and F = f(S_{1}, …, S_{k}), then there is a constant B_{4.14} = B_{4.14}(β, v, f) > 0 such that for all n ≥ k and x ∈ ℕ,

P^{x}[|F |z_{n}]λ_{+}^{−n} ≤ B_{4.14}(1 + |x|)^{p},  (4.14)
μ_{n}^{x}[|F |] ≤ B_{4.14}(1 + |x|)^{p}.  (4.15)

Proof: Given (4.7), the proof is straightforward and hence is omitted. □

4.2 Proof of Theorem 1.1 (b)

We prove (1.15) in the following generalized form.

Lemma 4.4 For β > β_{c}, there are constants C_{4.16} = C_{4.16}(β, v) > 0 and ε_{4.16} = ε_{4.16}(β, v) ∈ (0, 1) as follows; if l ≥ k + (1 + x)C_{4.16}, f : ℕ^{k} → ℝ and h : ℕ → ℝ are polynomially bounded, F = f(S_{1}, …, S_{k}), then

|μ_{l}^{x}[F h(S_{l})] − m_{β,x+l}[h]μ_{l}^{x}[F]| ≤ B_{4.16}(f, h)ε_{4.16}^{l−k}.  (4.16)

where B_{4.16}(f, h) is a constant which depends only on β, v, f and g.

Proof : We will use the following abbreviated notation in the proof; for k ∈ ℤ and a polynomially bounded function f : ℕ → ℝ,

⟨ f ⟩_{β,k} = ⟨ f, ρ_{+} + (−1)^{k}ρ_{−} ⟩_{β}.  (4.17)

It is easy to see from (4.7) that there is a constant C_{4.18} = C_{4.18}(β, v) > 0 such that

1/C_{4.18} ≤ \frac{λ_{+}^{−n}Z_{n}^{x}}{ρ_{+}(x)} ≤ C_{4.18} if n ≥ (1 + x)C_{4.18}.  (4.18)

We take C_{4.16} ≥ C_{4.18}. We see from (4.7) that

|P^{y}[h(S_{l})z_{l}] − m_{β,y+l}[h]Z_{l}^{y}|

≤ λ_{+}^{l} |λ_{+}^{−l}P^{y}[h(S_{l})z_{l}] − \frac{⟨ h ⟩_{β,y+l}}{⟨ ρ_{+}, ρ_{+} ⟩_{β}} ρ_{+}(y)| + λ_{+}^{l} |\frac{⟨ h ⟩_{β,y+l}}{⟨ 1 ⟩_{β,y+l}} (λ_{+}^{−l}Z_{l}^{y} − \frac{⟨ 1 ⟩_{β,y+l}}{⟨ ρ_{+}, ρ_{+} ⟩_{β}} ρ_{+}(y))|
≤ (1 + |y|)^{p}B_{4.19}λ_{+}^{l}ε_{4.7}^{l},  (4.19)

for some p = p(h) ≥ 0 and B_{4.19} = B_{4.19}(β, v, h) > 0. We have on the other hand that

μ_{l}^{x}[F h(S_{l})]
= \frac{1}{Z_{l}^{x}} ∑_{y∈ℕ}P^{x}[F z_{k}^{y}]P^{y}[h(S_{l−k})z_{l−k}]

= \frac{1}{Z_{l}^{x}} ∑_{y∈ℕ}P^{x}[F z_{k}^{y}] (P^{y}[h(S_{l−k})z_{l−k}] − m_{β,y+l−k}[h]Z_{l−k}^{y}) + m_{β,x+l}[h]μ_{l}^{x}[F]. (4.20)

Here, if r = 0, the first summation in (4.20) runs through y’s with k ≡ y −x (mod 2) for which m_{β,y+l−k} = m_{β,x+l}. By (4.19) and (4.20),

|μ_{l}^{x}[F h(S_{l})] − m_{β,x+l}[h]μ_{l}^{x}[F]| ≤ B_{4.19} \frac{λ_{+}^{−k}P_{k}^{x}[|F |(1 + |S_{k}|)^{p}z_{k}]}{λ_{+}^{−l}Z_{l}^{x}}ε_{4.7}^{l−k}

By (4.18) and (4.14), the right-hand-side is bounded from above by B_{4.16}ε_{4.16}^{l−k} for some B_{4.16} = B_{4.16}(β, v, f, h) > 0 and ε_{4.16} = ε_{4.16}(β, v) ∈ (0, 1) if C_{4.16} is large enough. This completes the proof of the lemma. □

4.3 Proof of Theorem 1.3

Case1: r > 0; We begin with an observation which can be seen from (4.7); there are constants C_{4.21} = C_{4.21}(β, v) > 0 and ε_{4.21} = ε_{4.21}(β, v) ∈ (0, 1) such that if n ≥ k + (1 + x)C_{4.21}, then

|\frac{λ_{+}^{−(n−k)}Z_{n−k}^{y}}{λ_{+}^{−n}Z_{n}^{x}} − \frac{ρ_{+}(y)}{ρ_{+}(x)}| ≤ C_{4.21}ε_{4.21}^{n−k} for all y ∈ ℕ.  (4.21)

We have

μ_{n}^{x}[F] = (Z_{n}^{x})^{−1} ∑_{y∈ℕ}P^{x}[F z_{k}^{y}]Z_{n−k}^{y}

= ∑_{y∈ℕ}P^{x}[F z_{k}^{y}]λ_{+}^{−k} (\frac{λ_{+}^{−(n−k)}Z_{n−k}^{y}}{λ_{+}^{−n}Z_{n}^{x}} − \frac{ρ_{+}(y)}{ρ_{+}(x)}) + λ_{+}^{−k} ∑_{y∈ℕ}P^{x}[F z_{k}^{y}]\frac{ρ_{+}(y)}{ρ_{+}(x)}. (4.22)

Since the second term of (4.22) equals ν^{x}[F], we see from (4.21), (4.22) and (4.14) that

|μ_{n}^{x}[F] − ν^{x}[F]| ≤ C_{4.21}λ_{+}^{−k}|P^{x}[F z_{k}]|ε_{4.21}^{n−k}
≤ C_{4.21}B_{4.14}(1 + |x|)^{p}ε_{4.21}^{n−k}

We therefore get (1.22) by taking C_{1.22} large enough.

Case2: r = 0; (4.21) is valid only when k ≡ y − x (mod 2). But the first summation in (4.22) runs through such y’s in this case. Therefore, the proof for the case r > 0 applies without change. □

4.4 Proof of Theorem 1.4

It is clear that (1.24) follows from (1.22) and (1.23). It is therefore, enough to prove (1.23).

We set h_{q}(y) = P^{y}[z_{q}g(S_{1}, …, S_{m})]λ_{+}^{−q}, q ≥ m. Then, by (4.14), h_{q} is polynomially bounded, uniformly in q ≥ m. It is easy to see that

μ_{n}^{x}[F G] = \frac{λ_{+}^{−l}Z_{l}^{x}}{λ_{+}^{−n}Z_{n}^{x}} μ_{l}^{x}[F h_{n−l}(S_{l})] and μ_{n}^{x}[G] = \frac{λ_{+}^{−l}Z_{l}^{x}}{λ_{+}^{−n}Z_{n}^{x}} μ_{l}^{x}[h_{n−l}(S_{l})].

Therefore,

μ_{n}^{x}(F G) − μ_{n}^{x}(F)μ_{n}^{x}(G)
= \frac{λ_{+}^{−l}Z_{l}^{x}}{λ_{+}^{−n}Z_{n}^{x}} (μ_{l}^{x}[F h_{n−l}(S_{l})] − μ_{l}^{x}[F]μ_{l}^{x}[h_{n−l}(S_{l})])  (4.23)
+ \frac{λ_{+}^{−l}Z_{l}^{x}}{λ_{+}^{−n}Z_{n}^{x}} μ_{l}^{x}[h_{n−l}(S_{l})] (μ_{l}^{x}[F] − μ_{n}^{x}[F]) .  (4.24)

We see from (1.22), (4.18) and (4.15) that if C_{1.23} is large enough, then the absolute value of (4.24) is not greater than B_{4.24}ε_{1.22}^{l−k} for some B_{4.24} = B_{4.24}(β, v, f, g) > 0.

We now turn to (4.23). We can write

μ_{l}^{x}[F h_{n−l}(S_{l})] − μ_{l}^{x}[F]μ_{l}^{x}[h_{n−l}(S_{l})]
= (μ_{l}^{x}[F h_{n−l}(S_{l})] − m_{β,x+l}[h_{n−l}]μ_{l}^{x}[F]) + μ_{l}^{x}[F] (m_{β,x+l}[h_{n−l}] − μ_{l}^{x}[h_{n−l}(S_{l})]) . (4.25)

We see from (4.16) and (4.15) that the both terms of (4.25) have the absolute values not greater than B_{4.25}ε_{1.15}^{l−k} for some B_{4.25} = B_{4.25}(β, v, f, g) > 0 if C_{1.23} is large enough. □

Acknowledgement: The authors thank Professor Y. Takahashi for suggesting the compact perturbation augument used in Lemma 4.2. They also thank Professor S. Watanabe for bringing a work of Iglehart [Igl74] into our interest.

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