Introduction

Divergence of the non-radom fluctuation inFirst-passage percolation
Shuta Nakajima (Nagoya University)

12th MSJ-SI 2019

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Introduction

In this talk, we discuss the behavior of fluctuations inFirst-passage percolation.
There are many results on the upper bound of fluctuations.
However, there are few results on the lower bound.
My motivation is to give a method to get the lower bound.

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Setting (FPP)
Ed = {{x, y }| x, y ∈ Zd , |x − y |1 = 1|}.
τ = {τe }e∈Ed : I.I.D non-negative random variables.
Γ(x, y ): the set of all paths from x to y .
First Passage time (x, y ∈ Zd )T(x, y ) := inf{∑}τe | γ ∈ Γ(x, y )e∈γ=:infT(γ).

γ∈Γ(x,y )optimal paths
O(x, y ) := {γ ∈ Γ(x, y )| T(γ) = T(x, y )} .

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

IntroductionFirst Passage TimeT(x, y ) = inf{∑}
τe | γ ∈ Γ(x, y )e∈γ•••••••••••••••••••••••••••••••••••x

γ: x → yy

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction“Law of large number” for T(x, y )For x, y ∈ Rd , T(x, y ) := T([x], [y ]), where [·] is a floor function.
Theorem 1 (Kingman ’68)
Suppose that E[τe ] < ∞. For any x ∈ Rd ,1T(0, nx) = g(x)n→∞ nlima.s.,

1
E[T(0, nx)] (time constant).
n∈N nwhere g(x) := inf

Proof.
Apply Kingman’s sub-additive ergodic theorem.
Q. How fast does it converge? (rate of conergence)

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Fluctuation exponent
Conjectures
There exists χ(d) ≥ 0 such that for any x ∈ Zd \{0},T(0, nx) − g(nx) grows like nχ(d) as n → ∞.
This χ(d) is called a fluctuation exponent.
χ(2) = 1/3.
lim χ(d) = 0.

d→∞Controversial Issue
For suﬃciently large d, χ(d) = 0?

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Random and non-random fluctuation

Kesten considered the following decompositon to estimate therate of convergence:
T(0, x) − g(x) = T(0, x) − ET(0, x) + ET(0, x) − g(x) .
|{z} |{z}randomnon−random

The key point is that we can estimate the non-random
fluctuation (from above) by using the estimate of the randomfluctuation (Kesten, Alexander, etc.).
In this talk, we only discuss the lower bound of thenon-random flucuation.

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Previous researches
In this slide, we suppose
the distribution is non-degenerate, (P(τe = a) < 1 ∀a ∈ R)
P(τe = 0) < pc (d), (subcritical regime)∃α > 0 such that Ee ατe < ∞. (finite exponential moment)
Theorem 2 (Kesten ’93)
For any x ∈ Zd \{0} and ϵ > 0, there exists c > 0 s.t.
ET(0, nx) − g(nx) ≥ cn−1−ϵ ,∀n ∈ N.

Theorem 3 (Auﬃnger-Damron-Hanson ’15)
For any x ∈ Zd \{0} and ϵ > 0, there are infinitely many n ∈ N s.t.
ET(0, nx) − g(nx) ≥ n− 2 −ϵ .
1

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Main result I

Theorem 4 (N)Suppose the distribution is non-degenerate and Eτe < ∞. Then,infx∈Zd \{0}(ET(0, x) − g(x)) > 0.

As before, we expect that there exists χ′ (d) such that′ET(0, nx) − g(nx) grows like nχ (d) .
The above result shows that χ′ (d) ≥ 0 if exists.

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Useful distributions
Let τ − be the infimum of the support of the distribution of τe .
Definition 1
defτ is useful ⇔ the following hold:there exists α > 0 such that Eτe2+α < ∞,{pc (d) if τ − = 0,P(τe = τ − ) <p⃗c (d) otherwise,
where pc (d) and p⃗c (d) are the critical probabilities of d-dim
percolation, oriented percolation model, resp.
Conjecture
Useful ⇔ Bd = {x ∈ Rd | g(x) ≤ 1} is compact & strictly convex.

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Main result IITheorem 5 (N)Suppose τ is useful. There exist c > 0 and a sequence (xn ) of Zd
such that |xn |1 = n,ET(0, xn ) − g(xn ) ≥ c(log log n)1/d .
Note that by Jensen’s inequality,E|T(0, xn ) − g(xn )| ≥ |ET(0, xn ) − g(xn )|≥ c(log log n)1/d .
⇒ Divergence of the fluctuation around the time constant.

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)

Introduction

Some open problems
We collect some open problems:Divergence of the random fluctuation for a fixed direction:
for any x ∈ Zd \{0},lim ET(0, nx) − g(nx) = ∞.

n→∞Divergence of the random fluctuation: for d ≥ 3,sup E|T(0, x) − E[T(0, x)]| = ∞.
x∈ZdThe existence of χ′ (d): there exists χ′ (d) such that′ET(0, nx) − g(nx) ≍ nχ (d) .

Divergence of the non-radom fluctuation in First-passage perco
Shuta Nakajima (Nagoya University)