x, y

d_{G}

, put

r = ⌊(d_{G} - 1) /4 ⌋

and

R = ⌊(d_{G} - 1) /2 ⌋

R

balls about

x

and

y

are disjoint, so one has stationary mass at most

1/2

; call its center

z

. Apply part b from radius

r

R

\frac{1}{2} \geq π^{G} (B (z, R)) \geq π_{*}^{G} (1 + Φ_{*}^{G})^{R - r} .

Here the exponent should be

R - r

, and

R - r \geq (d_{G} - 2) /4

. Therefore

Φ_{*}^{G} \leq (2 π_{*}^{G})^{- 4/ (d_{G} - 2)} - 1.

t_{rel}^{G} = 1/ γ^{G}

, part a yields

t_{rel}^{G} \geq \frac{1}{2 Φ _{*}^{G}} \geq \frac{1}{2 {( 2 π _{*}^{G} ) ^{- 4/ (d_{G} - 2)} - 1 }} .

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