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Janine LoBue
Analytic Methods and the Heat Kernel
1 Introduction
In this lecture, we employ some standard analytic methods from spectral geometry to better un
derstand the heat kernel. In particular, we determine upper and lower bounds for the difference
between the heat kernel and the steadystate vector.
2 Review of Facts about the Heat Kernel
Recall that the heat kernel with temperature
t
and seed
s
is deﬁned to be the vector
k
t,s
=
e

t
X
k
≥
0
t
k
k
!
sW
k
=
se

t
(
I

W
)
.
We have shown the following useful properties of the heat kernel:
H
t
=
e

t
L
∂
∂t
H
t
=
LH
t
Tr
(
H
t
) =
X
x
H
t
(
x,x
) =
X
i
e

λ
i
t
For a subset of the vertices
S
, deﬁne the function
f
S
(
u
) =
±
d
u
volS
u
∈
S
0
else
. Recall that
∂
∂t
k
t,f
S
(
S
) =
1
volS
X
u
∼
v
(
k
t/
2
,u
(
S
)

k
t/
2
,v
(
S
))
2
.
Since this is a sum of squares, in particular we have that
∂
∂t
k
t,f
S
(
S
)
is nonnegative.
1
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 Fall '08
 aterras
 Math, Geometry

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