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Off-Diagonal Dirichlet Heat Kernel for M = 4
with Torus Identifications
SPUR 2016
August 2, 2016
Note: All negative values with absolute value < 1 × 10−10 were treated as e−35. No negativevalues with absolute value ≥ 1× 10−10 were encountered.
1
1 Base Cell(’2000’, ’2000’)
1.1 t = 1 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2000', '2000'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2000', '2000'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2000', '2000'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 129
8
7
6
5
4
3
2
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2000', '2000'), Direction Horizontal Limited to 5 steps
2
1.2 t = 1 Vertical(Regular) Vertical
0 10 20 30 40 50 60 70 800.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2000', '2000'), Direction Vertical
(Log) Vertical
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2000', '2000'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2000', '2000'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 129
8
7
6
5
4
3
2
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2000', '2000'), Direction Vertical Limited to 5 steps
3
1.3 t = 0.5 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2000', '2000'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2000', '2000'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2000', '2000'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1212
10
8
6
4
2
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2000', '2000'), Direction Horizontal Limited to 5 steps
4
1.4 t = 0.5 Vertical(Regular) Vertical
0 10 20 30 40 50 60 70 800.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2000', '2000'), Direction Vertical
(Log) Vertical
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2000', '2000'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2000', '2000'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1212
10
8
6
4
2
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2000', '2000'), Direction Vertical Limited to 5 steps
5
1.5 t = 0.1 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2000', '2000'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2000', '2000'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2000', '2000'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1220
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2000', '2000'), Direction Horizontal Limited to 5 steps
6
1.6 t = 0.1 Vertical(Regular) Vertical
0 10 20 30 40 50 60 70 800.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2000', '2000'), Direction Vertical
(Log) Vertical
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2000', '2000'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2000', '2000'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1220
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2000', '2000'), Direction Vertical Limited to 5 steps
7
1.7 t = 0.01 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2000', '2000'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2000', '2000'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2000', '2000'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1235
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2000', '2000'), Direction Horizontal Limited to 5 steps
8
1.8 t = 0.01 Vertical(Regular) Vertical
0 10 20 30 40 50 60 70 800.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2000', '2000'), Direction Vertical
(Log) Vertical
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2000', '2000'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2000', '2000'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1235
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2000', '2000'), Direction Vertical Limited to 5 steps
9
2 Base Cell(’2200’, ’2200’)
2.1 t = 1 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2200', '2200'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2200', '2200'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2200', '2200'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 129
8
7
6
5
4
3
2
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2200', '2200'), Direction Horizontal Limited to 5 steps
10
2.2 t = 1 Vertical(Regular) Vertical
0 10 20 30 40 50 60 70 800.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2200', '2200'), Direction Vertical
(Log) Vertical
0 10 20 30 40 50 60 70 8040
35
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2200', '2200'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2200', '2200'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 129
8
7
6
5
4
3
2
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2200', '2200'), Direction Vertical Limited to 5 steps
11
2.3 t = 0.5 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2200', '2200'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2200', '2200'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2200', '2200'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1212
10
8
6
4
2
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2200', '2200'), Direction Horizontal Limited to 5 steps
12
2.4 t = 0.5 Vertical(Regular) Vertical
0 10 20 30 40 50 60 70 800.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2200', '2200'), Direction Vertical
(Log) Vertical
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2200', '2200'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2200', '2200'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1212
10
8
6
4
2
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2200', '2200'), Direction Vertical Limited to 5 steps
13
2.5 t = 0.1 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2200', '2200'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2200', '2200'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2200', '2200'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1220
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2200', '2200'), Direction Horizontal Limited to 5 steps
14
2.6 t = 0.1 Vertical(Regular) Vertical
0 10 20 30 40 50 60 70 800.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2200', '2200'), Direction Vertical
(Log) Vertical
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2200', '2200'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2200', '2200'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1220
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2200', '2200'), Direction Vertical Limited to 5 steps
15
2.7 t = 0.01 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2200', '2200'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2200', '2200'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2200', '2200'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1235
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2200', '2200'), Direction Horizontal Limited to 5 steps
16
2.8 t = 0.01 Vertical(Regular) Vertical
0 10 20 30 40 50 60 70 800.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2200', '2200'), Direction Vertical
(Log) Vertical
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2200', '2200'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2200', '2200'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1235
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2200', '2200'), Direction Vertical Limited to 5 steps
17
3 Base Cell(’2111’, ’2000’)
3.1 t = 1 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2111', '2000'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2111', '2000'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2111', '2000'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1210
9
8
7
6
5
4
3
2
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2111', '2000'), Direction Horizontal Limited to 5 steps
18
3.2 t = 1 Vertical(Regular) Vertical
0 5 10 15 20 250.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2111', '2000'), Direction Vertical
(Log) Vertical
0 5 10 15 20 2535
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2111', '2000'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.00
0.02
0.04
0.06
0.08
0.10
Torus Heat Kernel for M = 4,t = 1, Base Cell('2111', '2000'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1211
10
9
8
7
6
5
4
3
2
Log Torus Heat Kernel for M = 4,t = 1, Base Cell('2111', '2000'), Direction Vertical Limited to 5 steps
19
3.3 t = 0.5 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2111', '2000'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2111', '2000'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2111', '2000'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1214
12
10
8
6
4
2
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2111', '2000'), Direction Horizontal Limited to 5 steps
20
3.4 t = 0.5 Vertical(Regular) Vertical
0 5 10 15 20 250.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2111', '2000'), Direction Vertical
(Log) Vertical
0 5 10 15 20 2535
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2111', '2000'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.00
0.05
0.10
0.15
0.20
0.25
Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2111', '2000'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1214
12
10
8
6
4
2
0
Log Torus Heat Kernel for M = 4,t = 0.5, Base Cell('2111', '2000'), Direction Vertical Limited to 5 steps
21
3.5 t = 0.1 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2111', '2000'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2111', '2000'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2111', '2000'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1225
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2111', '2000'), Direction Horizontal Limited to 5 steps
22
3.6 t = 0.1 Vertical(Regular) Vertical
0 5 10 15 20 250.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2111', '2000'), Direction Vertical
(Log) Vertical
0 5 10 15 20 2535
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2111', '2000'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2111', '2000'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1225
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.1, Base Cell('2111', '2000'), Direction Vertical Limited to 5 steps
23
3.7 t = 0.01 Horizontal(Regular) Horizontal
0 10 20 30 40 50 60 70 800.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2111', '2000'), Direction Horizontal
(Log) Horizontal
0 10 20 30 40 50 60 70 8035
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2111', '2000'), Direction Horizontal
(Regular — Zoomed) Horizontal
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2111', '2000'), Direction Horizontal Limited to 5 steps
(Log — Zoomed) Horizontal
0 2 4 6 8 10 1235
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2111', '2000'), Direction Horizontal Limited to 5 steps
24
3.8 t = 0.01 Vertical(Regular) Vertical
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2111', '2000'), Direction Vertical
(Log) Vertical
0 5 10 15 20 2535
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2111', '2000'), Direction Vertical
(Regular — Zoomed) Vertical
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2111', '2000'), Direction Vertical Limited to 5 steps
(Log — Zoomed) Vertical
0 2 4 6 8 10 1235
30
25
20
15
10
5
0
Log Torus Heat Kernel for M = 4,t = 0.01, Base Cell('2111', '2000'), Direction Vertical Limited to 5 steps
25
