O -Diagonal Dirichlet Heat Kernel for M = 4 with Torus...

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