Verification run 2 -- C: the limit p -> 1
PASS  Theorem 1.2: P_1 is stochastic, e absorbing, block lower triangular; row sums of B_k are 1 - d^-(x)/d <= 1 - 1/d, d^- = 1 on level 1; N(x) B_k(x,y) symmetric   (exact, 40 graphs)
PASS  Theorem 1.2: every diagonal block with at most 12 vertices has only real eigenvalues, all in [-(1-1/d), 1-1/d]   (exact root count)
PASS  Theorem 1.2: 1 - 1/d is an eigenvalue of B_1   (exact)
PASS  Theorem 1.2: tau(1 - eps) -> d   (|tau(1 - 1e-24) - d| < 1e-10 on all graphs; 40 digits)
PASS  Remark 7.2(a): tau_abs(1 - eps) -> d as well
   name, n, d, diameter, tau(1-eps) - d for eps = 1e-3, 1e-6, 1e-12, 1e-24
   K_2 | 2 | 1 | 1 | -0.000999, -1.0e-6, -1.0e-12, -1.0e-24
   K_3 | 3 | 2 | 1 | -0.002, -2.0e-6, -2.0e-12, -2.0e-24
   K_4 | 4 | 3 | 1 | -0.003, -3.0e-6, -3.0e-12, -3.0e-24
   K_5 | 5 | 4 | 1 | -0.004, -4.0e-6, -4.0e-12, -4.0e-24
   K_6 | 6 | 5 | 1 | -0.005, -5.0e-6, -5.0e-12, -5.0e-24
   K_7 | 7 | 6 | 1 | -0.006, -6.0e-6, -6.0e-12, -6.0e-24
   C_3 | 3 | 2 | 1 | -0.002, -2.0e-6, -2.0e-12, -2.0e-24
   C_4 | 4 | 2 | 2 | -2.0e-6, -2.0e-12, -2.0e-24, -9.18e-41
   C_5 | 5 | 2 | 2 | 0.0442, 0.00141, 1.41e-6, 1.41e-12
   C_6 | 6 | 2 | 3 | 0.0452, 0.00141, 1.41e-6, 1.41e-12
   C_7 | 7 | 2 | 3 | 0.064, 0.002, 2.0e-6, 2.0e-12
   C_8 | 8 | 2 | 4 | 0.0645, 0.002, 2.0e-6, 2.0e-12
   C_9 | 9 | 2 | 4 | 0.0739, 0.00229, 2.29e-6, 2.29e-12
   C_10 | 10 | 2 | 5 | 0.0742, 0.00229, 2.29e-6, 2.29e-12
   C_11 | 11 | 2 | 5 | 0.0794, 0.00245, 2.45e-6, 2.45e-12
   C_12 | 12 | 2 | 6 | 0.0796, 0.00245, 2.45e-6, 2.45e-12
   C_13 | 13 | 2 | 6 | 0.0828, 0.00255, 2.55e-6, 2.55e-12
   Q_2 | 4 | 2 | 2 | -2.0e-6, -2.0e-12, -2.0e-24, -9.18e-41
   Q_3 | 8 | 3 | 3 | 0.000998, 1.0e-6, 1.0e-12, 1.0e-24
   Q_4 | 16 | 4 | 4 | 0.002, 2.0e-6, 2.0e-12, 2.0e-24
   Q_5 | 32 | 5 | 5 | 0.003, 3.0e-6, 3.0e-12, 3.0e-24
   S_3 all transpositions | 6 | 3 | 2 | -3.0e-6, -3.0e-12, -3.0e-24, 1.38e-40
   S_4 all transpositions | 24 | 6 | 3 | 0.002, 2.0e-6, 2.0e-12, 2.0e-24
   S_4 adjacent transpositions | 24 | 3 | 6 | 0.0794, 0.00245, 2.45e-6, 2.45e-12
   A_4 {(012)^+-,(013)^+-} | 12 | 4 | 3 | 0.00299, 3.0e-6, 3.0e-12, 3.0e-24
   S_4 {(0123)^+-,(01)} | 24 | 3 | 6 | 0.0794, 0.00245, 2.45e-6, 2.45e-12
   D_5 {r,r^-1,s} | 10 | 3 | 3 | 0.0543, 0.00173, 1.73e-6, 1.73e-12
   D_6 {s,sr,sr^3} | 12 | 3 | 3 | 0.0772, 0.00245, 2.45e-6, 2.45e-12
   D_7 {r,r^-1,s} | 14 | 3 | 4 | 0.0782, 0.00245, 2.45e-6, 2.45e-12
   Z_3 x Z_3 {(+-1,0),(0,+-1)} | 9 | 4 | 2 | -3.0e-6, -3.0e-12, -3.0e-24, 3.67e-40
   Z_2 x Z_4 {(1,0),(0,+-1)} | 8 | 3 | 3 | 0.000998, 1.0e-6, 1.0e-12, 1.0e-24
   Z_2 x Z_6 {(1,0),(0,+-1),(1,3)} | 12 | 4 | 2 | 0.0028, 2.81e-6, 2.81e-12, 2.81e-24
   Z_9 {+-1,+-2} | 9 | 4 | 2 | -2.4e-6, -2.41e-12, -2.41e-24, -7.35e-40
   Z_10 {+-1,5} | 10 | 3 | 3 | 0.0543, 0.00173, 1.73e-6, 1.73e-12
   Z_12 {+-1,+-5} | 12 | 4 | 3 | 0.00133, 1.33e-6, 1.33e-12, 1.33e-24
   Z_13 {+-1,+-5} | 13 | 4 | 2 | 0.00448, 4.5e-6, 4.5e-12, 4.5e-24
   Z_15 {+-1,+-4} | 15 | 4 | 3 | 0.00497, 5.0e-6, 5.0e-12, 5.0e-24
   Z_16 {+-1,+-3} | 16 | 4 | 4 | 0.0605, 0.002, 2.0e-6, 2.0e-12
   Z_16 {+-3,+-5,8} | 16 | 5 | 4 | -0.0018, -1.8e-6, -1.8e-12, -1.8e-24
   Petersen (vertex-transitive, not Cayley) | 10 | 3 | 2 | 0.0769, 0.00245, 2.45e-6, 2.45e-12
PASS  Remark 7.3: on C_n, 5 <= n <= 14, tau(1-eps) - 2 > 0 behaves like sqrt(eps)  (measured exponents 0.5000, 0.5000, 0.5000, 0.5000, 0.5000, 0.5000, 0.5000, 0.5000, 0.5000, 0.5000)
PASS  Corollary 4.1: on C_n, tau(0) = 1/(1 - cos(2 pi/n)) is < 2 for n <= 5, = 2 for n = 6, > 2 for n >= 7
PASS  Corollary 4.1(a): tau(1-)/tau(0) = d (1 - lambda_2)  (numerically, all graphs)
PASS  Remark 4.2: on Q_d (d = 2..6) the Metropolis chain for theta^|x| has beta_2 = 1 - (1+theta)/d, and for d <= 5 all eigenvalues are 1 - (j/d)(1+theta) with multiplicity C(d,j)   (largest deviation 6.89e-41)
PASS  Remark 4.2: on S_N with all transpositions (N = 3, 4, 5) beta_2 = 1 - 2 theta/N - 2/(N(N-1))   (largest deviation 1.42e-25)
PASS  Remark 4.2: relaxation time N(N-1)/(2(theta(N-1)+1)), equal to (N-1)/2 at theta = 1 and to C(N,2) at theta = 0
PASS  Remark 4.2: the walk on S_N with all transpositions has relaxation time (N-1)/2 and degree C(N,2)  (N = 3, 4, 5)
PASS  Remark 4.2: for the targets theta^|x| the limit matrix has symmetric diagonal blocks with row sums <= 1 - 1/d (equality on level 1)
PASS  Remark 4.2: relaxation time for theta^|x| at theta = 1e-20 is within 1e-8 of d on all graphs

checks passed: 14   TOTAL failures: 0
