9. Convergence and Monte Carlo Errors

Measuring Convergence to Equilibrium Variation distance where P 1 and P 2 are two probability distributions, A is a set of states, i is a single state.

Eigenvalue Problem Consider the matrix S defined by [S] ij = p i ½ W(i->j) p j -½ then S is real and symmetric and eigenvalues of S satisfy | n | ≤ 1 One of the eigenvalue must be 0 =1 with eigenvector p j ½.

Spectrum Decomposition Then we have U T SU = Λ, or S = U Λ U T where Λ is a diagonal matrix with diagonal elements k and U is orthonormal matrix, U U T = I. W can be expressed in U, P, and Λ as W = P -½ UΛU T P ½

Evolution in terms of eigen-states P n = P 0 W n = P 0 P -½ UΛU T P ½ P -½ UΛU T P ½ … = P 0 P -½ UΛ n U T P ½ In component form, this means P n (j) = ∑ i P 0 (i) p i -½ p j ½ ∑ k k n u ik u jk

Discussion In the limit n goes to ∞, P n (j) ≈ ∑ i P 0 (i) p i -½ p j ½ u i0 u j0 = p j The leading correction to the limit is P n (j) ≈ p j + a 1 n = p j + a e -n/ 

Exponential Correlation Time We define  by the next largest eigenvalue  = - 1/log 1 This number characterizes the theoretical rate of convergence in a Markov chain.

Measuring Error Let Q t be some quantity of interest at time step t, then sample average is Q N = (1/N) ∑ t Q t We treat Q N as a random variable. By central limit theorem, Q N is normal distributed with a mean = and variance σ N 2 = - 2. standards for average over the exact distribution.

Confidence Interval The chance that the actual mean is in the interval [ Q N – σ N, Q N + σ N ] is about 68 percents. σ N cannot be computed (exactly) in a single MC run of length N.

Estimating Variance The calculation of var(Q) = - 2 and  int can be done in a single run of length N.

Error Formula The above derivation gives the famous error estimate in Monte Carlo as: where var(Q) = - 2 can be estimated by sample variance of Q t.

Time-Dependent Correlation function and integrated correlation time We define and

Circular Buffer for Calculating f(t) Q t, Current time t Q t-1 Previous time t-1 Earliest time t-(M-1) We store the values of Q s of the previous M-1 times and the current value Q t QsQs

An Example of f(t) Time-dependent correlation function for 3D Ising at T c on a 16 3 lattice; Swendsen-Wang dynamics. From J S Wang, Physica A 164 (1990) 240.

Efficient Method for Computing  int We compute  int by the formula  int = N σ N 2 /var(Q) For small value N and then extrapolating N to ∞. From J S Wang, O Kozan and R H Swendsen, Phys Rev E 66 (2002)

Exponential and integrated correlation times where 1 < 1 is the second largest eigenvalue of W matrix. This result says that exponential correlation time  (=-1/log 1 ) is related to the largest integrated correlation time.

Critical Slowing Down TcTc T  The correlation time becomes large near T c. For a finite system  (T c )  L z, with dynamical critical exponent z ≈ 2 for local moves

Relaxation towards Equilibrium Time t Magnetization m T < T c T = T c T > T c Schematic curves of relaxation of the total magnetization as a function of time. At T c relaxation is slow, described by power law: m  t -β/(zν)

Jackknife Method Let n be the number of independent samples Let c be some estimate using all n samples Let c i be the same estimate but using n-1 samples, with i-th sample removed Then Jackknife error estimate is