TVDTemperingLift.py

From Werner KRAUTH

import random
import pylab
import numpy as np
for n in [10, 20, 40, 80, 160, 320]:
    ReplicaChange = 0.1
    const = 4.0 / n ** 2
    PiStat = {}
    Table = []
   
    for x in range(1, n + 1):
        Table.append((x, -1, 0))
        Table.append((x,  1, 0))
        Table.append((x, -1, 1))
        Table.append((x,  1, 1))
#
#   factor of 1/4 because the total must be normalized, with two liftings and
#   two replicas  
#
        PiStat[(x, -1,  0)] = 1.0 / float(n) / 4.0
        PiStat[(x,  1,  0)] = 1.0 / float(n) / 4.0
        PiStat[(x, -1, 1)] = const * abs( (n + 1) / 2 - x) / 4.0
        PiStat[(x,  1, 1)] = const * abs( (n + 1) / 2 - x) / 4.0
    PiStat[(0, -1, 0)] = 0.0 
    PiStat[(0,  1, 0)] = 0.0 
    PiStat[(0, -1, 1)] = 0.0
    PiStat[(0,  1, 1)] = 0.0
    PiStat[(n + 1, -1,  0)] = 0.0
    PiStat[(n + 1,  1,  0)] = 0.0
    PiStat[(n + 1, -1, 1)] = 0.0
    PiStat[(n + 1,  1, 1)] = 0.0
    Position = (1,  1, 0)
    PTrans   = np.zeros((4 * n, 4 * n))
    Pi = np.zeros([4 * n])
    for x in range(1, n + 1):
        for Dir in [-1, 1]:
            for Rep in [0, 1]: 
                i = Table.index((x, Dir, Rep))
                Pi[i] = PiStat[(x, Dir, Rep)]
                k = Table.index((x, -Dir, Rep))
                if PiStat[(x + Dir, Dir, Rep)] > 0.0:
                    j = Table.index((x + Dir, Dir, Rep))
                    PTrans[i, j] = min(1.0, PiStat[(x + Dir, Dir,  Rep)] / PiStat[(x, Dir, Rep)])
                    PTrans[i, k] = 1.0 - PTrans[i, j]
                else:
                    PTrans[i, k] = 1.0  

    PReplica = np.zeros((4 * n, 4 * n)) 
    for x in range(1, n + 1):
        for Dir in [-1, 1]:
            i = Table.index((x, Dir, 0))
            j = Table.index((x, Dir, 1))
            PReplica[i, j] = ReplicaChange * min(1.0, PiStat[(x, Dir, 1)] / PiStat[(x, Dir, 0)])
            PReplica[i, i] = 1.0 - PReplica[i, j]
            PReplica[j, i] = ReplicaChange * min(1.0, PiStat[(x, Dir,  0)] / PiStat[(x, Dir, 1)])
            PReplica[j, j] = 1.0 - PReplica[j, i]
    PResampling = np.zeros((4 * n, 4 * n))
    for x in range(1, n + 1):
        for Dir in [-1, 1]: 
            i = Table.index((x, Dir, 0))
            j = Table.index((x, -Dir, 0))
            k = Table.index((x, Dir, 1))
            l = Table.index((x, -Dir, 1))
            PResampling[i, j] = 1.0 / (4.0 * n)
            PResampling[i, i] =  1 - 1. / (4.0 * n)
            PResampling[k, l] = 1.0 / (4.0 * n)
            PResampling[k, k] =  1 - 1. / (4.0 * n)

    P = PTrans @ PReplica @ PResampling
    Pit = np.zeros([4 * n])
    Pit[0] = 1.0
    xvalues = []
    yvalues = []
    iter = 0
    while True:
        iter += 1
        Pit = Pit @ P
        TVD = sum(np.absolute(Pi - Pit) / 2.0)
        xvalues.append(iter / float(n))
        yvalues.append(TVD)
        if TVD < 0.1: break
    pylab.plot(xvalues,yvalues, label='$n =$ '+str(n))
pylab.legend(loc='upper right')
pylab.xlabel("$t/ n$ (rescaled time) ")
pylab.ylabel("TVD")
pylab.title("TVD lift tempering on the path graph of $n$ sites")
pylab.show()