def _cftp(self, start_row):
"""
Implement coupling from the past.
ALGORITHM:
The set of Gelfand-Tsetlin patterns can partially ordered by
elementwise domination. The partial order has unique maximum
and minimum elements that are computed by the methods
:meth:`_cftp_upper` and :meth:`_cftp_lower`. We then run the Markov
chain that randomly toggles each element up or down from the
past until the state reached from the upper and lower start
points coalesce as described in [Propp1997]_.
EXAMPLES::
sage: G = GelfandTsetlinPatterns(3, 5)
sage: G._cftp(0) # random
[[5, 3, 2], [4, 2], [3]]
sage: G._cftp(0) in G
True
"""
from sage.misc.randstate import current_randstate
from sage.misc.randstate import seed
from sage.misc.randstate import random
count = self._n * self._k
seedlist = [(current_randstate().long_seed(), count)]
upper = []
lower = []
while True:
upper = self._cftp_upper()
lower = self._cftp_lower()
for currseed, count in seedlist:
with seed(currseed):
for _ in range(count):
for row in range(start_row, self._n):
for col in range(self._n - row):
direction = random() % 2
self._toggle_markov_chain(
upper, row, col, direction)
self._toggle_markov_chain(
lower, row, col, direction)
if all(
all(x == y for x, y in zip(l1, l2))
for l1, l2 in zip(upper, lower)):
break
count = seedlist[0][1] * 2
seedlist.insert(0, (current_randstate().long_seed(), count))
return GelfandTsetlinPattern(upper)
def _cftp(self, start_row):
"""
Implement coupling from the past.
ALGORITHM:
The set of Gelfand-Tsetlin patterns can partially ordered by
elementwise domination. The partial order has unique maximum
and minimum elements that are computed by the methods
:meth:`_cftp_upper` and :meth:`_cftp_lower`. We then run the Markov
chain that randomly toggles each element up or down from the
past until the state reached from the upper and lower start
points coalesce as described in [Propp1997]_.
EXAMPLES::
sage: G = GelfandTsetlinPatterns(3, 5)
sage: G._cftp(0) # random
[[5, 3, 2], [4, 2], [3]]
sage: G._cftp(0) in G
True
"""
from sage.misc.randstate import current_randstate
from sage.misc.randstate import seed
from sage.misc.randstate import random
count = self._n * self._k
seedlist = [(current_randstate().long_seed(), count)]
upper = []
lower = []
while True:
upper = self._cftp_upper()
lower = self._cftp_lower()
for currseed, count in seedlist:
with seed(currseed):
for _ in range(count):
for row in range(start_row, self._n):
for col in range(self._n - row):
direction = random() % 2
self._toggle_markov_chain(upper, row, col, direction)
self._toggle_markov_chain(lower, row, col, direction)
if all(all(x == y for x,y in zip(l1, l2)) for l1, l2 in zip(upper, lower)):
break
count = seedlist[0][1] * 2
seedlist.insert(0, (current_randstate().long_seed(), count))
return GelfandTsetlinPattern(upper)