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A Mysterious Cluster Expansion Associated to the Expectation Value of the Permanent of 0–1 Matrices
We consider two ensembles of 0 - 1 n × n matrices. The first is the set of all n × n matrices with entries zeroes and ones such that all column sums and all row sums equal r , uniformly weighted. The second is the set of n × n matrices with zero and one entries where the probability that any given e...
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| Published in: | Journal of statistical physics 2017-06, Vol.167 (6), p.1489-1495 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | We consider two ensembles of
0
-
1
n
×
n
matrices. The first is the set of all
n
×
n
matrices with entries zeroes and ones such that all column sums and all row sums equal
r
, uniformly weighted. The second is the set of
n
×
n
matrices with zero and one entries where the probability that any given entry is one is
r
/
n
, the probabilities of the set of individual entries being i.i.d.’s. Calling the two expectation values
E
and
E
B
respectively, we develop a formal relation
E
(
perm
(
A
)
)
=
E
B
(
perm
(
A
)
)
e
∑
2
T
i
.
(
A
1
)
We use two well-known approximating ensembles to
E
,
E
1
and
E
2
. Replacing
E
by either
E
1
or
E
2
we can evaluate all terms in (A1). For either
E
1
or
E
2
the terms
T
i
have amazing properties. We conjecture that all these properties hold also for
E
. We carry through a similar development treating
E
(
perm
m
(
A
)
)
, with
m
proportional to
n
, in place of
E
(
perm
(
A
)
)
. |
|---|---|
| ISSN: | 0022-4715 1572-9613 |
| DOI: | 10.1007/s10955-017-1787-x |