A Mysterious Cluster Expansion Associated to the Expectation Value of the Permanent of 0–1 Matrices

Abstract

We consider two ensembles of $0-1$ $n\times n$ matrices. The first is the set of all $n\times n$ 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 $n \times n$ 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

$$\begin{aligned} E({{\mathrm{perm}}}(A)) = E_B({{\mathrm{perm}}}(A)) e^{\sum _2 T_i}.\quad \quad \quad \quad \mathrm{(A1)} \end{aligned}$$

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 $E({{\mathrm{perm}}}_m(A))$, with m proportional to n, in place of $E({{\mathrm{perm}}}(A))$.

Appendix

We will derive Eq. (9), (7) being a special case of Eq. (9). We work in a slightly different setting than in the paper, but one having the same computational details. We have a measure space with expectation, e. A, B, and C are $n \times n$ matrices whose entries are random variables. We assume $A =B + C$ and that the entries of B are statistically independent of those of C. Greek letters label a subset of the indices of the rows and an equal sized subset of the columns. If there are r elements in each of the subsets determined by $\alpha$, then $A_\alpha$ is an $r \times r$ submatrix of A, and we write $s(\alpha ) = r$. We assume

$$\begin{aligned} e(perm(A_\alpha )) = a ( s(\alpha )) \end{aligned}$$

(39)

$$\begin{aligned} e(perm(B_\alpha )) = b( s(\alpha )) \end{aligned}$$

(40)

$$\begin{aligned} e(perm(C_\alpha )) = c( s(\alpha )) \end{aligned}$$

(41)

That is, the expectations of the permanent of a submatrix of a given matrix depends only on the size of the submatrix. We note then

$$\begin{aligned} e(perm_m(A)) = {\left( {\begin{array}{c}n\\ m\end{array}}\right) }^2 a(m) \end{aligned}$$

(42)

$$\begin{aligned} e(perm_m(B)) = {\left( {\begin{array}{c}n\\ m\end{array}}\right) }^2 b(m) \end{aligned}$$

(43)

$$\begin{aligned} e(perm_m(C)) = {\left( {\begin{array}{c}n\\ m\end{array}}\right) }^2 c(m) \end{aligned}$$

(44)

Analagous to Eq. (6) we have that

$$\begin{aligned} perm(B_\gamma +C_\gamma )= \sum _{\alpha \subset \gamma }perm(B_\alpha ) perm(C_{\bar{\alpha }}) \end{aligned}$$

(45)

where $\bar{\alpha }$ is the set of indices of rows and columns inside the sets $\gamma$ complementary to the rows and columns of $\alpha$. Taking expectations., and setting $s \equiv s(\gamma )$, we get

$$\begin{aligned} a(s) = \sum _{i=0}^s \left( {\begin{array}{c}s\\ i\end{array}}\right) ^2 b(i) c (s-i). \end{aligned}$$

(46)

Substituting (42)–(44) into (46) we obtain Eq. (9).