On the Approximate Asymptotic Statistical Independence of the Permanents   of 0-1 Matrices

Paul Federbush

arXiv:1705.01868·math.CO·September 5, 2017

On the Approximate Asymptotic Statistical Independence of the Permanents of 0-1 Matrices

Paul Federbush

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TL;DR

This paper investigates the approximate statistical independence of permanents of 0-1 matrices with fixed row and column sums, providing new bounds and computational evidence for the approximation measure E_1.

Contribution

It proves a tighter asymptotic independence result for the measure E_1 compared to the measure E, with improved error bounds and computational validation.

Findings

01

E_1 measure closely approximates E with an error of O(1/n^2)

02

Computational evidence supports the asymptotic independence under E_1

03

New theoretical bounds improve understanding of permanents in structured matrices

Abstract

We consider the ensemble of n x n 0 - 1 matrices with all column and row sums equal r. We give this ensemble the uniform weighting to construct a measure E. We know from the work of Wanless and Pernici that E(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E(perm_{m_i}(A)) * (1+ O(1/n^4)) In this paper we prove E_1(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E_1(perm_{m_i}(A)) * (1+ O(1/n^2)) where E_1 is the measure constructed on the ensemble of n x n 0 - 1 matrices with non-negative integer entries realized as the sum of r random permutation matrices. E_1 is often used as an "approximation" to E. We have computer evidence for E_1(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E_1(perm_{m_i}(A)) * (1+ O(1/n^4)).

Equations113

E (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E (perm_{m_{i}} (A))) (1 + O (1/ n^{4}))

E (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E (perm_{m_{i}} (A))) (1 + O (1/ n^{4}))

E_{1} (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E_{1} (perm_{m_{i}} (A))) (1 + O (1/ n^{2}))

E_{1} (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E_{1} (perm_{m_{i}} (A))) (1 + O (1/ n^{2}))

E_{1} (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E_{1} (perm_{m_{i}} (A))) (1 + O (1/ n^{4}))

E_{1} (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E_{1} (perm_{m_{i}} (A))) (1 + O (1/ n^{4}))

E (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E (perm_{m_{i}} (A))) (1 + O (1/ n^{4}))

E (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E (perm_{m_{i}} (A))) (1 + O (1/ n^{4}))

E_{1} (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E_{1} (perm_{m_{i}} (A))) (1 + O (1/ n^{2}))

E_{1} (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E_{1} (perm_{m_{i}} (A))) (1 + O (1/ n^{2}))

E_{1} (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E_{1} (perm_{m_{i}} (A))) (1 + O (1/ n^{4}))

E_{1} (i = 1 \prod N (perm_{m_{i}} (A))) = i = 1 \prod N (E_{1} (perm_{m_{i}} (A))) (1 + O (1/ n^{4}))

I = k = 1 \prod N \sum_{i} m_{k, i} = m_{k} m_{k, i} \sum \frac{1}{( n ! ) ^{r}} (m _{k} n)^{2} m_{k}! \frac{m _{k} !}{m _{k, 1} ! \dots m _{k, r} !} i \prod (n - m_{k, i})!

I = k = 1 \prod N \sum_{i} m_{k, i} = m_{k} m_{k, i} \sum \frac{1}{( n ! ) ^{r}} (m _{k} n)^{2} m_{k}! \frac{m _{k} !}{m _{k, 1} ! \dots m _{k, r} !} i \prod (n - m_{k, i})!

I I = \frac{1}{( n ! ) ^{r}} k = 1 \prod N \sum_{i} m_{k, i} = m_{k} m_{k, i} \sum (m _{k} n - \sum _{t < k} m _{t})^{2} m_{k}! \frac{m _{k} !}{m _{k, 1} ! \dots m _{k, r} !} i \prod (n - t \sum m_{t, i})!

I I = \frac{1}{( n ! ) ^{r}} k = 1 \prod N \sum_{i} m_{k, i} = m_{k} m_{k, i} \sum (m _{k} n - \sum _{t < k} m _{t})^{2} m_{k}! \frac{m _{k} !}{m _{k, 1} ! \dots m _{k, r} !} i \prod (n - t \sum m_{t, i})!

I = \sum k I

I = \sum k I

I I = \sum k I I

I I = \sum k I I

I I - I = \sum (k I I - k I) = \sum k I (\frac{k I I}{k I} - 1)

I I - I = \sum (k I I - k I) = \sum k I (\frac{k I I}{k I} - 1)

\frac{k I I}{k I}

\frac{k I I}{k I}

= e^{α}

α ≅

α ≅

+ i \sum (t \sum m_{t, i})^{2} - k \sum i \sum (m_{k, i})^{2}

α ≅ \frac{1}{2 n} [- 4 k \sum t < k \sum m_{t} m_{k} + 2 i \sum k \sum t < k \sum m_{t, i} m_{k, i}]

α ≅ \frac{1}{2 n} [- 4 k \sum t < k \sum m_{t} m_{k} + 2 i \sum k \sum t < k \sum m_{t, i} m_{k, i}]

I I - I ≅ \sum (k I) α

I I - I ≅ \sum (k I) α

k I ≅ k \prod [\frac{1}{m _{k, 1} ! \dots m _{k, r} !} n^{m_{k}}]

k I ≅ k \prod [\frac{1}{m _{k, 1} ! \dots m _{k, r} !} n^{m_{k}}]

n \to \infty lim k = 1 \prod N [\frac{1}{( n ! ) ^{r}} \frac{( n ! ) ^{2}}{( ( n - m _{k} ) ! ) ^{2}} i \prod (n - m_{k, i})! \frac{1}{n ^{m_{k}}}]

n \to \infty lim k = 1 \prod N [\frac{1}{( n ! ) ^{r}} \frac{( n ! ) ^{2}}{( ( n - m _{k} ) ! ) ^{2}} i \prod (n - m_{k, i})! \frac{1}{n ^{m_{k}}}]

=

=

I I - I ≅ \sum (k = 1 \prod N (\frac{n ^{m_{k}}}{m _{k, 1} ! \dots m _{k, r} !})) \frac{1}{2 n} [- 4 k \sum t < k \sum m_{t} m_{k} + 2 i \sum k \sum t < k \sum m_{t, i} m_{k, i}]

I I - I ≅ \sum (k = 1 \prod N (\frac{n ^{m_{k}}}{m _{k, 1} ! \dots m _{k, r} !})) \frac{1}{2 n} [- 4 k \sum t < k \sum m_{t} m_{k} + 2 i \sum k \sum t < k \sum m_{t, i} m_{k, i}]

I I - I ≅ [k = 1 \prod N (\frac{n ^{m_{k}} r ^{m_{k}}}{m _{k} !})] \cdot \frac{1}{2 n} [- 4 k \sum t < k \sum m_{t} m_{k} + 2 i \sum k \sum t < k \sum \frac{m _{t}}{r} \frac{m _{k}}{r}]

I I - I ≅ [k = 1 \prod N (\frac{n ^{m_{k}} r ^{m_{k}}}{m _{k} !})] \cdot \frac{1}{2 n} [- 4 k \sum t < k \sum m_{t} m_{k} + 2 i \sum k \sum t < k \sum \frac{m _{t}}{r} \frac{m _{k}}{r}]

I I - I = [k = 1 \prod N (\frac{n ^{m_{k}} r ^{m_{k}}}{m _{k} !})] \frac{1}{2 n} [- 4 k \sum t < k \sum (m_{t} m_{k}) (1 - \frac{1}{2 r})] + I O (1/ n^{2})

I I - I = [k = 1 \prod N (\frac{n ^{m_{k}} r ^{m_{k}}}{m _{k} !})] \frac{1}{2 n} [- 4 k \sum t < k \sum (m_{t} m_{k}) (1 - \frac{1}{2 r})] + I O (1/ n^{2})

I I - I = I \cdot \frac{1}{2 n} [- 4 k \sum t < k \sum (m_{t} m_{k}) (1 - \frac{1}{2 r})] + I O (1/ n^{2})

I I - I = I \cdot \frac{1}{2 n} [- 4 k \sum t < k \sum (m_{t} m_{k}) (1 - \frac{1}{2 r})] + I O (1/ n^{2})

\sum m_{k, i} = m_{k} m_{k, i} \sum \frac{1}{( n ! ) ^{r}} (m _{k} n)^{2} m_{k}! \frac{m _{k} !}{m _{k, 1} ! \dots m _{k, r} !} i \prod (n - m_{k, i}) \cdot (m_{k, j} - \frac{m _{k}}{r}) = 0

\sum m_{k, i} = m_{k} m_{k, i} \sum \frac{1}{( n ! ) ^{r}} (m _{k} n)^{2} m_{k}! \frac{m _{k} !}{m _{k, 1} ! \dots m _{k, r} !} i \prod (n - m_{k, i}) \cdot (m_{k, j} - \frac{m _{k}}{r}) = 0

I I I =

I I I =

\cdot \sum m_{j, i} = m_{j} - 1 m_{j, i} \sum (m _{j} - 1 n - \sum _{t < j} m _{t})^{2} (m_{j} - 1)! \frac{( m _{j} - 1 ) !}{m _{j, 1} ! \dots m _{j, r} !} \cdot

k = j + 1 \prod N \sum m_{k, i} = m_{k} m_{k, i} \sum (m _{k} n - \sum _{t < k} m _{t} + 1)^{2} m_{k}! \frac{m _{k} !}{m _{k, 1} ! \dots m _{k, r} !} \cdot

i \prod (n - t \sum m_{t, i})! m_{i, s}

= s \sum i < j \sum Q_{j} i \prod (n - t \sum m_{t, i}) m_{i, s}

= s \sum i < j \sum Q_{j} i \prod (n - t \sum m_{t, i}) m_{i, s}

I I I = s \sum i < j \sum I I_{j}^{'} m_{i, s}

I I I = s \sum i < j \sum I I_{j}^{'} m_{i, s}

I I_{j}^{'} ≅

I I_{j}^{'} ≅

≅

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TopicsRandom Matrices and Applications · Advanced Combinatorial Mathematics · Markov Chains and Monte Carlo Methods

Full text

On the Approximate Asymptotic Statistical Independence of the Permanents of 0-1 Matrices

Paul Federbush

Department of Mathematics

University of Michigan

Ann Arbor, MI, 48109-1043

Abstract

We consider the ensemble of $n\times n$ $0\mkern 2.0mu\mathchar 45\relax 1$ matrices with all column and row sums equal $r$ . We give this ensemble the uniform weighting to construct a measure $E$ . We know from work of Wanless and Pernici that

[TABLE]

In this paper we prove

[TABLE]

where $E_{1}$ is the measure constructed on the ensemble of $n\times n$ matrices with non-negative integer entries realized as the sum of $r$ random permutation matrices. $E_{1}$ is often used as an “approximation” to $E$ . We have computer evidence for

[TABLE]

1 Introduction

We consider the ensemble of $n\times n$ $0\mkern 2.0mu\mathchar 45\relax 1$ matrices whose row and column sums all equal $r$ . We define the uniform measure in this ensemble, calling it $E$ . We know from the work of Wanless [5] and Pernici [2] that

[TABLE]

We let $E_{1}$ be the measure on $n\times n$ matrices with non-negative integer entries, constructed as the uniform measure on a sum of $r$ random permutations of $n$ objects. We here prove

[TABLE]

In Section 8 we present algebraic ( rigorous ) computer computations for some cases with $N=2,r=2$ and $N=2,r=3$ supporting

[TABLE]

These same calculations show (1.3) is not true with $\mathcal{O}\left(1/n^{4}\right)$ replaced by $\mathcal{O}\left(1/n^{5}\right)$ .

$E_{1}$ is often used as an ‘approximation’ to $E$ . In [1] and [4] certain aesthetic relations seem to hold in the same form for $E$ and $E_{1}$ expectations. Similarly we believe eq (1.1) presages the truth of eq (1.3). However Section 7 has some results that may cause one some hesitation.

We briefly consider the Bernoulli random matrix ensemble where each entry independently has a probability $p=r/n$ of being one, and is zero otherwise. We let $E_{B}$ be the associated measure. $E_{B}$ is a less worthy ”approximation” to $E$ , and following along the lines of the calculation of this paper it is not difficult to show if one replaces $E_{1}$ in eq(1.2) by $E_{B}$ the resulting relation does not hold.

In a future paper I plan to relate eq (1.1) to graph positivity, [3]. In particular we plan to use these equations to prove a weak form of the positivity of $\Delta^{k}d(i)$ , see Section II of [3]. It was a study of such ‘graph positivity’ that got me involved with the conjecture of this paper.

One can only appreciate the magic of eq (1.1), or eq (1.2) , by seeing the complicated calculations and cancellation involved in the proofs. We do not have any understanding why the necessary cancellations take place!

The reader will be forced to embed himself or herself in the world of [1] to follow developments within. But better yet, find another way to attack the study of this conjecture!

A final note before plunging into the calculation, is the observation that both the conjecture of this paper, and the computation in [1], can be viewed as an asymptotic statistical independence of the permanents of $0\mkern 2.0mu\mathchar 45\relax 1$ matrices.

2 The Strategy

We depend on the reader being familiar with the first five pages of [1]. In that paper one studies a product of two permanents, here we deal with a product of $N$ permanents, but the ideas are the same. A single permanent $\operatorname{perm}_{m}(A)$ , we view as a sum of ‘terms’, $\binom{n}{m}^{2}m!$ such, each term a product of $m$ ‘entries’, $\prod_{i=1}^{N}\operatorname{perm}_{m_{i}}(A)$ we view as a sum of ‘multiterms’, each multiterm a product of $N$ ‘subterms’. Given a fixed multiterm, let $1\leq i<j\leq N$ , and look at the subterms $i$ and $j$ . Each entry in the $j$ subterm is in class $1,2,3,4$ , or $5$ with respect to the $i$ subterm, in the language of [1]. In this paper we study those types of multiterms in the expansion of $E_{1}\left(\prod_{i=1}^{N}\operatorname{perm}_{m_{i}}(A)\right)$ that contribute a value larger than $\mathcal{O}\left(1/n^{2}\right)E_{1}\left(\prod_{i=1}^{N}\operatorname{perm}_{m_{i}}(A)\right)$ . More exactly, the sum of all the multiterms of the excluded types is bounded by $\mathcal{O}\left(1/n^{2}\right)E_{1}\left(\prod_{i=1}^{N}\operatorname{perm}_{m_{i}}(A)\right)$ . The result is that we need consider multiterms that have at most one pair $i<j$ of subterms with one entry of subterm $j$ that is in class other than class 1 to the subterm $i$ . In other words, of all the $\sum_{j=1}^{N}m_{j}$ entries in the multiterm we consider those that have at most two that share a row or a column! In the following sections we will treat each of the types of multiterms that might contribute to a $1/n$ correction in eq (1.2): only class 1 entries, a single class 2 entry, a single class 3 or class 4 entry. Loosely speaking, as is easy to believe, each time two entries share a row or a column one loses a power of $n$ .

3 Pure Class 1 Multiterms

We first consider the multiterms where no two entries share a row or a column. In the factor $\left(1+\mathcal{O}\left(1/n^{2}\right)\right)$ in eq (1.2), these multiterms give rise to the $1$ , as well as $\mathcal{O}\left(1/n\right)$ corrections canceled to order $1/n^{2}$ by contributions of the other type multiterms.

We write down the exact expression for $\prod_{i=1}^{N}E_{1}\left(\operatorname{perm}_{m_{i}}(A)\right)$ , which we call $I$ .

[TABLE]

$II$ is then the contribution of all pure class 1 multiterms to $E_{1}\left(\prod_{i=1}^{N}\left(\operatorname{perm}_{m_{i}}(A)\right)\right)$ .

[TABLE]

We now take all the sums out of the terms $I$ and $II$ writing

[TABLE]

where $k~{}I$ and $k~{}II$ are the ‘kernels’ of the sums. We then write

[TABLE]

where

[TABLE]

To evaluate $\alpha$ we turn to the lemma in the appendix, noting that to calculate the $1/n$ terms we need only keep the first term on the right side of eq (A7). This yields up to $\mathcal{O}\left(1/n^{2}\right)$ .

[TABLE]

or, with some thought

[TABLE]

In general we write $a\cong b$ to mean $a=b\left(1+\mathcal{O}\left(1/n\right)\right)$ . Substituting back into eq (3.5)

[TABLE]

To compute the right side of eq (3.10) up to $\frac{1}{n}\cdot I$ we can approximate $k~{}I$ as follows

[TABLE]

Since

[TABLE]

where in fact each of the expressions in brackets in eq (3.13) approaches $1$ with $n$ going to infinity. So finally

[TABLE]

We now use eq (A2) from the appendix to arrive at our final result

[TABLE]

or

[TABLE]

Alternatively we could avoid the discussion from eq (3.11) to eq (3.17), going directly from (3.10) to

[TABLE]

using

[TABLE]

that follows from the symmetry of the $r$ colors.

4 A Single Entry of Class 2

We now study the contribution to $E_{1}\left(\prod_{i=1}^{N}\operatorname{perm}_{m_{i}}(A)\right)$ of multiterms with all entries but one of class 1 and the single other entry of class 2. We sum over the color $s$ of this entry, the subterm $j$ it is in, and the subterm $i$ it is in class 2 relative to. We let this contribution be called $III$ .

[TABLE]

The final $m_{i,s}$ represents the choice of which entry of color $s$ in subterm $i$ agrees with the entry in subterm $j$ . In subterm $j$ there are just $m_{j}-1$ entries to be summed over once the class 2 entry has been selected. We write (4.1) as (we will use $Q_{j}$ later)

[TABLE]

and here also define $II_{j}^{\prime}$

[TABLE]

where $II_{j}^{\prime}$ is exactly $II$ of eq (3.2) with $m_{j}$ replaced by $m_{j}-1$ . We now use the argument as in eq (3.11)-(3.14) to write

[TABLE]

by eq (A.1) from the appendix. Substituting into eq (4.3) we get

[TABLE]

equivalently

[TABLE]

5 A Single Class 3 or Class 4 Entry

We calculate the contribution of the sum of these two types of multiterms, multiplying by 2 the contribution of the multiterms with a single class 3 entry. We let $IV$ be the sum of these two types. We specify all the class 1 entries in the multiterm and at the last choice select the single class 3 entry, arriving at

[TABLE]

see eq (4.2)

The 2 is for the sum of the two classes, $b$ is the color of the class 3 or class 4 entry, $a$ is the color of the entry in subterm $i$ that shares a row or column with this entry. $m_{i,a}$ selects the color $a$ entry in subterm $i$ sharing the row or column, $(n-\sum_{t}m_{t}+1)$ selects the column (row) for the class 3 (class 4) entry.

[TABLE]

$(r-1)$ arises choosing the color of the class 3 or 4 entry. Finally

[TABLE]

6 Summing the Terms

From eq (3.17) we get

[TABLE]

From eq (4.6) we have

[TABLE]

and from eq (5.3) we have

[TABLE]

Therefore

[TABLE]

We remark to the reader that a single class 5 entry makes no contribution to order $1/n$ . If one looks at the details of the computations of this paper addressed for getting out corrections to order $1/n$ , one can see how complex it will be to get the corrections to order $1/n^{2}$ . We have used integer arithmetic computer computations for $N=2,r=2$ and $r=3$ , to help check the correctness of analytic expressions.

7 A Cautionary Calculation

Whereas we have referred to a couple instances where $E_{1}$ expectations have the same behavior as $E$ expectations, we here present a blatant contrast. In [2], eq (11) and Appendix A, there is presented an expansion in descending powers of $n$

[TABLE]

(M. Pernici has shown me how to alternatively derive these equations using the formalism of [4], assuming Conjecture 1 therein.)

It is easy to check that the corresponding expansion for $E_{1}\left(\operatorname{perm}_{m}(A)\right)$ agrees in the first two terms, but not in the third, checked by computer to some high order. We take this expression for $E_{1}\left(\operatorname{perm}_{m}(A)\right)$ as established. This is a clash to the same power of $n$ as pursued in the sequel, paper II.

8 Some Computer Calculations

We note that by the result of this paper (1.2) and the expressions of Section 7 one has

[TABLE]

where $c$ depends on $r$ and the $m_{i}$ . Similarly from (1.1) and from the expressions from Section 7, one has

[TABLE]

with the value of $c$ the same in (8.1) and (8.2).

We have computed exactly, in the case $r=3$ ( and $N=2$ )

[TABLE]

for all $m_{1},m_{2}<7$ and found that the highest non-vanishing term in the expansion in powers of $n$ was $m_{1}+m_{2}-4$ . We have done the same for $r=2$ ( and $N=2$ ) for all $m_{1},m_{2}<9$ and found that the highest non-vanishing term in the expansion in powers of $n$ was $m_{1}+m_{2}-4$ . For r=2 the expectations in (8.3) are polynomials in n, for r=3 rational functions of n.

We will study expectations of $perm_{5}(A)perm_{3}(A)$ as an example. We are working with an $N=2$ example so that we can compute the $E_{1}$ expectation exactly, and with $r=2$ . ( It is possible to study expectations in the $E_{1}$ measure for $N>2$ , but at a tremendous increase in computational complexity. ) We define

[TABLE]

For $Q1$ we have an exact expression

[TABLE]

Similarly we have an exact expression for $Q2$

[TABLE]

Appendix

[TABLE]

Lemma Suppose

[TABLE]

then

[TABLE]

The lemma follows from eq(A.5) with a little calculation.

Bibliography5

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Federbush, P., Asymptotic Behavior of the expectation value of permanent products , ar Xiv: 1407.6545.
2[2] Pernici, M., 1/n expansion for the number of matchings on regular graphs and monomer-dimer entropy . J Stat Phys (2017) doi: 10.1007/s 10955-017-1819-6
3[3] Butera, P., Federbush P., and Pernici, M., A positivity property of the dimer entropy of graphs , Physica A 421 (2015) 208.
4[4] Federbush, P., A mysterious cluster expansion associated to the expectation value of the permanent of 0 − 1 0 1 0\mkern 2.0mu\mathchar 45\relax 1 matrices , J. Stat. Phys (2017).
5[5] Wanless, I.M., Counting matchings and tree-like walks in regular graphs , Combin. Probab. Comput. 19, 463 (2010).