Random covers of finite homogeneous lattices
We develop and extend some results for the scheme of independent random elements distributed on a finite lattice. In particular, we introduce the concept of the cover of a homogeneous lattice Ln of rank n and derive the exact equations and estimations for the number of covers with a given number o...
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Alekseychuk, A.N. 2009-11-10T14:48:05Z 2009-11-10T14:48:05Z 2006 Random covers of finite homogeneous lattices / A.N. Alekseychuk // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 12–19. — Бібліогр.: 10 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4437 519.21 We develop and extend some results for the scheme of independent random elements distributed on a finite lattice. In particular, we introduce the concept of the cover of a homogeneous lattice Ln of rank n and derive the exact equations and estimations for the number of covers with a given number of blocks and for the total covers number of the lattice Ln. A theorem about the asymptotic normality of the blocks number in a random equiprobable cover of the lattice Ln is proved. The concept of the cover index of the lattice Ln, that extend the notion of the cover index of a finite set by its independent random subsets, is introduced. Applying the lattice moments method, the limit distribution as n→∞ for the cover index of a subspace lattice of the n-dimensional vector space over a finite field is determined. en Інститут математики НАН України Random covers of finite homogeneous lattices Article published earlier |
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Random covers of finite homogeneous lattices |
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Random covers of finite homogeneous lattices Alekseychuk, A.N. |
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Random covers of finite homogeneous lattices |
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Random covers of finite homogeneous lattices |
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Random covers of finite homogeneous lattices |
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Random covers of finite homogeneous lattices |
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random covers of finite homogeneous lattices |
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Alekseychuk, A.N. |
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Alekseychuk, A.N. |
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2006 |
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English |
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Інститут математики НАН України |
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Article |
| description |
We develop and extend some results for the scheme of independent random elements
distributed on a finite lattice. In particular, we introduce the concept of the cover of
a homogeneous lattice Ln of rank n and derive the exact equations and estimations
for the number of covers with a given number of blocks and for the total covers
number of the lattice Ln. A theorem about the asymptotic normality of the blocks
number in a random equiprobable cover of the lattice Ln is proved. The concept of
the cover index of the lattice Ln, that extend the notion of the cover index of a finite
set by its independent random subsets, is introduced. Applying the lattice moments
method, the limit distribution as n→∞ for the cover index of a subspace lattice of
the n-dimensional vector space over a finite field is determined.
|
| issn |
0321-3900 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/4437 |
| citation_txt |
Random covers of finite homogeneous lattices / A.N. Alekseychuk // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 12–19. — Бібліогр.: 10 назв.— англ. |
| work_keys_str_mv |
AT alekseychukan randomcoversoffinitehomogeneouslattices |
| first_indexed |
2025-11-26T20:52:07Z |
| last_indexed |
2025-11-26T20:52:07Z |
| _version_ |
1850774772496990208 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 1–2, 2006, pp. 12–19
UDC 519.21
A. N. ALEKSEYCHUK
RANDOM COVERS OF FINITE HOMOGENEOUS LATTICES
We develop and extend some results for the scheme of independent random elements
distributed on a finite lattice. In particular, we introduce the concept of the cover of
a homogeneous lattice Ln of rank n and derive the exact equations and estimations
for the number of covers with a given number of blocks and for the total covers
number of the lattice Ln. A theorem about the asymptotic normality of the blocks
number in a random equiprobable cover of the lattice Ln is proved. The concept of
the cover index of the lattice Ln, that extend the notion of the cover index of a finite
set by its independent random subsets, is introduced. Applying the lattice moments
method, the limit distribution as n → ∞ for the cover index of a subspace lattice of
the n-dimensional vector space over a finite field is determined.
1. Basic notions and preliminary results
In this paper, we use notions and results from [1, 2]. We refer also to [3, 4] for the
terminology and detailed exposition of finite lattices theory.
Let L = {Ln : n = 0, 1, ...} be a sequence of finite lattices. Denote the Moebius func-
tion, the maximal and minimal elements of the lattice Ln by μn, 1n, and 0n respectively.
We assume that the sequence L satisfies the following homogeneity conditions (see [4]):
(a) Ln is a graduate lattice with rank function r, where r(1n) = n for any n = 0, 1, ...;
(b) for any X ∈ Ln such that r(X) = n−k, k ∈ 0, n, the interval [X, 1n] is isomorphic
to the lattice Lk.
Let
(1) w(n, k) =
∑
a∈Ln:r(a)=k
μn(0n, a), W (n, k) =
∑
a∈Ln:r(a)=k
1,
where k ∈ 0, n, n = 0, 1, . . . ; w(n, k) = W (n, k) = 0 otherwise. The numbers w(n, k) and
W (n, k) are called the k-th level numbers of the lattice Ln of the first kind and of the
second kind respectively [3, 4].
By χn(z), we denote the characteristic polynomial of the lattice Ln,
(2) χn(z) =
n∑
k=0
w(n, n − k)zk, n = 0, 1, . . . .
In what follows, we assume that there exists a sequence of real numbers ai ≥ 1 (i =
1, 2, . . . ) such that
(3) χn(z) =
n∏
i=1
(z − ai), n = 1, 2, . . . , χ0(z) ≡ 1.
2000 AMS Mathematics Subject Classification. Primary 60D05.
Key words and phrases. Finite homogeneous lattice, random cover, cover index, lattice moments.
12
RANDOM COVERS OF FINITE HOMOGENEOUS LATTICES 13
Note that condition (3) holds for the characteristic polynomial of any finite super-
solvable geometric lattice (see [4, 5]).
We consider the following most important examples of homogeneous lattices which
satisfy condition (3). Other examples can be found in [3, 4, 5].
1. Ln = B(n), where B(n) is the set of all subsets of Nn = {1, 2, . . . , n}. The
rank function, level numbers, and characteristic polynomial of the lattice B(n) are equal,
respectively, to r(X) = #X (where #X denotes the cardinality of a set X ∈ B(n)),
w(n, k) = (−1)k
(
n
k
)
, W (n, k) =
(
n
k
)
, χn(z) = (z − 1)n, k ∈ 0, n, n = 0, 1, . . . .
2. Ln = L(n, q) is a subspace lattice of the n-dimensional vector space V (n, q) over a
field with q elements. In this case, the rank r(X) of a subspace X ∈ Ln is equal to the
dimension of X ,
w(n, k) = (−1)kq(
k
2)
[n
k
]
q
, W (n, k) =
[n
k
]
q
, χn(z) =
n∏
i=1
(z − qi−1),
where [n
k
]
q
=
(q−1)n
(q−1)k(q−1)n−k
qk(n−k)
is the Gauss coefficient (the number of k-dimensional subspaces of the vector space
V (n, q), k ∈ 0, n), (q−1)n = (1 − q−1)(1 − q−2) . . . (1 − q−n), n = 0, 1, . . . .
3. Ln = �(n + 1) is the lattice of all partitions of the set Nn+1 = {1, 2, . . . , n + 1}.
The rank of a partition π ∈ �(n + 1) is equal to r(π) = n + 1 − b(π), where b(π) is the
number of blocks in π. The level numbers and the characteristic polynomial of the lattice
�(n + 1) are, respectively, equal to
w(n, k) = s(n + 1, n + 1 − k), W (n, k) = S(n + 1, n + 1 − k), χn(z) =
n∏
i=1
(z − i),
where k ∈ 0, n, n = 0, 1, . . . , s(·, ·) and S(·, ·) are Stirling numbers of the first kind and
of the second kind, respectively.
Let’s consider a sequence of random variables ξ0, ξ1, . . . , where ξn takes values in the
set {0, 1, . . . , n}, n = 0, 1, . . . . Let pn,k = P{ξn = k}, Bn,k = EW (ξn, ξn − k), k ∈
0, n, n = 0, 1, . . . . We call Bn,k the k-th lattice moment of the random variable ξn [2].
The following statement was proved in [2].
Statement 1. Let L be a sequence of finite homogeneous lattices with the level numbers
(1) and the characteristic polynomials (2) that satisfies condition (3). Then
1. We have the following mutually inverse relations:
(4) Bn,l =
n∑
k=l
pn,kW (k, k − l), pn,l =
n∑
k=l
Bn,kw(k, k − l), l ∈ 0, n, n = 0, 1, . . . .
2. For the generating function of the random variable ξn, the following equalities hold:
pn(z) =
n∑
l=0
pn,lz
l =
n∑
k=0
Bn,kχk(z), n = 0, 1, . . . ,
(5)
2ν+1∑
k=0
Bn,kχk(z) ≤ pn(z) ≤
2ν∑
k=0
Bn,kχk(z), z ∈ [0, 1],
14 A. N. ALEKSEYCHUK
where ν = 0, 1, . . . .
Assuming that z = 0 in (5), we obtain the following inequalities:
2ν+1∑
k=0
Bn,kw(k, k) ≤ pn,0 ≤
2ν∑
k=0
Bn,kw(k, k), ν = 0, 1, . . . ,
and, for ν = 0,
(6) 1 − Bn,1 ≤ pn,0 ≤ 1.
2. Random covers of the lattice Ln
By Λn,T , we denote the collection of all sets X = {X1, . . . , XT } such that X1 . . .XT are
different non-zero elements of the lattice Ln. Let’s assign the equiprobable distribution
to the Λn,T , assuming that
(7) p(X) =
(
λn − 1
T
)−1
, X = {X1, . . . , XT } ∈ Λn,T ,
where λn = #Ln. Put
(8) λn(Y ) = #[0n, Y ], Y ∈ Ln,
(9) λ(n−1) = max{λn(Y ) : Y ∈ Ln, r(Y ) = n − 1}, n = 0, 1, . . . .
Definition 1. A set X = {X1, . . . , XT } ∈ Λn,T will be called a T -block cover of the
lattice Ln (and its elements be called blocks of the cover X), if X1 ∨ . . . ∨ XT = 1n.
By rn,T = n − r(X1 ∨ . . . ∨ XT ), we denote the random variable equal to the co-rank
of the join of X1, . . . , XT , where X = {X1, . . . , XT } is a random element distributed
according to (7). Let’s denote
p
(T )
n,l = P{rn,T = l}, B
(T )
n,l = EW (rn,T , rn,T − l), l ∈ 0, n, n = 0, 1, . . . .
We denote the T -block cover number and the total block cover number of the lattice
Ln by Dn,T and Dn =
∑λn−1
T=1 Dn,T , respectively.
For the case Ln = B(n), exact formulas and estimations of the numbers Dn,T , Dn are
obtained in [6, p. 269]. The following statement extends these results.
Statement 2. The following relations hold:
(10) Dn,T =
(
λn − 1
T
) n∑
k=0
B
(T )
n,k w(k, k), Dn =
n∑
k=0
w(k, k)
λn−1∑
T=1
(
λn − 1
T
)
B
(T )
n,k ,
(11)
(
λn − 1
T
)
− B
(T )
n,1
(
λn − 1
T
)
≤ Dn,T ≤
(
λn − 1
T
)
,
(12) 2λn−1 − 1 −
λn−1∑
T=1
(
λn − 1
T
)
B
(T )
n,1 ≤ Dn,T ≤ 2λn−1 − 1.
As this takes place, the following equation holds for any l ∈ 0, n, T ∈ 1, λn − 1:
(13) B
(T )
n,l =
(
λn − 1
T
)−1 ∑
Y ∈Ln:
r(Y )=n−l
(
λn(Y ) − 1
T
)
, l ∈ 0, n, T ∈ 1, λn − 1.
RANDOM COVERS OF FINITE HOMOGENEOUS LATTICES 15
Proof. The first equality in (10) is immediate from (4) and the equality
Dn,T =
(
λn − 1
T
)
p
(T )
n,0 ;
the second equality in (10) follows from the first one. Inequalities (11) follow from (6)
and the first equation in (10); inequalities (12) are obtained by the summation of (11)
over T ∈ 1, λn − 1. Finally, the proof of (13) is similar to the proof of Theorem 1 [1]
applying the Moebius inversion formula.
3. Asymptotic behavior of the block number distribution
in a random equiprobable cover of the lattice Ln
Let ζn denote the random variable equal to the block number in a random equiprobable
cover of the lattice Ln. Let’s prove the following theorem generalizing a result from [7].
Theorem 1. Suppose that
(14) lim
n→∞λ(n−1)(λn)−1 < 1,
where λ(n−1) is defined by (9). Then
(15) Dn = 2λn−1(1 + o(1)), n → ∞,
(16) P{ζn = T } =
Dn,T
Dn
=
1√
π
2 (λn − 1)
exp
{
− (xn,T )2
2
}
(1 + o(1)),
provided n and T tend to infinity in such a way that
(17) xn,T
def=
T − 1
2 (λn − 1)
1
2
√
λn − 1
= o(λ
1
6
n ).
Under condition (14), the remainder term on the right-hand side of (16) tends uni-
formly to zero for all T such that xn,T lies in any fixed finite interval.
Proof. First we show that, under assumption (14),
(18) 2−(λn−1)
∑
Y ∈Ln:
r(Y )=n−1
(2λn(Y )−1 − 1) = o(1), n → ∞.
Applying the second equality from (1), we obtain
λn = #Ln =
n∑
k=0
W (n, k) > W (n, n − 1).
Whence and from (14), we have
2−(λn−1)
∑
Y ∈Ln:
r(Y )=n−1
(2λn(Y )−1 − 1) <
∑
Y ∈Ln:
r(Y )=n−1
2λn(Y ) ≤ 2λ(n−1)−λnW (n, n − 1) <
(19) < λn2λ(n−1)−λn = 2−λn(1−λ(n−1)
λn
− log λn
λn
) = o(1), n → ∞.
So equality (18) is proved.
16 A. N. ALEKSEYCHUK
Now, taking into account (12) and (13), we obtain
2λn−1 − 1 −
λn−1∑
T=1
(
λn − 1
T
)
B
(T )
n,1 = 2λn−1 −
∑
Y ∈Ln:
r(Y )=n−1
(2λn(Y )−1 − 1) ≤ Dn ≤ 2λn−1 − 1.
So, applying (18), we arrive at (15).
To prove equality (16), we set
α(n, T ) =
1√
π
2 (λn − 1)
exp
{
− (xn,T )2
2
}
.
Then ∣∣∣∣Dn,T
Dn
α(n, T )−1 − 1
∣∣∣∣ ≤
(20) ≤
∣∣∣∣Dn,T
Dn
α(n, T )−1 − 1
Dn
(
λn − 1
T
)
α(n, T )−1
∣∣∣∣ +
∣∣∣∣ 1
Dn
(
λn − 1
T
)
α(n, T )−1 − 1
∣∣∣∣ .
From (11) and (13), we obtain that the augend on the right-hand side of (20) is not
greater than
α(n, T )−1 1
Dn
∑
Y ∈Ln:
r(Y )=n−1
(
λn(Y ) − 1
T
)
< α(n, T )−1 1
Dn
λn2λ(n−1)
=
α(n, T )−1λn2λ(n−1)−(λn−1)(1 + o(1)), n → ∞
[see relations (19) and (15)]. Further, applying (17) and (19), we obtain
α(n, T )−1λn2λ(n−1)−(λn−1) = O
(
exp
{
− (xn,T )2
2
}
(λn)
3
2 2λ(n−1)−λn
)
= o(1), n, T → ∞.
Due to these relations, the augend on the right-hand side of (20) tends to zero as
n, T → ∞, and this convergence is uniform for all T , for which xn,T lies in any fixed
finite interval.
To estimate the addend on the right-hand side of (20), we employ equality (15) and
the Moivre – Laplace local theorem. Thus, we obtain that∣∣∣∣ 1
Dn
(
λn − 1
T
)
α(n, T )−1 − 1
∣∣∣∣ = o(1), n, T → ∞,
where o(1) tends to zero uniformly for all T , for which xn,T lies in any fixed finite interval.
So equality (16) is completely proved, and so is the theorem.
Corollary. Under condition (14), the sequence of random variables {ζn : n = 0, 1, . . . }
is asymptotically normal with parameters 1
2 (λn − 1), 1
2
√
λn − 1:
P
{
ζn − 1
2 (λn − 1)
1
2
√
λn − 1
≤ x
}
→ 1√
2π
x∫
−∞
e−
t2
2 dt, n → ∞.
Notice that condition (14) is fulfilled if Ln is one of the lattices B(n), L(n, q), �(n+1)
from Section 1. For B(n), Theorem 1 and its Corollary were earlier obtained by Sachkov
[7]. It is evident that, in this case, the limit on the left-hand side of (14) equals 1/2.
RANDOM COVERS OF FINITE HOMOGENEOUS LATTICES 17
If Ln = L(n, q), n = 0, 1, . . . , then λn = Gn, λ(n−1) = Gn−1, where Gn =
∑n
k=0
[
n
k
]
q
is the total subspace number of the vector space V (n, q) (the Galois number) [6]. In this
case, inequality (14) follows from the asymptotic formula [8]
(21) Gn − 1 = θn(2)(q)q
n2
4 (1 + O(q−
n
2 )), n → ∞,
where n(2) denotes the residue n modulo 2,
(22) θ0(q) =
1
(q−1)∞
∞∑
n=−∞
q−n2
, θ1(q) =
1
(q−1)∞
∞∑
n=−∞
q−(n− 1
2 )2 ,
(q−1)∞ =
∞∏
m=1
(1 − q−m).
In the case of Ln = �(n + 1), n = 0, 1, . . . , (14) follows from the asymptotic formula
for Bell numbers (see, for example, [6, p. 297]).
4. Probability distribution asymptotic behavior of
the cover index of a finite homogeneous lattice
Let Ξ = X1, X2, . . . be a sequence of independent random elements of the lattice Ln.
Definition 2. The cover index of the lattice Ln by elements of the sequence Ξ is the
least θ = θ(n, Ξ) ∈ N such that
(23) X1 ∨ . . . ∨ Xθ = 1n.
This definition extends the concept of the n-set cover index by its independent random
subsets (see [9]).
Taking into account the equality {θ(n, Ξ) ≤ T } = {rn,T = 0}, where rn,T = n −
r(X1∨ . . .∨XT ) is the co-rank of the join of T first elements of the sequence Ξ, we can to
apply the lattice moments method [2] for obtaining the limit distribution of the random
sequence θ(n, Ξ).
For any natural n, T and l ∈ 0, n, we set
(24) B
(T )
n,l =
∑
Y ∈Ln:
r(Y )=n−l
P{X1 ≤ Y } . . .P{XT ≤ Y }.
Theorem 2. Let n and T = T (n) be such that, for any l = 0, 1, . . . ,
(25) B
(T )
n,l → Bl < ∞, n → ∞.
Then, if the series
(26) B(z) def=
∞∑
l=0
Blχ(z) =
∞∑
l=0
Bl
(
l∑
k=0
w(l, l − k)zk
)
is uniformly convergent in a neighborhood of zero, then
(27) lim
n→∞P{θ(n, Ξ) ≤ T } =
∞∑
l=0
w(l, l)Bl.
Proof. It follows from Theorem 2 in [1] that numbers (24) are equal to the lattice moments
of the random variable rn,T . Thus, expression (27) follows directly from equalities (25)
and {θ(n, Ξ) ≤ T } = {rn,T = 0} and the statement of Theorem 2 in [2].
18 A. N. ALEKSEYCHUK
Theorem 2 yields the expression for the limit distribution law (as n → ∞) of the
subspace lattice cover index of the n-dimensional vector space over a field with q elements.
Theorem 3. Let Ξ be a sequence of independent and equiprobable random subspaces in
the vector space V (n, q). Then
(28) lim
n→∞:
n≡i( mod 2)
P{θ(n, Ξ) ≤ 2} =
1
2
(
1 + Ci
(−q−1/2)∞
+
1 − Ci
(q−1/2)∞
)
,
where
(29) i ∈ {0, 1}, C0 = 4q−1/2, C1 = 1/4q1/2, (x)∞ =
∞∏
m=0
(1 − xq−m).
In addition,
(30) lim
n→∞P{θ(n, Ξ) ≤ 3} = 1.
Proof. From (24), we get
(31) B
(T )
n,l =
[n
l
]
q
(
Gn−l
Gn
)T
, n, T ∈ {1, 2, . . .}, l ∈ 0, n.
Hence, applying (21), it is easy to obtain that, for T = 3,
B
(3)
0
def= lim
n→∞B
(3)
n,0 = 1, B
(3)
l = lim
n→∞B
(3)
n,l = 0, l = 1, 2, . . . .
Note that series (26) corresponding to values Bl = B
(3)
l , l = 0, 1, . . . is uniformly con-
vergent over the complex plane. Hence, we obtain (30) from (27).
For T = 2, applying (31) and (21), we get
B
(2)
n,l =
q−l2/2
(q−1)l
(
θ(n−l)(2)
θn(2)
)2
(1 + o(1)), n → ∞, l = 0, 1, . . . ,
Hence, taking into account the equality θ1(q) = 2q−1/4θ0(q) emerging from the Jacobi
identity (see, for example, [10]), we obtain
B0,l
def= lim
n→∞:
n≡0( mod 2)
B
(2)
n,l =
q−l2/2
(q−1)l
, if l ≡ 0( mod 2),
(32) B0,l = 4q−1/2 q−l2/2
(q−1)l
, if l ≡ 1( mod 2),
B1,l
def= lim
n→∞:
n≡1( mod 2)
B
(2)
n,l =
q−l2/2
(q−1)l
, if l ≡ 0( mod 2),
(33) B1,l = 1/4q1/2 q−l2/2
(q−1)l
, if l ≡ 0( mod 2);
As far as series (26) corresponding to each sequence {Bl = Bi,l : l = 0, 1, . . .}, i ∈
{0, 1}, is uniformly convergent over the whole complex plane, we obtain, by applying
(27), the following equalities:
(34) lim
n→∞:
n≡i( mod 2)
P{θ(n, Ξ) ≤ 2} =
∞∑
l=0
w(l, l)Bi,l =
∞∑
l=0
(−1)lq(
l
2 )Bi,l, i ∈ {0, 1}.
RANDOM COVERS OF FINITE HOMOGENEOUS LATTICES 19
Substituting (32), (33), respectively, in (34) and applying the equalities
∑
k≥0:
k≡0( mod 2)
(−1)k q−k/2
(q−1)k
=
1
2
(
1
(−q−1/2)∞
+
1
(q−1/2)∞
)
,
∑
k≥0:
k≡1( mod 2)
(−1)k q−k/2
(q−1)k
=
1
2
(
1
(−q−1/2)∞
− 1
(q−1/2)∞
)
following from the identity
∑∞
k=0
tk
(q−1)k
= 1
(t)∞
, |t| < 1 (see [10]), we obtain equality
(28) after simple transformations.
So, the theorem is proved.
The author would like to thank the anonymous referee for useful censorious remarks.
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E-mail : alex-crypto@mail.ru.
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