A threshold effect for systems of random equations in finite fields

Kolchin, V. F.

doi:10.1515/dma.1999.9.4.355

Cited by 2 publications

(2 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first part of our Theorem 2.7 represents a slight generalization of the results just mentioned above because it allows for any class of sequences W n provided W n → r in probability. The case of finite fields of order q ≥ 3 has also been studied: see for instance [4,6,25].…”

Section: Previous Results On Threshold Valuesmentioning

confidence: 99%

Rank deficiency in sparse random GF$[2]$ matrices

Darling

Penrose

Wade

et al. 2014

Electron. J. Probab.

View full text Add to dashboard Cite

Let M be a random m×n matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let N (n, m) denote the number of left null vectors in {0, 1} m for M (including the zero vector), where addition is mod 2. We take n, m → ∞, with m/n → α > 0, while the weight distribution may vary with n but converges weakly to a limiting distribution on {3, 4, 5, . . .}; let W denote a variable with this limiting distribution. Identifying M with a hypergraph on n vertices, we define the 2-core of M as the terminal state of an iterative algorithm that deletes every row incident to a column of degree 1.We identify two thresholds α * and α, and describe them analytically in terms of the distribution of W . Threshold α * marks the infimum of values of α at which n −1 log E[N (n, m)] converges to a positive limit, while α marks the infimum of values of α at which there is a 2-core of non-negligible size compared to n having more rows than non-empty columns.We have 1/2 ≤ α * ≤ α ≤ 1, and typically these inequalities are strict; for example when W = 3 almost surely, numerics give α * = 0.88949 . . . and α = 0.91793 . . . (previous work on this model has mainly been concerned with such cases where W is non-random). The threshold of values of α for which N (n, m) ≥ 2 in probability lies in [α * , α] and is conjectured to equal α.The random row weight setting gives rise to interesting new phenomena not present in the non-random case that has been the focus of previous work.Note that for a fixed n and a given realization of the sequence of rows X 1 , X 2 , . . ., the numbers N (n, m) are nondecreasing as m increases.Suppose that n, m → ∞, with m/n → α > 0. Our goal is to examine the limiting behaviour of the expected number E[N (n, m)] of left null vectors, and the limiting probability P[σ(n, m) > 0] of a mod-2 linear dependency of the rows of M (n, m), as a function of the parameter α, and especially to derive computable thresholds at which phase transitions occur. We also study the rate of exponential decay of the probability that 1 : = (1, 1, . . . , 1) is a null vector.The probabilistic setting that we consider has the rows X 1 , X 2 , . . . , X m being independent and identically distributed (i.i.d.) with the law of a random vector X = X(n) ∈ {0, 1} n . The problem has different flavours depending on the underlying law of X, and several regimes have received considerable attention in the literature, including:(a) The dense regime in which X has order n non-zero components; the standard model studied in this regime has X distributed uniformly over {0, 1} n .(b) The classical sparse regime in which X has order log n non-zero components.(c) The uniformly (very) sparse regime in which X has O(1) non-zero components.The main focus of the present paper is regime (c) (albeit our 'O(1)' may be random for each row, and might not even have a mean); in Section 2.7 below we briefly discuss other models that have been studi...

show abstract

Section: Previous Results On Threshold Valuesmentioning

confidence: 99%

Rank deficiency in sparse random GF$[2]$ matrices

Darling

Penrose

Wade

et al. 2014

Electron. J. Probab.

View full text Add to dashboard Cite

show abstract

“…The consistency and the mean number of solutions of various types of random systems of linear equations were studied in [2,[4][5][6][7][8][9][10][11]. G. V. Balakin in late 1970 estimated E .c/ n for c m D 1, m 1, and obtained a sufficient condition for validity of the estimate P. n > 0/ 1.…”

Section: Formulation Of Resultsmentioning

confidence: 99%

Characteristics of random systems of linear equations over a finite field

Шаповалов¹

2008

Discrete Mathematics and Applications

View full text Add to dashboard Cite

We consider random and a priori consistent random systems of equations over a finite field with q elements in n unknowns. A random system consists of M D M.n/ equations, each of which can depend on 2; 3; : : : ; m variables, which are obtained by sampling without replacement. We obtain limit distributions and estimates of moments for the numbers of solutions of random systems of equations provided that n ! 1 and the relation between the parameters n and M , the number of vertices and the number of hyperedges, falls into the subcritical domain of the evolution of random hypergraphs which describe the random systems of equations. The form and parameters of the limit distributions are determined by the characteristics of the limit distributions of the number of cycles of a special form in the corresponding random hypergraphs.in n unknowns x 1 ; : : : ; x n , the choice of the ordered set of the indices of unknowns fs 1 ; : : : ; s i g is realised by equiprobable sampling with replacement from all .n/ i possible ordered sets of i indices, i D 2; : : : ; m. For i D 2, the choice of the values 1 and 2 of the index j (that is, the choice of the functions y 1 y 2 and y 1 C y 2 ) is realised with probabilities c 2;1 =c 2 and c 2;2 =c 2 . For i D 2, j D 2, the ordered samples fx s 1 ; x s 2 g and fx s 2 ; x s 1 g

show abstract

A threshold effect for systems of random equations in finite fields

Cited by 2 publications

References 4 publications

Rank deficiency in sparse random GF$[2]$ matrices

Rank deficiency in sparse random GF$[2]$ matrices

Characteristics of random systems of linear equations over a finite field

Contact Info

Product

Resources

About