Sharp singularity rates for symmetric random sign matrices

1 · The essentials

Imagine a square table filled with $+ 1$ and $- 1$ , with matching entries on opposite sides of its diagonal. The entries on and above that diagonal are chosen independently; the remaining entries are fixed by the matching rule. The question is how often the table fails to have an inverse. When both signs are equally likely, the main manuscript shows that this chance has exponential rate $1/2$ as the table grows. A companion manuscript treats a fixed unequal chance of choosing the two signs and gives the corresponding rate. In that setting, two identical rows already occur often enough to account for the reported rate. These statements describe large tables at the exponential scale, not exact probabilities for every size.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, think of a square seating chart marked with plus and minus signs, where every mark across the diagonal must mirror its partner. Two identical rows are like two indistinguishable seating instructions: they illustrate one way the chart can lose information, but the analogy does not capture every way a matrix can fail to have an inverse.

Why it matters

The results identify the exponential rate of singularity rather than merely saying that singularity becomes rare. Together, the manuscripts describe both equally likely signs and any fixed bias strictly between the two deterministic endpoints. In the biased setting, the reported rate also has a concrete explanation: the event that two rows agree.

2 · Going further

The precise problem concerns a symmetric random matrix $A_{n}$ of size $n \times n$ . Entries on and above the diagonal are independent signs, and symmetry determines those below it. Singularity is the event $det A_{n} = 0$ . For uniform signs, the main manuscript shows that $Pr (det A_{n} = 0) = (1/2 + o (1))^{n}$ , where $o (1)$ denotes a quantity tending to zero as $n \to \infty$ . This specifies the limiting exponential base, not an exact finite-size formula.

The companion manuscript considers entries equal to $+ 1$ with fixed probability $p \in (0, 1) ∖ {1/2}$ and to $- 1$ otherwise. It shows in the manuscript that $Pr (det A_{n} = 0) = (p^{2} + (1 - p)^{2} + o (1))^{n}$ . Thus the two manuscripts address the same singularity question under different sign distributions.

The only argument mechanism identified in the supplied abstracts is that, in the biased case, the event that two rows agree attains the stated exponential rate. The abstracts do not describe how other causes of singularity are controlled, nor do they provide the main argument for the uniform case. They also give no historical context or earlier bounds, so the supplied material supports a statement of the results and this one mechanism, rather than a reconstruction of their full arguments.

Useful notions

Square matrices and inverses
Symmetry across a matrix diagonal
Independent random choices
Determinants and singularity
Exponential rates and asymptotic notation

What was known before / what is new

Before

The supplied material does not state earlier singularity estimates or identify a previously known theorem. It presents the uniform and biased models together, but gives no chronology establishing which results were known before these manuscripts.

New

The main manuscript reports the sharp exponential singularity rate $1/2$ for symmetric matrices with independent uniform signs on and above the diagonal. The companion manuscript reports the rate $p^{2} + (1 - p)^{2}$ for fixed nonuniform bias $p \in (0, 1) ∖ {1/2}$ . It also states that agreeing rows attain this biased rate.

Limits and assumptions

The hypotheses require symmetry and independence of the signs on and above the diagonal. The biased result assumes a fixed $p$ strictly between 0 and 1 and different from $1/2$ ; it does not state a result for a bias varying with matrix size. Both formulas concern the exponential rate as $n$ tends to infinity and do not specify exact finite-size probabilities or the size of the vanishing correction. The main result is not formalized in Lean and has not been formally verified. The supplied abstracts do not identify further open problems or provide the full arguments.

Mini glossary

Symmetric matrix: A square matrix whose entries match across the diagonal.
Singularity: The failure of a square matrix to have an inverse, expressed here by its determinant being zero.
Bias: The probability p of choosing the sign +1, with equal sign probabilities corresponding to p = 1/2.
Exponential rate: The limiting base c in a probability written as $(c + o (1))^{n}$ .
o(1): A quantity that tends to zero as the matrix size tends to infinity.

Keywords : Symmetric random matrices · Singularity · Exponential rates

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give $Pr (det A_{n} = 0) = (1/2 + o (1))^{n}$ ; for fixed bias $p \in (0, 1) ∖ {1/2}$ , the rate is $(p^{2} + (1 - p)^{2} + o (1))^{n}$ . In the biased case, agreeing rows attain this rate.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

The sharp exponential rate of singularity for symmetric Bernoulli matrices

03/10/2026

To be verified

Let A_n be a symmetric $n \times n$ matrix whose entries on and above the diagonal are independent uniform signs. We prove $Pr (det A_{n} = 0) = (\frac{1}{2} + o (1))^{n} .$

PDF GitHub folder

The sharp singularity rate for biased symmetric sign matrices

04/10/2026

To be verified

Let A_n be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability $p \in (0, 1) ∖ {1/2}$ . We prove that $P (det A_{n} = 0) = (p^{2} + (1 - p)^{2} + o (1))^{n}$ . The rate is attained by the event that two rows agree.

PDF GitHub folder