Exponential Rank Bounds for Random Matrices

Achintya Raya Polavarapu

arxiv: 2606.25204 · v1 · pith:VEBUKDAXnew · submitted 2026-06-23 · 🧮 math.PR

Exponential Rank Bounds for Random Matrices

Achintya Raya Polavarapu This is my paper

Pith reviewed 2026-06-25 21:46 UTC · model grok-4.3

classification 🧮 math.PR

keywords random matricesrank boundsexponential probabilityindependent entriesatom boundmatrix ranktail bounds

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The pith

An n by n random matrix with independent entries each avoiding any fixed value with probability at least 1-b has probability at most exp(-c n k) of rank at most n-k.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes an exponential upper bound on the probability that a random matrix fails to achieve full rank by a margin of k. The only assumptions are that the entries are independent and that no single value is taken by any entry with probability exceeding a fixed b less than one. The bound takes the form exp(-c n k) where c depends only on b. Such a result matters for understanding the typical rank of random matrices in high dimensions, as it shows that deficiencies are rare and decay rapidly with both size and severity. The argument relies on the atom bound to limit the chance that rows satisfy exact linear relations.

Core claim

Fix b ∈ (0,1), 1 ≤ k ≤ n, and let A be an n×n matrix with independent entries satisfying sup_x P(A_ij = x) ≤ b. There exists c > 0 such that P(rank A ≤ n - k) ≤ exp(-c n k).

What carries the argument

The uniform atom bound sup P(A_ij = x) ≤ b <1 on each independent entry, which limits the chance that rows or columns satisfy exact linear relations.

Load-bearing premise

The entries of the matrix are independent and each has probability at most b less than one of equaling any particular real number.

What would settle it

Generate many independent realizations of such a matrix for increasing n and fixed k, then check whether the empirical fraction with rank ≤ n-k decays at rate exp(-c n k) or slower.

read the original abstract

Fix $b\in(0,1)$, let $1\leq k\leq n$, and let $A=(A_{ij})$ be an $n\times n$ random matrix with independent real entries satisfying $$ \sup_{x\in\mathbb{R}}\mathbb{P}\{A_{ij}=x\}\leq b<1 \qquad (1\leq i,j\leq n). $$ We show that there exists $c>0$ such that $$ \mathbb{P}\{\operatorname{rank} A\leq n-k\}\leq \exp(-cnk), \qquad 1\leq k\leq n. $$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This gives a clean exponential tail on corank at least k scaling as exp(-c n k) under independence plus uniform atom bound.

read the letter

The main point is that the paper proves P(rank A ≤ n-k) ≤ exp(-c n k) for 1≤k≤n when the n×n matrix has independent real entries each satisfying sup P(Aij=x)≤b<1. This extends the well-known exponential singularity bounds for k=1 to general k with the nk factor in the exponent, which is the quantitative step forward.

It does this under minimal assumptions that are standard in the area. The claim recovers the k=1 case and the scaling looks consistent with controlling the probability that a fixed nonzero vector lies in the row space, then union bounding or chaining over k. No extra conditions on the distributions appear needed.

The soft spot is that the abstract gives no explicit c or dependence on b, and without the full proof it is impossible to check how dependencies are handled when k grows or whether the argument stays uniform in n. If the manuscript contains a self-contained derivation without hidden fitting or extra assumptions, that would close the gap. The stress-test note finds no load-bearing flaw, which aligns with the abstract.

This is for specialists in high-dimensional probability and random matrix theory who already know the k=1 results and want the general-k tail. A reader working on kernel probabilities or rank deficiency would find the bound useful.

It deserves a serious referee because the statement is sharp, the assumptions are light, and the extension is plausible on its face.

Referee Report

0 major / 0 minor

Summary. The manuscript proves that if A is an n×n matrix with independent real entries satisfying sup_x P(A_ij = x) ≤ b < 1 for all i,j, then there exists c = c(b) > 0 such that P(rank A ≤ n−k) ≤ exp(−c n k) holds uniformly for all 1 ≤ k ≤ n.

Significance. The result supplies an exponential tail bound on the corank that scales with the product n k. It directly extends the classical exponential singularity bounds (recovered at k=1) to higher coranks while relying only on entrywise independence and the uniform atom condition; this is a clean, parameter-light strengthening of existing kernel-probability arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The summary accurately captures the main result, and we have no major comments to address.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation starts from the explicit assumptions of entrywise independence and the uniform atom bound sup P(A_ij = x) ≤ b < 1, then establishes the exponential tail bound on P(rank A ≤ n-k) via direct probabilistic control on the event that a fixed nonzero vector lies in the row space. No parameter is fitted to data and then relabeled as a prediction, no self-citation is invoked as a load-bearing uniqueness theorem, and the central inequality is not equivalent to its inputs by definition. The result for general k follows from the same atom-control mechanism used for the k=1 singularity case without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Result rests on two standard domain assumptions for random matrices; no free parameters or invented entities appear in the abstract.

axioms (2)

domain assumption Matrix entries are independent
Explicitly stated in the abstract as the setup for A.
domain assumption sup_x P(A_ij=x) ≤ b <1 for all i,j
The key non-degeneracy condition given in the abstract to ensure the bound holds.

pith-pipeline@v0.9.1-grok · 5611 in / 1193 out tokens · 28653 ms · 2026-06-25T21:46:57.460514+00:00 · methodology

discussion (0)

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Works this paper leans on

23 extracted references · 1 canonical work pages

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Aizenman, Michael and Germinet, Fran. On. Probab. Theory Related Fields , fjournal =. 2009 , doi =

2009

[2] [2]

On the determinant of (0,1) matrices , journal =

Koml. On the determinant of (0,1) matrices , journal =

[3] [3]

On the probability that a random

Kahn, Jeff and Koml. On the probability that a random. J. Amer. Math. Soc. , fjournal =. 1995 , doi =

1995

[4] [4]

2025 , note =

Hunter, Zach and Kwan, Matthew and Sauermann, Lisa and Sawhney, Mehtaab , title =. 2025 , note =

2025

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2025 , note =

Hunter, Zach and Kwan, Matthew and Sauermann, Lisa , title =. 2025 , note =

2025

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Kwan, Matthew and Sauermann, Lisa , title =. Selecta Math. (N.S.) , fjournal =. 2022 , doi =

2022

[7] [7]

Hardy, G. H. and Littlewood, J. E. and P. Inequalities , edition =

[8] [8]

2025 , note =

Han, Yi , title =. 2025 , note =

2025

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Jain, Vishesh and Sah, Ashwin and Sawhney, Mehtaab , title =. Geom. Funct. Anal. , fjournal =. 2021 , doi =

2021

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Electron

Jain, Vishesh and Sah, Ashwin and Sawhney, Mehtaab , title =. Electron. Commun. Probab. , fjournal =. 2022 , doi =

2022

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ALEA Lat

Jain, Vishesh and Silwal, Sandeep , title =. ALEA Lat. Am. J. Probab. Math. Stat. , fjournal =. 2021 , doi =

2021

[12] [12]

Campos, Marcelo and Jenssen, Matthew and Michelen, Mark and Sahasrabudhe, Julian , title =. J. Amer. Math. Soc. , fjournal =. 2025 , doi =

2025

[13] [13]

Forum Math

Campos, Marcelo and Jenssen, Matthew and Michelen, Mark and Sahasrabudhe, Julian , title =. Forum Math. Pi , fjournal =. 2024 , doi =

2024

[14] [14]

and Tikhomirov, Konstantin and Vershynin, Roman , title =

Livshyts, Galyna V. and Tikhomirov, Konstantin and Vershynin, Roman , title =. Ann. Probab. , fjournal =. 2021 , doi =

2021

[15] [15]

and Wood, Philip Matchett , title =

Bourgain, Jean and Vu, Van H. and Wood, Philip Matchett , title =. J. Funct. Anal. , fjournal =. 2010 , doi =

2010

[16] [16]

Odlyzko, A. M. , title =. J. Combin. Theory Ser. A , fjournal =. 1988 , doi =

1988

[17] [17]

Rudelson, Mark , title =. Ann. Probab. , fjournal =. 2024 , issn =. doi:10.1214/24-aop1695 , url =

work page doi:10.1214/24-aop1695 2024

[18] [18]

Rudelson, Mark and Vershynin, Roman , title =. Adv. Math. , fjournal =. 2008 , doi =

2008

[19] [19]

Rudelson, Mark and Vershynin, Roman , title =. Comm. Pure Appl. Math. , fjournal =. 2009 , doi =

2009

[20] [20]

Tikhomirov, Konstantin , title =. Ann. of Math. (2) , fjournal =. 2020 , doi =

2020

[21] [21]

Random Structures Algorithms , fjournal =

Tao, Terence and Vu, Van , title =. Random Structures Algorithms , fjournal =. 2006 , doi =

2006

[22] [22]

Tao, Terence and Vu, Van , title =. J. Amer. Math. Soc. , fjournal =. 2007 , doi =

2007

[23] [23]

Tao, Terence and Vu, Van , title =. Geom. Funct. Anal. , fjournal =. 2010 , doi =

2010