In this paper, we study the distribution of the cokernels of random p-adic matrices with fixed zero entries. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> be a random <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> matrix over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {Z}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> in which some entries are fixed to be zero and the other entries are i.i.d. copies of a random variable <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\xi \in \mathbb {Z}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. We consider the minimal number of random entries of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$X_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> required for the cokernel of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$X_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> to converge to the Cohen–Lenstra distribution. When <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> is given by the Haar measure, we prove a lower bound of the number of random entries and prove its converse-type result using random regular bipartite multigraphs. When <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> is a general random variable, we determine the minimal number of random entries. Let <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$M_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> be a random <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$n \times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> matrix over <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {Z}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> with k-step stairs of zeros and the other entries given by independent random <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-balanced variables valued in <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathbb {Z}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>. We prove that the cokernel of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$M_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> converges to the Cohen–Lenstra distribution under a mild assumption. This extends Wood’s universality theorem on random p-adic matrices.

Random p-adic matrices with fixed zero entries and the Cohen–Lenstra distribution

期刊论文
发表日期：

2026年7月17日

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Dong Yeap Kang,
Jungin Lee,
Myungjun Yu

摘要

In this paper, we study the distribution of the cokernels of random p-adic matrices with fixed zero entries. Let $X_n$ $X_{n}$ be a random $n \times n$ $n \times n$ matrix over $\mathbb {Z}_{p}$ $Z_{p}$ in which some entries are fixed to be zero and the other entries are i.i.d. copies of a random variable $\xi \in \mathbb {Z}_{p}$ $ξ \in Z_{p}$ . We consider the minimal number of random entries of $X_n$ $X_{n}$ required for the cokernel of $X_n$ $X_{n}$ to converge to the Cohen–Lenstra distribution. When $\xi $ $ξ$ is given by the Haar measure, we prove a lower bound of the number of random entries and prove its converse-type result using random regular bipartite multigraphs. When $\xi $ $ξ$ is a general random variable, we determine the minimal number of random entries. Let $M_n$ $M_{n}$ be a random $n \times n$ $n \times n$ matrix over $\mathbb {Z}_{p}$ $Z_{p}$ with k-step stairs of zeros and the other entries given by independent random $\epsilon $ $ϵ$ -balanced variables valued in $\mathbb {Z}_{p}$ $Z_{p}$ . We prove that the cokernel of $M_n$ $M_{n}$ converges to the Cohen–Lenstra distribution under a mild assumption. This extends Wood’s universality theorem on random p-adic matrices.