Let $d=(n-1)^2+1$, the dimension of the real linear span of the $n\times n$ permutation matrices. We prove that $d$ independent uniformly random permutation matrices fail to form a basis with probability $(1+o(1))n^2(1-1/n)^d=(e^{3/2}+o(1))n^2e^{-n}$. The same asymptotic holds for a uniformly random $d$-element subset, confirming a conjecture of Kushwaha and Tripathi and identifying the leading obstruction: a matrix position avoided by every sample. More generally, for any fixed number of additional samples, we determine the first three exponential orders of the failure probability. To prove these results, we develop support estimates valid over arbitrary fields, derive Fourier bounds from permanental minors, and introduce a counting argument for concentrated assignment functionals over large prime fields. We also give a sparse lifting argument showing that real rank deficiency without an annihilating functional of small support has probability $o(e^{-Cn})$ for every fixed $C>0$.
We study the distribution of the zeros of $\det P_N(z)$ where $P_N(z)$ is a random monic polynomial matrix, i.e., $P_N(z)=z^dI-\sum_{j=0}^{d-1}A_{j,N}z^j$ for possibly coupled random matrices $A_{j,N}$, scaled to have entrywise variance $O(1/N)$. We provide general conditions under which this distribution almost-surely...
We consider the probability that a discrete random matrix $M_n(\xi)$ is \emph{strongly non-singular}, meaning all its leading principal submatrices are non-singular. This property is equivalent to the existence of an LU factorization. We show that for any discrete random variable $\xi$ with finite support and $|\xi|_\i...
S. Mateo, John Urschel, Nicholas West· 2 citations
Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establis...
The present work builds on the differential-equation method, itself a limiting form of the auxiliary-receiver approach in network information theory using a continuum of degraded receivers, and gives a computer-assisted proof of the Courtade--Kumar conjecture.
Zi-Jie Chen, Amin Gohari, Adel Javanmard et al.· 1 citation
We study the linearization of the random $p$-adic orthogonal matrix model. Let $A_n$ and $B_n$ be Haar-random $n\times n$ special orthogonal and alternating matrices over $\mathbb{Z}_p$, respectively. For an odd prime $p$, we prove that $\mathrm{cok}(A_n-I_n)$ and $\mathrm{cok}(B_n)$ have the same limiting distribution...
The rows and columns of the character table of the symmetric group $S_n$ are both naturally indexed by partitions of $n$. Let $D(n)$ denote the number of conjugacy classes of $S_n$ whose column contains no zero entry. The identity column is always zero-free, so $D(n)\geq 1$. It is known that $D(n)\ll n^2$. We prove tha...
Colin Defant, S. Hariharan, Kenny Lau et al.· 0 citations
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