Toeplitz matrices from permutation displacements and the triangular kernel

Abstract

Toeplitz matrices arise naturally in harmonic analysis, operator theory, and numerical analysis. In this note we investigate Toeplitz matrices whose coefficients depend on the matrix size through a scaled kernel $a_k=f(k/n)$. We show that the empirical mean of their eigenvalues converges to a weighted integral of $f$, where the weight $1-|x|$ reflects the density of diagonals in Toeplitz matrices. We then introduce a combinatorial construction associating a Toeplitz matrix to a permutation via its displacement counts. For a uniformly random permutation, the expected matrix converges to the Toeplitz matrix generated by the triangular kernel $1-|x|$. Interestingly, the triangular kernel also appears as the covariance function of the integrated Brownian motion, providing a probabilistic interpretation of the same operator. Finally, we analyze the integral operator with kernel $(1-|x-y|)$ on $[0,1]$ and determine its eigenfunctions and eigenvalues explicitly. This operator describes the limiting spectral structure associated with the averaged Toeplitz matrices arising from permutation displacements. These results highlight a natural bridge between Toeplitz matrix theory, permutation statistics, and classical integral operators.