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.