Block Jacobi matrices, Barycentric limits and Manifolds

TL;DR

The paper studies isospectral deformations of block Jacobi/Dirac operators on finite simplicial complexes, using a Lax pair formulation $\frac{d}{dt}D_t=[B_t,D_t]$ together with a QR-flow $e^{-t g(D_0)}=Q_t R_t$ to generate a family $D_t=c_t+c_t^*+m_t$ whose Betti numbers remain invariant while the spectrum evolves. By coupling these deformations with Barycentric refinements, the work establishes a multi-scale framework in which the density of states $dk(G_n)$ converges to a universal limit depending only on the maximal dimension $q$, revealing a central limit-type phenomenon for discrete Laplacians. The authors develop discrete manifold theory in this setting, showing level surfaces $G_f$ induced by colorings are manifolds of dimension $m-k$ and proving a discrete Gauss–Bonnet theorem via localized curvature $K(v)$ whose sum recovers $\chi(G)$. Practical computer algebra implementations illustrate isospectral deformation, barycentric refinement, and level-set analysis, highlighting connections to Connes distance and potential applications to geometric analysis and statistical mechanics on networks.

Abstract

We deform block triangular Jacobi matrices appearing in geometry, look at multi-scale Barycentric limits of geometries and droplet boundary manifolds in Potts networks.

Figures (5)

• Figure 1: The matrix $D$ and Laplacian $L$ before and after the deformation.
• Figure 2: Barycentric refinements of $G$ and the Hodge spectrum of $G_4$.
• Figure 3: Two co-dimension 2 surfaces of a 4-manifold $G$. They are 2-manifolds and subgraphs of the Barycentric refinement $G_1$.
• Figure 4: The $64$-cell is the discrete $5$-dimensional cross-polytop. It is a discrete 5-sphere. It leads to a simplicial complex $G$ with $f$-vector $f(G) = (12, 60, 160, 240, 192, 64)$ and Betti vector $b(G) = (1,0,0,0,0,1)$ and Euler characteristic $0$.
• Figure 5: We see the Dirac matrix $D$ that belongs to the already mentioned 64 cell. It is a $728 \times 728$ block triangular Jacobi matrix with 6 diagonal blocks. The blocks are the incidence matrices which Poincaré already introduced. The isospectral deformation $c_t+c_t^*+m_t$ to the right has a block diagonal part $m_t$. The blocks $c_t$ define new exterior derivatives, leading to to operators $C_t^2=(c_t+c_t^*)^2$ with the same Betti numbers. Like in the Witten deformation, the spectrum of $C_t$ changes but the kernel of $C_t$ does not.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• proof