Limiting eigenvalue distribution and entropy of multi-Toeplitz matrices

Abstract: We state and prove a version of Szegő's first limit theorem for multi-Toeplitz operators acting in the full Fock space in $d\geq 2$ letters. In particular we show that the eigenvalue distributions of truncated multi-Toeplitz operators converge to a limiting distribution. We compute this limiting distribution and its associated entropy in some special cases. Even in the simplest examples the limiting distribution is quite different from the classical case (corresponding to $d=1$); in these examples we obtain a purely atomic measure, which admits a probabilistic interpretation in connection with a percolation process associated to a simple tridiagonal random matrix model.

• The paper proves that finite-rank truncations of any bounded self-adjoint multi-Toeplitz operator on full Fock space with d ≥ 2 have a unique weak limiting eigenvalue distribution, using Schatten-norm estimates and moment convergence.
• Explicit examples, including L₁ + L₁* and homogeneous multi-letter symbols, produce purely atomic measures formed from geometrically weighted spectra of finite tridiagonal blocks, unlike the absolutely continuous laws in the classical one-variable case.
• The paper connects these atomic limits to Bernoulli percolation and computes Popescu entropy for 2 + L₁ + L₁*, obtaining approximately 0.507 when d = 2 while identifying general-symbol formulas as an open problem.

Overview

This paper, by Michael T. Jury and Arya Gayathri Memana (2608.17859), establishes a version of Szegő's first limit theorem for multi-Toeplitz operators acting on the full Fock space $\mathcal{F}_d$ in $d \geq 2$ letters. The authors prove that the empirical eigenvalue distributions of finite-rank truncations of a bounded self-adjoint multi-Toeplitz operator converge weakly to a unique limiting probability measure $\mu_T$, and they compute $\mu_T$ explicitly in several families of examples. The central finding is that in these examples the limiting measure is purely atomic, in sharp contrast to the classical one-variable case, where Szegő's theorem yields an absolutely continuous measure given by the push-forward of normalized Lebesgue measure on the circle under the symbol. The paper also connects this atomic measure to a Bernoulli percolation process arising from a random tridiagonal matrix model, and computes the associated Popescu entropy in a special case.

Background and setting

The full Fock space $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$ carries the left and right $d$-shifts $L_i$ and $R_i$. An operator $T$ is $R$-multi-Toeplitz if $d \geq 2$0; polynomials of the form $d \geq 2$1 are canonical examples. The authors focus on positive multi-Toeplitz operators with polynomial symbol

$d \geq 2$2

whose matrix representation relative to the word basis $d \geq 2$3 has a block-Toeplitz structure built from tensor powers of identity matrices. The natural truncations are compressions $d \geq 2$4 onto words of length at most $d \geq 2$5, of rank $d \geq 2$6.

In the classical case ($d \geq 2$7), Szegő's theorem identifies the weak limit of the empirical eigenvalue measures of $d \geq 2$8 as $d \geq 2$9, the push-forward of Lebesgue measure under the symbol — an absolutely continuous measure supported on the spectrum. The noncommutative analogue asks whether an analogous limit exists and what it looks like.

Existence of the limiting distribution

The first main result (Theorem 1) asserts that for any bounded self-adjoint multi-Toeplitz operator $\mu_T$0 in $\mu_T$1 variables, there exists a unique probability measure $\mu_T$2 on $\mu_T$3 such that for every continuous $\mu_T$4,

$\mu_T$5

and $\mu_T$6 is supported on the smallest interval containing the spectrum of $\mu_T$7.

The proof is direct and self-contained: after reducing to the positive case by adding a constant, it suffices to show convergence of moments. For fixed $\mu_T$8, the quantity $\mu_T$9 equals the Schatten $\mu_T$0-norm $\mu_T$1. The key structural observation is that $\mu_T$2 is obtained from $\mu_T$3 by appending a new first row and column, which perturbs the Schatten norm only by an $\mu_T$4 amount:

$\mu_T$5

Dividing by $\mu_T$6 shows that successive differences of $\mu_T$7 are summable precisely because $\mu_T$8, so the sequence is Cauchy. Existence of $\mu_T$9 then follows from the Riesz–Markov–Kakutani theorem.

The authors note explicitly that this argument fails when $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$0: without the geometric decay factor, the gaps between successive terms decay only like $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$1, which does not guarantee convergence. This is not merely a technical artifact; it reflects the genuinely different spectral behavior in several variables. The result could alternatively be deduced from Popescu's work on entropy of multi-Toeplitz kernels [popescu-entropy-2001], though it is not stated explicitly there; the value added here is a self-contained proof that does not rely on free Schur parameter theory. Notably, neither classical arguments nor $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$2-algebraic approaches apply here because the projections $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$3 fail to form a Følner sequence — a structural obstruction distinguishing the full Fock space from the Drury–Arveson setting discussed in [memana-2026].

As a corollary, the Popescu entropy of the kernel $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$4 admits the clean formula

$$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$5

linking the noncommutative entropy directly to the limiting spectral distribution.

Explicit computation: purely atomic limits

The second main contribution computes $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$6 for concrete operators, revealing behavior radically different from the classical theorem. For $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$7 acting on $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$8, let $$\mathcal{F}_d = \bigoplus_{n \geq 0} (\mathbb{C}^d)^{\otimes n}$$9 denote the tridiagonal matrix with ones on the off-diagonals, whose spectral measure is $d$0 with $d$1. Then

$d$2

The proof exploits sparsity: the matrix of $d$3 is the incidence matrix of a graph on words of length at most $d$4, with edges connecting words differing by appending or deleting the letter $d$5. This graph decomposes into isolated vertices and linear chains, each chain being a copy of some $d$6. Counting chains of length $d$7 — asymptotically proportional to $d$8 — and normalizing by $d$9 yields the formula above, with convergence in total variation norm.

Two features deserve emphasis. First, the measure is purely atomic even though the symbol is as simple as possible; in the classical case ($L_i$0) the same operator $L_i$1 produces the arcsine law, an absolutely continuous measure. Second, the family $L_i$2 extends continuously to a one-parameter family $L_i$3 for $L_i$4, and as $L_i$5 these converge weak-* to the arcsine distribution — recovering the classical answer at the boundary. The physical cases correspond to $L_i$6, far from the arcsine regime.

The argument generalizes along two directions. For $L_i$7 with $L_i$8 analytic, the chains carry Toeplitz blocks $L_i$9 instead of $R_i$0, and the same geometric weighting gives a purely atomic limit (Theorem 3). For homogeneous sums $R_i$1 with normalization $R_i$2, two unitary reduction lemmas show that the truncations behave exactly like those of $R_i$3 on $R_i$4, giving

$R_i$5

Thus atomicity persists across this entire class, with the effective alphabet size $R_i$6 controlling the weight distribution.

Entropy computation

For the positive operator $R_i$7 on $R_i$8, the support of $R_i$9 touches zero, so continuity of $T$0 on the support is not automatic; the authors verify the entropy formula nonetheless holds in this case. Using the product identity $T$1 (equivalently, $T$2), they reduce $T$3 to $T$4 and obtain

$T$5

which evaluates numerically to approximately 0.507 for $T$6 (first 20 partial sums). Via Theorem 4, the calculation extends to any positive operator of the form $T$7.

Connection to random tridiagonal matrices

A striking coincidence emerges: the limiting measure for $T$8 coincides with the almost-sure limiting eigenvalue distribution of the Bernoulli tridiagonal model studied in [popescu-2025] — self-adjoint tridiagonal matrices with independent entries equal to 1 with probability $T$9 and 0 otherwise — at $R$0:

$R$1

The heuristic explanation is combinatorial. The deterministic multi-Toeplitz graph is a percolation process on the rooted $R$2-ary tree in which only "left" edges survive; it decomposes into linear clusters whose size distribution is asymptotically geometric with ratio $R$3. The random tridiagonal model induces a percolation process on $R$4 that likewise decomposes into linear clusters with geometrically distributed sizes, parameter $R$5. Matching cluster statistics yield matching spectral limits. The authors stress that this explanation is heuristic and does not extend to symbols involving multiple creation and annihilation terms, where the underlying graphs no longer consist of simple linear chains.

Limitations and open questions

The paper is candid about scope. The existence theorem covers bounded self-adjoint multi-Toeplitz operators, but the explicit computations cover only restricted classes: single-letter symbols $R$6 and homogeneous degree-$R$7 sums. No general formula expressing $R$8 in terms of the polynomial symbol exists — the analogue of the push-forward description in the classical case remains unknown, and the authors state plainly that "a full picture of the limiting measure for general multi-Toeplitz operators is still lacking." The probabilistic interpretation via percolation applies only to the sparsest examples and breaks down once the cluster structure ceases to be linear. Additionally, the entropy computation requires special handling when the support includes zero, and its validity beyond the specific case treated is asserted rather than proven in general. Whether the atomicity phenomenon persists for generic multi-Toeplitz symbols, or whether absolutely continuous components can appear, is left open.

Conclusion

This paper delivers a self-contained Szegő-type limit theorem for multi-Toeplitz operators on the full Fock space, proves it by a Schatten-norm telescoping argument that fundamentally requires $R$9, and computes the resulting limiting measures explicitly in tractable families. The principal qualitative discovery is that the limiting distributions are purely atomic — governed by geometrically weighted sums of spectra of small tridiagonal blocks — contradicting the intuition carried over from the classical one-variable theory, and admitting a percolation-theoretic explanation shared with a Bernoulli random tridiagonal model. The general identification of $d \geq 2$00 from the symbol, and the extension of the probabilistic picture beyond linear-cluster regimes, remain open problems.