Tensorial Free Additive Convolution

• Tensorial free additive convolution is a framework that generalizes free probability by incorporating operator-valued and tensorial methods to analyze eigenvalue distributions of random matrices and tensors.
• It employs analytic subordination and combinatorial cumulants to systematically address selfadjoint polynomial problems and capture higher-order tensor behaviors.
• In finite free probability, the p⊞ₜ q operation preserves real-rootedness via expected characteristic polynomial techniques, unifying diverse convolution frameworks.

Tensorial free additive convolution is a family of related constructions that extend free additive convolution while retaining non-scalar structure. In operator-valued free probability, it is the free additive convolution with amalgamation over a fixed subalgebra $B$, typically a matrix algebra such as $M_N(\mathbb C)$, and it serves as the analytic mechanism behind the computation of asymptotic eigenvalue distributions for selfadjoint polynomials in independent random matrices (Belinschi et al., 2013). In recent tensor-probabilistic frameworks, the same expression denotes an additive operation linearized by tensorial free cumulants indexed either by combinatorial maps or by colored permutation-tuples, producing higher-order analogues of semicircular and free Poisson laws and cumulant-based convolution rules for LU-invariant tensors (Bonnin, 2024, Buc-d'Alché et al., 3 May 2026). A third usage appears in finite free probability, where the “tensorial” or asymmetric additive convolution of polynomials is the operation $p \boxplus_t q$, realized as an expected characteristic polynomial of $(A+RBQ)(A+RBQ)^*$ and related to rectangular free additive convolution after the lift $S(p)(x)=p(x^2)$ (Marcus et al., 2015).

1. Operator-valued additive convolution and analytic subordination

In the operator-valued formulation, the basic object is an operator-valued non-commutative probability space $(M,E,B)$, where $M$ is a unital Banach algebra, $B\subset M$ a unital Banach subalgebra, and $E:M\to B$ a unit-preserving conditional expectation; throughout the main results one works in the $C^*$-setting, with $M_N(\mathbb C)$0 a $M_N(\mathbb C)$1-subalgebra and $M_N(\mathbb C)$2 completely positive. Freeness is freeness with amalgamation over $M_N(\mathbb C)$3: subalgebras $M_N(\mathbb C)$4 containing $M_N(\mathbb C)$5 are free over $M_N(\mathbb C)$6 if $M_N(\mathbb C)$7 whenever $M_N(\mathbb C)$8, $M_N(\mathbb C)$9 for all $p \boxplus_t q$0, and $p \boxplus_t q$1. The analytic domain is the operator upper half-plane

$p \boxplus_t q$2

For a selfadjoint $p \boxplus_t q$3-valued random variable $p \boxplus_t q$4, the Cauchy transform, reciprocal transform, and $p \boxplus_t q$5-transform are

$p \boxplus_t q$6

On $p \boxplus_t q$7, $p \boxplus_t q$8 is holomorphic, $p \boxplus_t q$9 maps $(A+RBQ)(A+RBQ)^*$0 into itself, and one has the positivity bound $(A+RBQ)(A+RBQ)^*$1, hence $(A+RBQ)(A+RBQ)^*$2 (Belinschi et al., 2013).

If $(A+RBQ)(A+RBQ)^*$3 and $(A+RBQ)(A+RBQ)^*$4 are free over $(A+RBQ)(A+RBQ)^*$5, there exist unique Fréchet analytic self-maps $(A+RBQ)(A+RBQ)^*$6 such that for all $(A+RBQ)(A+RBQ)^*$7,

$(A+RBQ)(A+RBQ)^*$8

$(A+RBQ)(A+RBQ)^*$9

and

$S(p)(x)=p(x^2)$0

Equivalently,

$S(p)(x)=p(x^2)$1

This is the operator-valued Bercovici–Voiculescu identity on the full operator upper half-plane, established globally as a Fréchet analytic subordination theorem and without any traciality assumption.

The same theory yields a fixed-point description suitable for computation. For each $S(p)(x)=p(x^2)$2, define

$S(p)(x)=p(x^2)$3

Then $S(p)(x)=p(x^2)$4 is the unique fixed point of $S(p)(x)=p(x^2)$5, and for any $S(p)(x)=p(x^2)$6,

$S(p)(x)=p(x^2)$7

The analogous statement holds for $S(p)(x)=p(x^2)$8 after swapping $S(p)(x)=p(x^2)$9 and $(M,E,B)$0. On tubes of the form $(M,E,B)$1, the Earle–Hamilton theorem gives a strict Carathéodory contraction, and Lemma 2.3 provides the explicit bound

$(M,E,B)$2

which supplies boundedness and stability on invariant subsets.

2. The tensorial viewpoint in random matrix polynomial problems

The operator-valued theory becomes “tensorial” when the amalgamation algebra is itself a tensor or matrix algebra. For polynomials in independent random matrices, the natural choice is

$(M,E,B)$3

Here $(M,E,B)$4 collapses the second tensor factor to scalars while preserving the operator structure on the matrix factor. If $(M,E,B)$5 is a linearization of a polynomial $(M,E,B)$6, then $(M,E,B)$7 is a $(M,E,B)$8-valued random variable, and its decomposition into summands $(M,E,B)$9 turns the original polynomial problem into an operator-valued free additive convolution problem over $M$0 (Belinschi et al., 2013).

The key bridge is Anderson’s selfadjoint linearization trick. For any selfadjoint noncommutative polynomial $M$1, there exists a selfadjoint block matrix

$M$2

with $M$3, of the form

$M$4

such that

$M$5

Applying the Schur complement to $M$6, where $M$7 is the block-diagonal embedding of $M$8, yields

$M$9

Thus the scalar resolvent of $B\subset M$0 is obtained as the $B\subset M$1-entry of an operator-valued resolvent.

This leads to a concrete algorithm. One computes $B\subset M$2 on $B\subset M$3, typically by iterating the operator-valued subordination maps for the free summands of $B\subset M$4. The scalar Cauchy transform is then extracted by

$B\subset M$5

and the limiting spectral measure is recovered by Stieltjes inversion. In tensorial cases of the form $B\subset M$6, the operator-valued Cauchy transform can be written as the Bochner integral

$B\subset M$7

so the nonlinear part of the computation is carried entirely by matrix inversions in $B\subset M$8.

The anticommutator example makes the mechanism explicit. For

$B\subset M$9

a selfadjoint linearization is

$E:M\to B$0

Evaluated on independent Wigner matrices, this becomes an element of $E:M\to B$1, amalgamated over $E:M\to B$2 with $E:M\to B$3. The limiting eigenvalue distribution is then obtained by the same subordination procedure, and the paper reports agreement with simulations.

3. High-order tensor freeness and convolution of measures

A different line of work develops tensorial free additive convolution directly for random tensors of higher order. In this framework, the tensor space is $E:M\to B$4 or $E:M\to B$5, and trace invariants are indexed by combinatorial maps. If $E:M\to B$6 is a map with boundary sequence $E:M\to B$7, then for $E:M\to B$8,

$E:M\to B$9

while for $C^*$0,

$C^*$1

with $C^*$2 the number of connected components. Freeness is defined by means of a non-crossing poset on maps generated by switches, together with chromatic conditions on components. Tensorial free cumulants $C^*$3 are introduced by Möbius inversion on that poset: $C^*$4 For even families, tensorial freeness is characterized by vanishing mixed cumulants: for every connected non-monochromatic map $C^*$5, one has $C^*$6. In the matrix case $C^*$7, connected maps are cycles, the poset becomes the non-crossing partitions lattice, and the construction reduces to classical free cumulants (Bonnin et al., 2024).

In this map-based setting, tensorial free additive convolution is defined for tensors of the same order $C^*$8. If $C^*$9 are centered and free, then for every connected $M_N(\mathbb C)$00-regular map $M_N(\mathbb C)$01,

$M_N(\mathbb C)$02

This is the exact tensorial analogue of cumulant additivity in ordinary free probability. The same framework also proves a free CLT for tensors: if $M_N(\mathbb C)$03 are centered, free, and uniformly moment-bounded, and if the two-vertex melonic moments $M_N(\mathbb C)$04 are independent of $M_N(\mathbb C)$05, then

$M_N(\mathbb C)$06

converges in distribution to a tensorial Gaussian law with cumulants supported only on the melons $M_N(\mathbb C)$07 (Bonnin et al., 2024).

The compactly supported measure version refines this picture by aggregating cumulants over connected rooted $M_N(\mathbb C)$08-regular trace maps. For a tensorial distribution $M_N(\mathbb C)$09, one sets

$M_N(\mathbb C)$10

and defines generating series

$M_N(\mathbb C)$11

They satisfy the functional equation

$M_N(\mathbb C)$12

For compactly supported measures $M_N(\mathbb C)$13, tensorial free additive convolution is denoted $M_N(\mathbb C)$14 and is defined by the rule

$M_N(\mathbb C)$15

The associated tensorial $M_N(\mathbb C)$16-transform is

$M_N(\mathbb C)$17

and it linearizes the convolution: $M_N(\mathbb C)$18 For $M_N(\mathbb C)$19, this recovers Voiculescu’s free additive convolution (Bonnin, 2024).

The same paper introduces higher-order analogues of the semicircular and free Poisson laws. The higher-order semicircular law $M_N(\mathbb C)$20 has moments

$M_N(\mathbb C)$21

where $M_N(\mathbb C)$22, and cumulants

$M_N(\mathbb C)$23

The higher-order free Poisson law $M_N(\mathbb C)$24 satisfies, for even $M_N(\mathbb C)$25,

$M_N(\mathbb C)$26

and one has

$M_N(\mathbb C)$27

4. LU-invariant tensors, permutation-indexed cumulants, and HCIZ/BGW asymptotics

A third tensorial framework organizes moments and cumulants by colored permutation data. A mixed tensor $M_N(\mathbb C)$28 has $M_N(\mathbb C)$29 inputs and $M_N(\mathbb C)$30 outputs,

$M_N(\mathbb C)$31

while a pure tensor is a pair $M_N(\mathbb C)$32. Local unitary invariance means invariance in distribution under $M_N(\mathbb C)$33 with $M_N(\mathbb C)$34. Trace-invariants are indexed by $M_N(\mathbb C)$35; for mixed tensors,

$M_N(\mathbb C)$36

and analogous formulas hold in the pure case. The connectedness of $M_N(\mathbb C)$37 is encoded by the join $M_N(\mathbb C)$38 of the cycle-partitions of $M_N(\mathbb C)$39, with $M_N(\mathbb C)$40 (Buc-d'Alché et al., 3 May 2026).

Finite-$M_N(\mathbb C)$41 tensorial cumulant precursors $M_N(\mathbb C)$42 are defined as classical cumulants of the $M_N(\mathbb C)$43-observables. In the mixed case,

$M_N(\mathbb C)$44

and there are analogous formulas for pure tensors. The asymptotic tensorial free cumulants are the rescaled limits

$M_N(\mathbb C)$45

with scaling exponent $M_N(\mathbb C)$46 determined by the asymptotic class. For independent LU-invariant tensors $M_N(\mathbb C)$47, additivity holds already at finite $M_N(\mathbb C)$48: $M_N(\mathbb C)$49 and asymptotically this becomes

$M_N(\mathbb C)$50

Accordingly, tensorial free additive convolution is defined by

$M_N(\mathbb C)$51

for all $M_N(\mathbb C)$52 and all $M_N(\mathbb C)$53.

In this approach, tensorial $M_N(\mathbb C)$54-transform data arise from free-energy derivatives of tensor HCIZ and BGW integrals. In the mixed matrix-product scaling regime,

$M_N(\mathbb C)$55

and in the pure Gaussian or $M_N(\mathbb C)$56-scaling regime,

$M_N(\mathbb C)$57

In the melonic first-order sector, addition of these generating series mirrors addition of tensorial cumulants.

This formalism also distinguishes between invariance classes. Under a coarser LU grouping $M_N(\mathbb C)$58, cumulants collapse via the canonical embedding $M_N(\mathbb C)$59. In the global unitary invariant case $M_N(\mathbb C)$60, tensorial free cumulants coincide with matricial free cumulants if and only if all $M_N(\mathbb C)$61 are equal; otherwise they vanish. The paper emphasizes that Gaussian pure tensors with non-trivial covariance provide concrete examples with genuinely non-trivial tensorial cumulants, including explicit formulas when the covariance is a tensor product of matrices.

5. The finite polynomial operation $M_N(\mathbb C)$62

In finite free probability, tensorial free additive convolution appears as an operation on monic polynomials of degree at most $M_N(\mathbb C)$63. If

$M_N(\mathbb C)$64

their asymmetric or tensorial additive convolution is

$M_N(\mathbb C)$65

Equivalently, if $M_N(\mathbb C)$66 and $M_N(\mathbb C)$67, then

$M_N(\mathbb C)$68

The random-matrix model is

$M_N(\mathbb C)$69

for square $M_N(\mathbb C)$70 matrices $M_N(\mathbb C)$71 such that $M_N(\mathbb C)$72 and $M_N(\mathbb C)$73. The paper states explicitly that this “tensorial” convolution is not a Kronecker-sum (Marcus et al., 2015).

The basic spectral property is real-rootedness. If $M_N(\mathbb C)$74 and $M_N(\mathbb C)$75 have nonnegative real roots, then $M_N(\mathbb C)$76 also has nonnegative real roots. The transform inequality is formulated after the lift $M_N(\mathbb C)$77: $M_N(\mathbb C)$78 Equivalently,

$M_N(\mathbb C)$79

Equality holds if and only if $M_N(\mathbb C)$80 or $M_N(\mathbb C)$81 is $M_N(\mathbb C)$82.

The operation has characteristic differential identities. Degree reduction is controlled by the Laguerre derivative: $M_N(\mathbb C)$83 and if $M_N(\mathbb C)$84, then

$M_N(\mathbb C)$85

There is also a Laguerre identity: $M_N(\mathbb C)$86 The paper interprets this finite convolution as a finite-dimensional analogue of rectangular free additive convolution, with the $M_N(\mathbb C)$87-lift supplying the connection to the corresponding finite $M_N(\mathbb C)$88-transform bound.

6. Relations, distinctions, and open directions

The literature does not use the expression “tensorial free additive convolution” in a single uniform sense. In one usage, it means operator-valued free additive convolution over a matrix or tensor algebra $M_N(\mathbb C)$89, with analytic subordination on $M_N(\mathbb C)$90 and direct application to selfadjoint random matrix polynomials. In another, it means the cumulant-additive convolution of higher-order tensors indexed by combinatorial maps or by colored permutation-tuples. In a third, it denotes the finite polynomial operation $M_N(\mathbb C)$91. A plausible implication is that results about transforms, limits, or combinatorics cannot be transferred between these settings without additional structure.

A related but distinct algebraic viewpoint is due to Cebron, who enlarges polynomial calculus to a tensorial algebra $M_N(\mathbb C)$92 with trace slots and a trace-product $M_N(\mathbb C)$93. In that calculus, additive free convolution is realized by a derivation $M_N(\mathbb C)$94 built from free cumulants, and the central formula is

$M_N(\mathbb C)$95

For semicircular noise $M_N(\mathbb C)$96, this gives the semigroup

$M_N(\mathbb C)$97

which realizes the transition $M_N(\mathbb C)$98. This is a tensorial realization of scalar free convolution rather than a theory of high-order random tensors, but it shows that tensorial calculi also arise on the operator side of free probability (Cébron, 2013).

The principal limitations are framework-specific. In the operator-valued theory, the global analytic subordination results are set up for selfadjoint variables in a $M_N(\mathbb C)$99-operator-valued probability space; for non-selfadjoint polynomials one must pass to Brown measure. In the map-based tensor theory, analytic Cauchy or resolvent transforms and subordination are not developed, and extending such tools is explicitly left open. In the compactly supported high-order theory, even $p \boxplus_t q$00 simplifies the poset structure through uniqueness of the minimal map, while odd $p \boxplus_t q$01 requires additional care. In the LU-invariant permutation framework, one open direction is the full equivalence between different proposals for tensorial free cumulants, especially beyond connected cases and for broader invariance classes (Belinschi et al., 2013, Bonnin et al., 2024, Bonnin, 2024, Buc-d'Alché et al., 3 May 2026).

These distinctions also clarify a common misconception. Tensorial free additive convolution is not a single universal object that has merely been expressed in different notation. The operator-valued theory is analytic and resolvent-based; the high-order tensor theories are combinatorial and cumulant-based; the finite polynomial theory is an expected-characteristic-polynomial construction. They agree in certain reductions—most notably at $p \boxplus_t q$02, where the tensor frameworks collapse to classical free probability—but the surrounding structures, admissible observables, and computational techniques remain different.