Law of Large Numbers for Roots of Finite Free Multiplicative Convolution of Polynomials

We provide the law of large numbers for roots of finite free multiplicative convolution of polynomials which have only non-negative real roots. Moreover, we study the empirical root distributions of limit polynomials obtained through the law of large numbers of finite free multiplicative convolution when their degree tends to infinity.


Introduction 1.Free probability theory
Denote by P and P + the set of all probability measures on R and [0, ∞), respectively.Moreover, we define P c and P +,c as the set of all compactly supported probability measures on R and [0, ∞), respectively.The notation w − → means the weak convergence of sequences of probability measures.Voiculescu initiated free probability theory to attack problems related to the free product of operator algebras.One of the most important notions in this theory is the free independence of non-commutative random variables.In this paper, we say free random variables as freely independent non-commutative random variables for short.
The law of large numbers (LLN) is well-known as a result that a sample average of independent identically distributed random variables with finite mean concentrates on the theoretical mean when the sample size is sufficiently large.As the analogous result on classical probability, the LLN for free random variables was also established (see [5]).More precisely, for any µ ∈ P with mean α, we have D 1/n µ ⊞n w − → δ α as n → ∞, where (i) D c (ν) is the push-forward of a measure ν by the mapping x → cx for c ∈ R and (ii) µ ⊞ ν is called the free additive convolution, which is the probability distribution of addition X + Y of free random variables X and Y distributed as µ ∈ P and ν ∈ P, respectively, in particular µ ⊞n is the n-th power of free additive convolution of µ.
The LLN for multiplication of (classically or freely) independent positive random variables are also considered.In classical probability, it is easy to formulate and investigate the LLN of multiplication by considering the exponential mapping of those random variables.However, it is not easy to consider the LLN for multiplication in free probability since e X+Y ̸ = e X e Y for (non-commutative) random variables X and Y .In [8], the LLN for multiplication of free bounded positive random variables was obtained.After that, this LLN was extended to one for multiplication of free positive random variables (which are not necessary to be bounded) in [3].More precisely, the LLN for multiplication of free positive random variables can be formulated as the convergence of for µ ∈ P + , where (i) ν α denotes the push forward of a measure ν by the mapping x → x α for α ∈ R and (ii) µ ⊠ ν is called the free multiplicative convolution, which is the probability distribution of multiplication √ XY √ X of free random variables X ≥ 0 and Y distributed as µ ∈ P + and ν ∈ P, respectively, in particular µ ⊠n is the n-th power of free multiplicative convolution of µ (see [10] and [2] for more details).According to [3], the limit distribution of the sequence (1.1) always exists and is denoted by Φ(µ).For µ ̸ = δ 0 , the measure Φ(µ) ∈ P + is characterized by the S-transform (see Section 2.1 for details).

Finite free probability and main result
In [6] and [7], Marcus, Spielman and Srivastava investigated a link between polynomial convolutions and the sum of random matrices related to free probability theory.For monic polynomials p(x) = d i=0 (−1) i p i x d−i and q(x) = d i=0 (−1) i q i x d−i of degree d, the finite free additive convolution p ⊞ d q is defined by The finite free additive convolution plays an important role in studying characteristic polynomials of the sum of (random) matrices.More precisely, for d × d real symmetric matrices A and B with characteristic polynomials χ A and χ B , respectively, χ A ⊞ d χ B is given by where the expectation is taken over unitary matrices Q distributed uniformly on the unitary group in dimension d.Furthermore, the finite free additive convolution is very closely related to free additive convolution because it turned out to be that the empirical root distribution of p d ⊞ d q d converges weakly to µ ⊞ ν when d tends to infinity, where µ, ν ∈ P are limit laws of sequences of empirical root distribution of p d and q d , respectively.Moreover, Marcus [6] obtained the typical limit theorems (LLN, the central limit theorem and the Poisson's law of small numbers, etc.) for finite free additive convolution.According to the evidence above, we can treat finite free probability as a discrete approximation theory for free probability.
In this paper, we investigate the LLN for finite free multiplicative convolution.The finite free multiplicative convolution ⊠ d of monic polynomials p(x) = d i=0 (−1) i p i x d−i and q(x) = d i=0 (−1) i q i x d−i of degree d with non-negative real roots is defined by In particular, p ⊠ d n denotes the n-th power of finite free multiplicative convolution of p.We formulate the LLN for roots of finite free multiplicative convolution of polynomials as the convergence of a sequence of where λ (n) i is the i-th (non-negative) root of p ⊠ d n for a monic polynomial p of degree d with non-negative roots.
We then obtain the limit theorem for the sequence (1.2) as follows.
Theorem 1.1 (LLN for ⊠ d ).Let p be a monic polynomial of degree d with non-negative real roots Λ and let us set k = k(p) as the number of zeros in Λ.Then, The paper consists of 4 sections.In Section 2, we introduce some concepts and preliminary results on free probability and finite free probability theories.In Section 3, we study the roots of finite free multiplicative convolution of polynomials and provide a proof of our main result (Theorem 1.1).In Section 4, we investigate the behavior of the empirical root distribution of polynomials obtained by the LLN for finite free multiplicative convolution when their degree tends to infinity.In the last of this section, we give a conjecture related to a connection between LLNs for ⊠ d and ⊠ from evidence obtained by this section.
The LLN for free multiplicative convolution of a probability measure on [0, ∞) was obtained by Tucci [8] and Haagerup and Möller [3].
Proposition 2.1.Let us consider µ ∈ P + .As n → ∞, the sequence of µ ⊠n 1 n converges weakly to the measure Φ(µ) ∈ P + characterized by Moreover, the support of the measure Φ(µ) is the closure of the interval Example 2.2.

Finite free multiplicative convolution
In this section, we introduce some concepts and preliminary results on finite free probability that are used in the remainder of this paper; see [1,6,7] for more details.Definition 2.3.For monic polynomials p and q of degree d which have only non-negative real roots: the finite free multiplicative convolution p ⊠ d q is defined by In [7, Theorem 1.5], the finite free multiplicative convolution ⊠ d can be realized as a characteristic polynomial of a product of positive definite matrices.That is, if χ A and χ B are characteristic polynomials of d × d positive definite matrices A and B, respectively, then where the expectation is taken over unitary matrices Q distributed uniformly on the unitary group in dimension d.Moreover, if p and q have only non-negative real roots, then so is p ⊠ d q (see [7, Theorem 1.6]).
There is the following asymptotic relation between finite free multiplicative convolution and free multiplicative convolution by [1,Theorem 1.4].Let us consider p d and q d as real-rooted monic polynomials of degree d in which p d has only non-negative real roots.Assume the empirical root distributions of p d and q d converge weakly to µ ∈ P +,c and ν ∈ P c as d → ∞, respectively.Then the empirical root distribution of p d ⊠ d q d converges weakly to µ ⊠ ν as d → ∞.

Main result
In this section, we provide the LLN for finite free multiplicative convolution.First, we calculate the n-th power of finite free multiplicative convolution of polynomials which have only nonnegative real roots.Let Λ (n) be the set of roots of p ⊠ d n for n ≥ 1 and a monic polynomial p of degree d with non-negative real roots.We put Λ := Λ (1) , for short.Lemma 3.1.Let p be a monic polynomial of degree d with non-negative real roots Λ.Then we have In particular, the number of zeros in Λ is the same as the one in Λ (n) .
Proof .Note that then by Definition 2.3 This is equivalent to (3.1).The rest is because the number of zeros in Λ is equal to k if and only if be the set of non-negative real roots of p ⊠ d n .Then we get For i = 1, the equation (3.2) implies that we obtain and therefore λ Similar to the estimation above, we obtain Consequently, we have and therefore n by using (3.2).Taking the n-th root of each value in the above inequality, we get Hence, we obtain λ For a positive number α > 0 and a monic polynomial p(x) = d i=1 (x − λ i ) with non-negative real roots, we define Remark 3.4.According to Theorem 3.2, if p is a monic polynomial of degree d with nonnegative real roots Λ and k is the number of zeros in Λ, then x − e i (Λ) e i−1 (Λ) .
Thus, the LLN for finite free multiplicative convolution of polynomials is established.the i-th real root of p ⊠ d n for 1 ≤ i ≤ d.By Newton's inequality (see, e.g., [4]), we have where the equality holds if and only if However, the inequality (3.3) can be directly proven by Theorem 3.2 due to λ i+1 .Consequently, we find the following remarkable phenomenon; except for trivial cases, the limit roots of p ⊠ d n ( 1 n ) , not being zero, are all distinct.
In particular, we apply Theorem 3.2 to the (renormalized) Laguerre polynomial and a polynomial with two real roots.
Example 3.6 (case of the Laguerre polynomial).Consider d ≥ 1.We define where L α,d is the Laguerre polynomial which is defined by Let Λ be the set of positive real roots of p.Note that p has no zero roots since p(0) = d!(−d) −d ̸ = 0. Computing the polynomial p, we have where the last equality holds by changing variable to j = d − i.This implies that are non-negative real roots of p ⊠ d n .By Theorem 3.2, we obtain Example 3.7 (case of a polynomial with two roots).Given d ≥ 1, consider the following monic polynomial p of 2d degree: and ẽj (Λ) = 0 for d + 1 ≤ j ≤ 2d.Suppose that λ d are positive real roots of p ⊠ d n .By Theorem 3.2, we have At the end of this section, we mention the rate of convergence in the LLN for roots of finite free multiplicative convolution of polynomials.Remark 3.8.According to a proof of Theorem 3.2, it is easy to see that log λ as n → ∞.
We demonstrate an example in which the rate of convergence is of order 1/n and it is optimal.Consider d = 2 in Example 3.6, that is, p(x) = x 2 − 2x + 2 −1 .As a consequent result of a proof of Lemma 3.1, we obtain p ⊠ 2 n (x) = x 2 − 2x + 2 −n for n ∈ N. Hence (positive) real roots of p ⊠ 2 n are given by It is easy to see that lim n→∞ λ as n → ∞.Consequently, the order 1/n is optimal.
4 Relation to the LLN for free multiplicative convolution In this section, given a monic polynomial of degree d, we investigate how the empirical root distributions of their limit polynomial obtained by Theorem 3.2 (or Remark 3.4) converge weakly as d → ∞.For the reason above, we emphasize their degree as follows.
Let p d be a monic polynomial of degree d with non-negative real roots and let k d be the number of zeros in Λ d .Denote by R i (Λ d ) the limit roots of p as provided in Theorem 3.2.In the following, we investigate relationships between the empirical root distributions: as d → ∞, where Φ(µ) is defined in Proposition 2.1.
Proof .By the assumptions and Theorem 3.2, we get Moreover, we assume that 0 / ∈ K in the following.Note that the functions t → t −1 and t → log t are bounded and continuous on K.We then obtain According to [3, Proposition 1], the last integral equals to ∞ 0 (log t)Φ(µ)(dt), and therefore we get the convergence.■ We give examples of the weak limit laws of empirical root distributions 1

e
d−k (Λ) > 0 and e d−k+1 (Λ) = 0, where we understand e d+1 (Λ) = 0. ■ Due to the relation (3.1), we obtain the following LLN for roots of finite free multiplicative convolution of polynomials.Theorem 3.2.Consider a monic polynomial p of degree d with non-negative real roots Λ and let k = k(p) be the number of zeros in Λ.Let Λ(n)

Proposition 4 . 2 .
Let p d be a monic polynomial of degree d with non-negative real roots Λ d .Assume that there exist µ ∈ P +,c and a compact set K in [0, ∞), such that the measures µ d and µ are supported on K for all d ≥ 1, and such that µ d w − → µ as d → ∞.Then we obtain