Characteristic polynomials of modified permutation matrices at microscopic scale
We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens’ measures which are one-parameter deformations of the uniform distribution on the permutation group. We also look at some modifications of permutation matrices where the entries equal to one are replaced by i.i.d uniform variables on the unit circle. Once appropriately normalized and scaled, we show that the characteristic polynomial converges in distribution on every compact subset of to an explicit limiting entire function, when the size of the matrices goes to infinity. Our findings can be related to results by Chhaibi, Najnudel and Nikeghbali on the limiting characteristic polynomial of the Circular Unitary Ensemble [chhaibi2017circular].
1.1 Convergence of characteristic polynomials
Characteristic polynomials of random matrices have drawn much interest the last few decades. These objects encode the information of the whole spectrum of matrices. Moreover, in the case of unitarily invariant matrices (as Gaussian Unitary Ensemble or Circular Unitary Ensemble), the characteristic polynomial is believed to have a similar microscopic behavior as holomorphic functions which appear in number theory, as the Riemann zeta function. The characteristic polynomial of random matrices is also related to Gaussian fields, including the Gaussian multiplicative chaos introduced by Kahane [kahane1985chaos].
On the macroscopic scale, Keating and Snaith [keating2000random], Hugues Keating and O-Connell [hughes2001characteristic], and then Bourgade Hugues Nikeghbali and Yor [bourgade2008characteristic] study the logarithm of characteristic polynomial of unitary matrices following the Haar distribution, and prove in particular that its real and imaginary parts normalized by converge jointly in law to independent centred and reduced Gaussian random variables.
Hambly, Keevash, O-Connell and Stark [hambly2000characteristic] give a similar result for permutation matrices following the uniform measure. Zeindler [zeindler2010permutation] [zeindler2013central] generalizes this result for permutation matrices under Ewens measures, considering more general class functions than the characteristic polynomial, the so-called multiplicative class functions. Dehaye and Zeindler [dehaye2013averages], and Dang and Zeindler [dang2014characteristic] extend the study to some Weyl groups, and some wreath products involving the symmetric group.
On the microscopic scale, Chhaibi, Najnudel and Nikeghbali [chhaibi2017circular] show that the characteristic polynomial of unitary matrices following the Haar measure, suitably renormalized, converges to a limiting entire function. With the coupling of virtual isometries introduced by Bourgade, Najnudel and Nikeghbali [bourgade2012unitary], the authors get an almost sure convergence. Chhaibi, Hovhannisyan, Najnudel, Nikeghbali, and Rodgers [chhaibi2017limiting] extend the study to the special orthogonal group, the symplectic group, and give a related result for the Gaussian Unitary Ensemble.
Our motivation in this paper is to prove similar results on the characteristic polynomial of some particular unitary matrices related to random permutations. More precisely:
We focus on matrices belonging to two particular subgroups of the unitary group: the set of permutation matrices, and the wreath product (which can be seen as the set of permutation matrices where entries equals to one are replaced by complex numbers of modulus one).
We tackle a large family of measures on the symmetric group, which are invariant by conjugation and verify a certain property of decay over the cycle lengths. This family includes the family of Ewens measures, as we shall see.
We introduce a coupling method for generating sequences of modified permutations under these particular measures, by analogy of the notion of virtual isometries introduced in [bourgade2012unitary]. This coupling provides an almost sure convergence in our main result given below.
For all events and all random variables , we will denote by the conditional expectation of given .
We will write for the convergence in distribution of the sequence of random variables to the random variable .
We will use the arrow to denote the convergence in law on the space of continuous functions from to equipped with the topology of uniform convergence on compact sets.
Finally, for all real numbers , will denote the fractional part of , and the distance from to the nearest integer.
1.2 Main results and outline of the paper
Let be a random virtual permutation (we give the definition in the next section). Let be the sequence of random permutation matrices associated to , that is to say for each we define as the matrix whose coordinates are given by
Let be the random modified virtual permutation generated by and a sequence of i.i.d uniform variables on the unit circle independent of (see Corollary 10).
For all and , we consider the characteristic polynomials of and , respectively defined by
We are interested in the behavior as goes to infinity of
where is an irrational number between and .
and similarly, can be written as
We will give more details about these expressions in Section 3.
Finally, let us recall that the type of any real number is defined by
We say that is of finite type if is finite. A basic property is that if is irrational then its type is greater or equal to one (and can be infinite). See e.g. [hambly2000characteristic] for a little more details about finite type.
The main result of the present paper is the following:
Almost surely, converges uniformly on every compact set to an entire function defined by
Assume is an irrational number of finite type. Then
where is the same entire function as above.
Without the coupling of the modified virtual permutation, the theorem still holds replacing the first point by:
Note that the parameter is not allowed to be rational, otherwise some denominators in the product expression of could be zeros. Moreover, heuristically, the motivation to take irrational of finite type is to avoid a too fast accumulation of small denominators.
The article is organized as follows: In Section 2, we set out our preliminary definitions and results for generating sequences of random permutations and sequences of modified permutation matrices. In Section 3, we give a proof of Theorem 1 by showing the first point in Subsection 3.1, and the second point in Subsection 3.2. These two subsections are mutually independent. In Section 4, we give some estimates on the limiting function , and compare our results to the unitary case presented in [chhaibi2017circular]. Finally, in Section 5, we extend the study to more general central measures, removing the restriction to for the support of their corresponding distributions on (see (9)).
2 Generating random permutations
Before giving the construction of the random permutations we will deal with, let us recall the few following definitions and facts:
A virtual permutation is a sequence where for each , is an element of which can be derived from by simply removing the element from the decomposition into disjoint cycles of . Let denote the space of virtual permutations. There is a canonical projection of a measure on to a measure on . We call central if and only if each is central, that is to say
For each , it is easy to notice that every central measure on can be fully described by a distribution on the set
of partitions of the integer , and conversely, in such a way that there is a one-to-one correspondence. A highly less obvious result (Theorem 2.3 in [olshanski2011random]) is that there exists a natural one-to-one correspondence between the central measures on and the probability measures on
The following definition introduces a new notion which specifies the family of measures we are going to consider in the paper.
Let be a probability measure on .
We say that is a measure with exponential decay if it satisfies the following property: There exists and with , such that for all ,
We say that a distribution on is a central measure with exponential decay if its corresponding distribution on is a measure with exponential decay.
The Ewens measure [ewens1972sampling] of any arbitrary parameter on , denoted by , is a central measure with exponential decay.
Indeed, first recall that, given , one can define on thanks to the family of Ewens measures of parameter on , , denoted by , and defined by the probability functions
where denotes the total number of cycles of once decomposed into disjoint cycles. More precisely, the sequence of measures is coherent with the projections . In other words, if follows , then the random permutation obtained by removing the element from the cycle-decomposition of follows .
For each , the fact that is central on immediately derives from the fact that is central on for all . It is also well-known that the corresponding distribution on of the central measure is the Poisson-Dirichlet distribution of parameter (denoted by ).
Let be a random vector following . We know that has the same distribution as the order statistics of the random vector defined as follows: let be a sequence of i.i.d random variables (with density function given by ). For all , define , and . The distribution of is called . In the literature, this method for generating such a vector with i.i.d random variables is called residual allocation model [patil1977diversity] or stick-breaking process [kerov1997stick]. With this representation it is easy to compute that for all ,
with . Hence for any arbitrary ,
which is summable over , and then the Borel-Cantelli lemma applies and gives that the number of such that is almost surely finite. In other words, there exists a random number such that for all , . Finally, coming back to it remains to see that the same kind of inequality holds for its coordinates, which is a direct consequence of the fact that for all we have . Then the Ewens measure is a central measure with exponential decay.
Note that the Ewens measures are particular central measures whom corresponding distributions on are supported on
In the main body of the paper, we focus on central measures with exponential decay on whose corresponding distributions on are supported on .
Now, let us present the coupling we consider for generating random permutations, which is highly inspired from [tsilevich1997distribution], [najnudel2013distribution], and [najnudel2014flow].
Let be an element of , and let be the disjoint union of circles , where for all , has perimeter . Let . For all , one defines a permutation as follows: for all , there exists a unique such that . Let us follow the circle , counterclockwise, starting from . The image of by is the index of the first point in we encounter after . In particular, if is the only point in and , then is a fixed point of , because starting from we do a full turn of the circle before encountering again. To illustrate, if the equal and if the six first are distributed on as shown
The key feature of this construction is highlighted in the following proposition.
The sequence is a virtual permutation. Moreover, if follows any arbitrary distribution on , and if conditionally on the points are i.i.d following the uniform distribution on , then follows the central measure on corresponding to .
If follows the distribution and if conditionally on the points are i.i.d random variables uniformly distributed on , then follows .
Let be a distribution on . Let be a random vector following and let be the disjoint union of circles of perimeters . Assume that conditionally given , the are i.i.d random variables uniformly distributed on . Finally, introduce the array of random variables defined by
Then, as a consequence of Proposition 4, almost surely, converges in distribution to . Moreover, conditionally on , for all ,
by the strong law of large numbers.
In this paper we also consider some modifications of permutation matrices, that we will call modified permutation matrices, which are permutation matrices where the entries equal to one are replaced by complex numbers of modulus one. The set of modified permutation matrices of size has a group structure and can be identified to the wreath product , where denotes the unit circle. Let us denote by the subset of matrices of which do not have as an eigenvalue. The next lemma provides a construction of sequences of elements of , , by analogy to the notion of virtual isometries introduced by Bourgade, Najnudel and Nikeghbali in [bourgade2012unitary].
For all , for all , there exists a unique such that
Moreover, the permutation corresponding to derives from the one of by removing the element from its cycle-decomposition.
Before proving this result, let us give an insight with the basic example .
Let . First observe that one can write
where . Then implies that the first column and the second row of
are zeros, notably . Moreover, from this same rank condition we deduce
Conversely, the matrix satisfies , and lies in since by assumption on .
Proof of Lemma 6.
Let and . Write the cycle of the corresponding permutation of containing the element . There exist and some complex numbers of modulus one such that for all , and where is the canonical basis of .
Denote by the top-left minor of size of . Let .
If (i.e. is a fixed point of the associated permutation), then is an eigenvalue of . By hypothesis, this implies . Hence if and only if (since is the only non-zero entry of the last row and last column of ). Moreover in this case, as we have , and the procedure amounts to remove the fixed point from the associated permutation of .
If , then and with . The -th roots of are eigenvalues of . By hypothesis, it follows . Moreover, if and only if where is the -by- matrix with in row column , and zeros elsewhere. In this case, for all (not considered when ) and , so that the -th roots of are eigenvalues of (the corresponding cycle is ). As we deduce .
We say that a sequence of matrices is a modified virtual permutation if for all , and .
Note that every modified virtual permutation is in particular a virtual isometry.
Let be a modified virtual permutation. There exists a virtual permutation such that, for all , corresponds to the permutation and has a characteristic polynomial of the form
where is a sequence of elements of and the denote the cycle-lengths of . Moreover, for all and such that , can be defined as the product of the non-zero entries of corresponding to the cycle of .
Let . As in the proof of the previous lemma, let us denote by the cycle of containing the element , and by and the complex numbers of modulus one such that for all , and where is the canonical basis of . The characteristic polynomials of and satisfy the equality
Indeed, if , then can be written by the previous lemma, so that .
Otherwise, there exists a permutation matrix of size which fixes the element , such that and are block diagonal matrices where:
All the blocks are of the form or , with .
If is the number of blocks of , then has exactly blocks (including the bottom-right ), and the first blocks of and are equal.
The last block of is , hence with the help of the previous lemma the penultimate block of is , and we get
which gives (17). As a consequence, by analogy with the Chinese restaurant process (see e.g. [pitman2002combinatorial]), here the customers arrive one by one and choose a table according to its weight, regardless of the past, and when a new customer seats at a table (empty or not), it does not affect the element of corresponding to this table. Hence we can assign a to each table , independently of . ∎
We call random modified permutation matrix a random matrix such that:
corresponds to a random permutation generated with the procedure of Proposition 4 for a given distribution on .
The non-zero entries of are i.i.d random variables uniformly distributed on the unit circle.
Let be a random virtual permutation, and let be a sequence of i.i.d uniform variables on the unit circle, independent of . One can couple with a random modified virtual permutation such that, for all ,
is a random modified permutation matrix corresponding to .
Denoting by the cycle-lengths of , then for all and such that , is the product of the non-zero entries of corresponding to the cycle of .
This immediately derives from Proposition 8 and the fact that the projection via is coherent with respect to the sequence of probability measures defined for all as the law of a -by- random modified permutation matrix. Indeed, if the non-zero entries of , say , are i.i.d uniform variables on the unit circle, then the non-zero entries of , say , satisfy the following rule: There exists such that for all , , and . Consequently are i.i.d uniform variables on the unit circle. ∎
3 Proof of the main theorem
3.1 Quotient of characteristic polynomials related to modified permutation matrices
Consider a distribution with exponential decay on , giving a as in (10). Let be a random vector following the distribution .
Let be a sequence of modified random permutation matrices generated by the coupling given by Corollary 10.
Moreover, using the notations from (13) and (14), for all and such that , is the product of the non-zero entries of whom cycle is associated with the circle , so that does not depend on . Hence, by Corollary 10, can be reformulated with the help of the sequence which is independent of the , as
The next lemmas aim to handle the tail of the infinite product in the expression of , in order to apply a dominated convergence theorem and get the pointwise convergence of . Moreover, they provide a bound of uniformly on compact sets, allowing to conclude with Montel theorem.
Let . For all , set . Then a.s. there exists a random number such that for all ,
Let . Let be a random variable following the uniform distribution on .
Then for all ,
using the mean value inequality. Thus,
Applying Borel-Cantelli lemma we deduce that the number of such that is a.s. finite, i.e. a.s. there exists such that for all , . Finally, gives the claim. ∎
For all , a.s. there exists a random number , such that for all ,
Let . By the definition of in equation (14), we see that is the mean of i.i.d Bernoulli random variables of parameter .
If , then it is easy to check that the events and are equal, hence
If , then for any arbitrary we have the Chernoff bound
This inequality is optimized at point , which gives
By assumption of exponential decay (10), conditionally on , almost surely there exists a constant such that for all , . Thus, taking any arbitrary , for almost every there exists an integer such that for all , . Fix a given , and . Then, setting ,
for all sufficiently large and greater than , say for all ( dependent on ). Then for all ,
We deduce, for all ,