Spectral Statistics of Erdős-Rényi Graphs II:
Eigenvalue Spacing and the Extreme Eigenvalues
We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on vertices where every edge is chosen independently and with probability . We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least moments.
AMS Subject Classification (2010): 15B52, 82B44
Keywords: Erdős-Rényi graphs, universality, Dyson Brownian motion.
The Erdős-Rényi ensemble [20, 21] is a law of a random graph on vertices, in which each edge is chosen independently with probability . The corresponding adjacency matrix is called the Erdős-Rényi matrix. Since each row and column has typically nonzero entries, the matrix is sparse as long as . We shall refer to as the sparseness parameter of the matrix. In the companion paper , we established the local semicircle law for the Erdős-Rényi matrix for , i.e. we showed that, assuming , the eigenvalue density is given by the Wigner semicircle law in any spectral window containing on average at least eigenvalues. In this paper, we use this result to prove both the bulk and edge universalities for the Erdős-Rényi matrix under the restriction that the sparseness parameter satisfies
More precisely, assuming that satisfies (1.1), we prove that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble (GOE). In order to outline the statement of the edge universality for the Erdős-Rényi graph, we observe that, since the matrix elements of the Erdős-Rényi ensemble are either or , they do not satisfy the mean zero condition which typically appears in the random matrix literature. In particular, the largest eigenvalue of the Erdős-Rényi matrix is very large and lies far away from the rest of the spectrum. We normalize the Erdős-Rényi matrix so that the bulk of its spectrum lies in the interval . By the edge universality of the Erdős-Rényi ensemble, we therefore mean that its second largest eigenvalue has the same distribution as the largest eigenvalue of the GOE, which is the well-known Tracy-Widom distribution. We prove the edge universality under the assumption (1.1).
Neglecting the mean zero condition, the Erdős-Rényi matrix becomes a Wigner random matrix with a Bernoulli distribution when is a constant independent of . Thus for we can view the Erdős-Rényi matrix, up to a shift in the expectation of the matrix entries, as a singular Wigner matrix for which the probability distributions of the matrix elements are highly concentrated at zero. Indeed, the probability for a single entry to be zero is . Alternatively, we can express the singular nature of the Erdős-Rényi ensemble by the fact that the -th moment of a matrix entry is bounded by
For this decay in is much slower than in the case of Wigner matrices.
There has been spectacular progress in the understanding of the universality of eigenvalue distributions for invariant random matrix ensembles [5, 7, 8, 27, 28]. The Wigner and Erdős-Rényi matrices are not invariant ensembles, however. The moment method [31, 33, 32] is a powerful means for establishing edge universality. In the context of sparse matrices, it was applied in  to prove edge universality for the zero mean version of the -regular graph, where the matrix entries take on the values and instead of and . The need for this restriction can be ascribed to the two following facts. First, the moment method is suitable for treating the largest and smallest eigenvalues. But in the case of the Erdős-Rényi matrix, it is the second largest eigenvalue, not the largest one, which behaves like the largest eigenvalue of the GOE. Second, the modification of the moment method to matrices with non-symmetric distributions poses a serious technical challenge.
A general approach to proving the universality of Wigner matrices was recently developed in the series of papers [12, 13, 14, 15, 16, 17, 18, 19]. In this paper, we further extend this method to cover sparse matrices such as the Erdős-Rényi matrix in the range (1.1). Our approach is based on the following three ingredients. (1) A local semicircle law – a precise estimate of the local eigenvalue density down to energy scales containing around eigenvalues. (2) Establishing universality of the eigenvalue distribution of Gaussian divisible ensembles, via an estimate on the rate of decay to local equilibrium of the Dyson Brownian motion . (3) A density argument which shows that for any probability distribution of the matrix entries there exists a Gaussian divisible distribution such that the two associated Wigner ensembles have identical local eigenvalue statistics down to the scale . In the case of Wigner matrices, the edge universality can also be obtained by a modification of (1) and (3) . The class of ensembles to which this method applies is extremely general. So far it includes all (generalized) Wigner matrices under the sole assumption that the distributions of the matrix elements have a uniform subexponential decay. In this paper we extend this method to the Erdős-Rényi matrix, which in fact represents a generalization in two unrelated directions: (a) the law of the matrix entries is much more singular, and (b) the matrix elements have nonzero mean.
As an application of the local semicircle law for sparse matrices proved in , we also prove the bulk universality for generalized Wigner matrices under the sole assumption that the matrix entries have moments. This relaxes the subexponential decay condition on the tail of the distributions assumed in [17, 18, 19]. Moreover, we prove the edge universality of Wigner matrices under the assumption that the matrix entries have moments. These results on Wigner matrices are stated and proved in Section 7 below. We note that in  it was proved that the distributions of the largest eigenvalues are Poisson if the entries have at most moments. Numerical results  predict that the existence of four moments corresponds to a sharp transition point, where the transition is from the Poisson process to the determinantal point process with Airy kernel.
We remark that the bulk universality for Hermitian Wigner matrices was also obtained in , partly by using the result of  and the local semicircle law from Step (1). For real symmetric Wigner matrices, the bulk universality in  requires that the first four moments of every matrix element coincide with those of the standard Gaussian random variable. In particular, this restriction rules out the real Bernoulli Wigner matrices, which may be regarded as the simplest kind of an Erdős-Rényi matrix (again neglecting additional difficulties arising from the nonzero mean of the entries).
As a first step in our general strategy to prove universality, we proved, in the companion paper , a local semicircle law stating that the eigenvalue distribution of the Erdős-Rényi ensemble in any spectral window which on average contains at least eigenvalues is given by the Wigner semicircle law. As a corollary, we proved that the eigenvalue locations are equal to those predicted by the semicircle law, up to an error of order . The second step of the strategy outlined above for Wigner matrices is to estimate the local relaxation time of the Dyson Brownian motion [15, 16]. This is achieved by constructing a pseudo-equilibrium measure and estimating the global relaxation time to this measure. For models with nonzero mean, such as the Erdős-Rényi matrix, the largest eigenvalue is located very far from its equilibrium position, and moves rapidly under the Dyson Brownian motion. Hence a uniform approach to equilibrium is impossible. We overcome this problem by integrating out the largest eigenvalue from the joint probability distribution of the eigenvalues, and consider the flow of the marginal distribution of the remaining eigenvalues. This enables us to establish bulk universality for sparse matrices with nonzero mean under the restriction (1.1). This approach trivially also applies to Wigner matrices whose entries have nonzero mean.
Since the eigenvalue locations are only established with accuracy , the local relaxation time for the Dyson Brownian motion with the initial data given by the Erdős-Rényi ensemble is only shown to be less than . For Wigner ensembles, it was proved in  that the local relaxation time is of order . Moreover, the slow decay of the third moment of the Erdős-Rényi matrix entries, as given in (1.2), makes the approximation in Step (3) above less effective. These two effects impose the restriction (1.1) in our proof of bulk universality. At the end of Section 2 we give a more detailed account of how this restriction arises. The reason for the same restriction’s being needed for the edge universality is different; see Section 6.3. We note, however, that both the bulk and edge universalities are expected to hold without this restriction, as long as the graphs are not too sparse in the sense that ; for -regular graphs this condition is conjectured to be the weaker . A discussion of related problems on -regular graphs can be found in .
Acknowledgement. We thank P. Sarnak for bringing the problem of universality of sparse matrices to our attention.
2 Definitions and results
We begin this section by introducing a class of sparse random matrices . Here is a large parameter. (Throughout the following we shall often refrain from explicitly indicating -dependence.)
The motivating example is the Erdős-Rényi matrix, or the adjacency matrix of the Erdős-Rényi random graph. Its entries are independent (up to the constraint that the matrix be symmetric), and equal to with probability and with probability . For our purposes it is convenient to replace with the new parameter , defined through . Moreover, we rescale the matrix in such a way that its bulk eigenvalues typically lie in an interval of size of order one.
Thus we are led to the following definition. Let be the symmetric matrix whose entries are independent (up to the symmetry constraint ) and each element is distributed according to
Here is a scaling introduced for convenience. The parameter expresses the sparseness of the matrix; it may depend on . Since typically has nonvanishing entries, we find that if then the matrix is sparse.
We extract the mean of each matrix entry and write
where the entries of (given by ) have mean zero, and we defined the vector
Here we use the notation to denote the orthogonal projection onto , i.e. .
One readily finds that the matrix elements of satisfy the moment bounds
More generally, we consider the following class of random matrices with non-centred entries characterized by two parameters and , which may be -dependent. The parameter expresses how singular the distribution of is; in particular, it expresses the sparseness of for the special case (2.1). The parameter determines the nonzero expectation value of the matrix elements.
Definition 2.1 ().
We consider random matrices whose entries are real and independent up to the symmetry constraint . We assume that the elements of satisfy the moment conditions
for and , where is a positive constant. Here satisfies
for some positive constant .
Definition 2.2 ().
Let satisfy Definition 2.1. Define the matrix through
where is a deterministic number that satisfies
for some constants and .
For definiteness, and bearing the Erdős-Rényi matrix in mind, we restrict ourselves to real symmetric matrices satisfying Definition 2.2. However, our proof applies equally to complex Hermitian sparse matrices.
As observed in , Remark 2.5, we may take to be a Wigner matrix whose entries have subexponential decay by choosing .
We shall use and to denote generic positive constants which may only depend on the constants in assumptions such as (2.4). Typically, denotes a large constant and a small constant. Note that the fundamental large parameter of our model is , and the notations always refer to the limit . Here means . We write for .
After these preparations, we may now state our results. They concern the distribution of the eigenvalues of , which we order in a nondecreasing fashion and denote by . We shall only consider the distribution of the first eigenvalues . The largest eigenvalue lies far removed from the others, and its distribution is known to be normal with mean and variance ; see , Theorem 6.2, for more details.
First, we establish the bulk universality of eigenvalue correlations. Let be the probability density111Note that we use the density of the law of the eigenvalue density for simplicity of notation, but our results remain valid when no such density exists. of the ordered eigenvalues of . Introduce the marginal density
In other words, is the symmetrized probability density of the first eigenvalues of . For we define the -point correlation function (marginal) through
Similarly, we denote by the -point correlation function of the symmetrized eigenvalue density of an GOE matrix.
Theorem 2.5 (Bulk universality).
Suppose that satisfies Definition 2.2 with for some satisfying , and that additionally satisfies for some . Let and assume that
Let and take a sequence satisfying for some . Let and be compactly supported and continuous. Then
where we abbreviated
for the density of the semicircle law.
We also prove the universality of the extreme eigenvalues.
Theorem 2.7 (Edge universality).
Suppose that satisfies Definition 2.2 with for some satisfying . Let be an GOE matrix whose eigenvalues we denote by . Then there is a such that for any we have
as well as
for , where is independent of . Here denotes the law of the GOE matrix , and the law of the sparse matrix .
Theorem 6.4 can be easily extended to correlation functions of a finite collection of extreme eigenvalues.
We conclude this section by giving a sketch of the origin of the restriction in Theorem 2.5. To simplify the outline of the argument, we set in Theorem 2.5 and ignore any powers of . The proof of Theorem 2.5 is based on an analysis of the local relaxation properties of the marginal Dyson Brownian motion, obtained from the usual Dyson Brownian motion by integrating out the largest eigenvalue . As an input, we need the bound
where denotes the classical location of the -th eigenvalue (see (3.15) below). The bound (2.13) was proved in . In that paper we prove, roughly, that , from which (2.13) follows. The precise form is given in (3.16). We then take an arbitrary initial sparse matrix ensemble and evolve it according to the Dyson Brownian motion up to a time , for some . We prove that the local spectral statistics, in the first eigenvalues, of the evolved ensemble at time coincide with those of a GOE matrix , provided that
The precise statement is given in (4.9). This gives us the condition
Next, we compare the local spectral statistics of a given Erdős-Rényi matrix with those of the time-evolved ensemble by constructing an appropriate initial , chosen so that the first four moments of and are close. More precisely, by comparing Green functions, we prove that the local spectral statistics of and coincide if the first three moments of the entries of and coincide and their fourth moments differ by at most for some . (See Proposition 5.2.) Given we find, by explicit construction, a sparse matrix such that the first three moments of the entries of are equal to those of , and their fourth moments differ by at most ; see (5.6). Thus the local spectral statistics of and coincide provided that
3 The strong local semicircle law and eigenvalue locations
In this preliminary section we collect the main notations and tools from the companion paper  that we shall need for the proofs. Throughout this paper we shall make use of the parameter
We introduce the spectral parameter
where and . Let be a fixed but arbitrary constant and define the domain
with a parameter that always satisfies
For we define the Stieltjes transform of the local semicircle law
where the density was defined in (2.10). The Stieltjes transform may also be characterized as the unique solution of
satisfying for . This implies that
where the square root is chosen so that as . We define the resolvent of through
as well as the Stieltjes transform of the empirical eigenvalue density
For we define the distance to the spectral edge through
At this point we warn the reader that we depart from our conventions in . In that paper, the quantities and defined above in terms of bore a tilde to distinguish them from the same quantities defined in terms of . In this paper we drop the tilde, as we shall not need resolvents defined in terms of .
We shall frequently have to deal with events of very high probability, for which the following definition is useful. It is characterized by two positive parameters, and , where is given by (3.1).
Definition 3.1 (High probability events).
We say that an -dependent event holds with -high probability if
Similarly, for a given event , we say that holds with -high probability on if
Theorem 3.3 (Local semicircle law ).
Moreover, assume that
We have the local semicircle law: the event
holds with -high probability. Moreover, we have the following estimate on the individual matrix elements of . If instead of (3.9) satisfies
for some constant , then the event
holds with -high probability.
Next, we recall that the first eigenvalues of are close the their classical locations predicted by the semicircle law. Let denote the integrated density of the local semicircle law. Denote by the classical location of the -th eigenvalue, defined through
The following theorem compares the locations of the eigenvalues to their classical locations .
Theorem 3.4 (Eigenvalue locations ).
Suppose that satisfies Definition 2.2, and let be an exponent satisfying , and set . Then there is a constant – depending on and the constants in (2.4), (2.5), and (2.7) – as well as a constant such that the following holds.
We have with -high probability that
Moreover, for all we have with -high probability that
where we abbreviated .
Under the assumption the estimate (3.17) simplifies to
which holds with -high probability.
Moreover, from , Theorem 6.2, we get, with -high probability,
In particular, using (2.7) we get, with -high probability,
where is a constant spectral gap depending only on the constant from (2.7).
4 Local ergodicity of the marginal Dyson Brownian motion
In Sections 4 and 5 we give the proof of Theorem 2.5. Throughout Sections 4 and 5 it is convenient to adopt a slightly different notation for the eigenvalues of . In these two sections we shall consistently use to denote the ordered eigenvalues of , instead of used in the rest of this paper. We abbreviate the collection of eigenvalues by .
The main tool in the proof of Theorem 2.5 is the marginal Dyson Brownian motion, obtained from the usual Dyson Brownian motion of the eigenvalues by integrating out the largest eigenvalue . In this section we establish the local ergodicity of the marginal Dyson Brownian and derive an upper bound on its local relaxation time.
Let be a matrix satisfying Definition 2.2 with constants and . Let be a symmetric matrix of independent Brownian motions, whose off-diagonal entries have variance and diagonal entries variance . Let the matrix satisfy the stochastic differential equation
It is easy to check that the distribution of is equal to the distribution of
where is a GOE matrix independent of .
Let be a constant satisfying to be chosen later. In the following we shall consider times in the interval , where
One readily checks that, for any fixed as above, the matrix satisfies Definition 2.2, with constants
where all estimates are uniform for . Denoting by the largest eigenvalue of , we get in particular from (3.21) that
for some and .
From now on we shall never use the symbols and in their above sense. The only information we shall need about is (4.3). In this section we shall not use any information about , and in Section 5 we shall only need that uniformly in . Throughout this section will denote the joint eigenvalue density evolved under the Dyson Brownian motion. (See Definition 4.1 below.)
It is well known that the eigenvalues of satisfy the stochastic differential equation (Dyson Brownian motion)
where is a family of independent standard Brownian motions.
In order to describe the law of , we define the equilibrium Hamiltonian
and denote the associated probability measure by
where is a normalization. We shall always consider the restriction of to the domain
i.e. a factor is understood in expressions like the right-hand side of (4.6); we shall usually omit it. The law of the ordered eigenvalues of the GOE matrix is .
Define the Dirichlet form and the associated generator through
where is a smooth function of compact support on . One may easily check that
and that is the generator of the Dyson Brownian motion (4.4). More precisely, the law of is given by , where solves and is the law of .
Let to denote the solution of satisfying . It is well known that this solution exists and is unique, and that is invariant under the Dyson Brownian motion, i.e. if is supported in , so is for all . For a precise formulation of these statements and their proofs, see e.g. Appendices A and B in . In Appendix A, we present a new, simpler and more general, proof.
Fix and let be an increasing family of indices. Let be a continuous function of compact support and set
Let denote the classical locations of the first eigenvalues, as defined in (3.15), and set
Choose an . Then for any satisfying there exists a such that, for any , we have
for all . Here is the equilibrium measure of eigenvalues (GOE).
Note that, by definition, the observables in (4.9) only depend on the eigenvalues .
The rest of this section is devoted to the proof of Theorem 4.2. We begin by introducing a pseudo equilibrium measure. Abbreviate
Here we set for convenience, but one may easily check that the proof remains valid for any larger choice of . Define the probability measure
Next, we consider marginal quantities obtained by integrating out the largest eigenvalue . To that end we write
and denote by the marginal measure of obtained by integrating out . By a slight abuse of notation, we sometimes make use of functions , , and , defined as the densities (with respect to Lebesgue measure) of their respective measures. Thus,
For any function we introduce the conditional expectation
Throughout the following, we write . In order to avoid pathological behaviour of the extreme eigenvalues, we introduce cutoffs. Let be the spectral gap from (4.3), and choose to be smooth functions that satisfy
Define . One easily finds that
where the left-hand side is understood to vanish outside the support of .
Define the density
If is a probability measure and a density such that is also a probability measure, we define the entropy
The following result is our main tool for controlling the local ergodicity of the marginal Dyson Brownian motion.
Then for we have
First we note that
uniformly for , by (4.12). Dropping the time index to avoid cluttering the notation, we find
We find that
Bounding the Dirichlet form in terms of the entropy (see e.g. , Theorem 3.2), we find that
Thus we have
We therefore need to estimate
The second term of (4.19) is given by
Therefore (4.18) yields
The first term of (4.20) is given by
where we defined
Next, we estimate the error terms and