A unifying model for random matrix theory in arbitrary space dimensions
A sparse random block matrix model suggested by the Hessian matrix used in the study of elastic vibrational modes of amorphous solids is presented and analyzed. By evaluating some moments, benchmarked against numerics, differences in the eigenvalue spectrum of this model in different limits of space dimension , and for arbitrary values of the lattice coordination number , are shown and discussed. As a function of these two parameters (and their ratio ), the most studied models in random matrix theory (Erdos-Renyi graphs, effective medium, replicas) can be reproduced in the various limits of block dimensionality . Remarkably, the Marchenko-Pastur spectral density (which is recovered by replica calculations for the Laplacian matrix) is reproduced exactly in the limit of infinite size of the blocks, or , which for the first time clarifies the physical meaning of space dimension in these models. The approximate results for provided by our method have many potential applications in the future, from the vibrational spectrum of glasses and elastic networks, to wave-localization, disordered conductors, random resistor networks and random walks.
The eigenvalue spectrum of sparse random matrices is a fascinating subject with widespread applications in physics, from the energy levels of nuclei, to random resistor networks, random walks, the electronic density of states of disordered conductors, and many other topics handbook . It was investigated for several decades, from pioneering works pio to modern times mod .
In particular, random matrix theory has been applied extensively in recent years to the problem of the vibrational spectrum of glasses, where structural disorder leads to a number of puzzling effect in the vibrational density of states (DOS), such as the excess of soft low-energy modes (boson peak) with respect to Debye’s law schirmacher ; parisi ; bir ; vitelli ; milkus . This anomaly in the spectrum is related to well-know anomalies in the thermal properties at low temperatures phillips . This remains a famously unsolved problem because its mathematical description is plagued by the impossibility of analytically solving for the eigenvalue spectrum of the Hessian matrix of a disordered solid.
Recently, replica-symmetry breaking and allied techniques have been applied to the problem of vibrational eigenmodes of glasses, and produced results which recover the well-known Marchenko-Pastur (MP) distribution of eigenvalues of random Laplacian matrices parisi . The big question is about the applicability of these results: both MP and replica are generally thought to be valid for ”high-dimensional” systems, but what this means, in practice or in quantitative terms, has remained unanswered. This is clearly a central point of paramount relevance in the current debate on the theoretical description of glasses.
In this work, we clarify for the first time that MP and replica results are exactly valid in the case of random block Laplacian matrix where the dimension of the blocks is infinite. Furthermore, we show that while the lowest eigenvalue of the support is weakly dependent on the space dimension (which ensures that the scaling of the boson peak frequency in jammed solids and some models of glasses is rather well captured by high-dimensional models parisi ; beltukov ), instead the shape of the eigenvalue distribution changes significantly with and therefore high-dimensional methods such as MP and replica may not provide an accurate modelling of the vibrational DOS of disordered solids.
In all models or random spring networks, the elastic energy is a quadratic function of the displacements of the particles from their instantaneous “frozen” positions. The stiffness matrix or Hessian matrix is a Laplacian random symmetric matrix where each row is comprised of a small and random number of non-zero coefficients. The off-diagonal entries , , are identical independent random variables, whereas the diagonal entries . The latter requirement is dictated by enforcing mechanical equilibrium on every atom in the lattice.
The most typical model is the study of the spectrum of the Adjacency matrix or the Laplacian matrix of a Erdos-Renyi graph with vertices in the limit of large order of the matrices (the large limit).
The only parameter in the model is the probability of a link in the random graph to be present, whereas the dimension of the space of the amorphous material or the random spring model is absent.
In this work, we consider a block random matrix model which seems the simplest generalization of the above models, which retains a couple of relevant parameters.
We consider a real symmetric matrix of dimension where each row or column has random block entries , each being a matrix.
Every off-diagonal block has probability of being a null matrix and a probability of being a rank one matrix, where is a -dimensional random vector of unit length, chosen with uniform probability on the -dimensional sphere. Furthermore, is the usual matrix (or dyadic) product of a column vector times a row vector, which gives a rank-one matrix.
In the formulation of the stiffness matrix , the unit vector provides the direction between vertex and vertex (in a disordered solid or elastic network, between two atoms and ). For more details on the Hessian matrix of disordered solids see Refs.lemaitre ; scossa .
We study two prototypes of such block random matrices called the Adjacency block matrix and the Laplacian block matrix .
In both the above matrices, the set of , is a set of independent identically distributed random matrices and each is a rank-one matrix and a projector.
The study of the spectral density of the matrices , , in the limit , with fixed and fixed, is more difficult then the corresponding study with , the Erdos-Renyi graph, where all moments of both spectral functions are known bau , yet the
spectral distributions are not known.
Iii Evaluation of moments
Any symmetric matrix of order corresponds to a complete graph with vertices where the non-oriented link has the weight and is evaluated as the sum of the contributions associated to all paths of steps on the graph from vertex to itself. We used this familiar technique to evaluate the limiting moments. However in the present case, the contribution of each path is the product of matrices and the evaluation of moments of high order is laborious. We evaluated the first five limiting moments
which produce the following results:
Iv Results and discussion
The above evaluations are the main analytic task we performed. It involves to identify several non-equivalent classes of dominant paths, made of non-commuting sequences of blocks , which are dominant in the limit, to evaluate their cardinality, to average over the random unit vectors in the space, and to average over the probability of a block to be non-zero.
Eqs.(3),(4) are displayed in a way to point out that the lowest moments are polynomials in the variable whereas moments of higher order, starting with and , have additional terms involving just the space dimension .
We proceed to compare these moments, with the moments of three limiting cases, as it is schematically indicated in Fig.1.
Some relations are obvious but other are new and valuable.
First, in the limit our model reduces to the Erdos-Renyi graph. The moments of the spectral distributions of the Adjacency matrix and Laplacian matrix were determined by recurrence relations at every order bau . Those moments are reproduced by setting in
Eqs.(3),(4) and this is merely a consistency check of our evaluations.
A second limiting case is shown in Fig.1: the average connectivity is allowed to increase as the order of the matrices increase: with fixed. In this limit, the number of non-zero blocks in each row of the matrices increases in the limit, still keeping . It is sometimes referred as the dilute matrix limit. Many investigations found that in this limit the spectral distribution of the matrix is the same as a symmetric matrix with independent entries.
Let us consider the Wigner semi-circle distribution and its well known moments (Catalan coefficients)
These moments reproduce the highest powers of the polynomials of Eq.(3). Now let us consider the shifted semi-circle distribution and the first five moments
These moments reproduce the leading and the first non-leading powers of the polynomials of Eq.(4).
New and more relevant relations are related to the third limiting case: the limit , for fixed.
Semerjian and Cugliandolo semer evaluated the effective medium (EM) approximation for the spectral distribution of the ensemble of real symmetric matrices where the diagonal elements vanish and the off-diagonal entry , is zero with probability and it is one with probability :
where , and , are functions of .
We evaluated the moments of this spectral function from the Taylor expansion of the corresponding resolvent. One then obtains the moments in the table in Eq.(3) where the terms are absent and . That is, the limit with fixed.
Finally, the same limit, , with fixed, performed on the table in Eq.(4) leads to
The moments of the Marchenko-Pastur distribution
with the following definition of parameters:
where reproduce the above Eq.(7).
It is important to support the analytic indications of few moments with the full numerical evaluation of the spectral distributions. Large block-Adjacency matrices and block-Laplacian matrices, with and were generated according the probability distribution of our model and the eigenvalues were numerically evaluated. The obtained spectral distributions are in Fig.2. They support the conjectured limits indicated in Fig.1 and the emerging unifying picture.
Strikingly, while the difference between MP distribution and the numerical results for is of quantitative nature for the Laplacian, the difference between the EM approximation and the numerics for is of qualitative nature, especially around where the numerical results for show a delta-like peak whereas EM predicts a saddle.
In conclusion, the analytic evaluations of a few limiting moments and the numerical simulations support the conjecture of the relations schematically indicated in Fig.1 among different random matrix models. Since in the traditional models of disordered systems through random matrices and replica approach, the space dimension does not enter in the formulation of the model, the argument that the Effective Medium approximation (for the Adjacency matrix) and the Marchenko-Pastur distribution (for the Laplacian matrix) are valid for infinite space dimension is rather indirect and not well defined. The proposed relations and the systematic numerical results presented in this work substantiate these arguments by clarifying the role of space dimension for the various random matrix models, and suggest new ways to investigate disordered systems in finite space dimension.
In regard to the theory of random matrices, the present model explores ensembles of blocks random matrices with two different probabilities: the probability of independent identically distributed blocks to occur and the probability of the entries in the blocks. This structure is new, very promising, and of great relevance for physics applications.
The conjectured relations schematically indicated in Fig.1 indicate that this structure interpolates among all best studied spectral distributions.
We are also confident that the limiting moments here evaluated will be useful in the search for suitable approximate analytic representations of the eigenvalue distributions of physical models in finite space dimensions.
Appendix A Definition of the model
It is useful to recall the well known correspondence between any real symmetric matrix of order and the corresponding non-directed graphs with vertices. Between a generic pair of vertices of the graph there is a link, or edge, with the weight . The edge is absent if the corresponding matrix entry is zero. Edges where the extrema of the edge is the same vertex correspond to the diagonal entries of the matrix. The matrix element of a power of the matrix, say may be evaluated as the sum of the contributions of weighted paths of steps from vertex and vertex on the graph.
The sparse random block matrix we study in this work, is an ensemble of real symmetric matrices of dimension .
The generic matrix of the ensemble is a block matrix, with blocks in each row and column. Each block is a real symmetric matrix of order
The blocks are independent identically distributed random matrices.
The graph corresponding to the matrix has vertices, the weight of the (non-directed) edge connecting the pair of vertices is a matrix . It is still useful to evaluate elements of powers of the matrix in terms of the weighted paths connecting the vertices. Since the weight of a path is a product of non-commuting blocks, the order of them is relevant.
The Adjacency matrix has a zero block on the diagonal entries.
One easily evaluates traces of powers in terms of classes of non-equivalent paths cic . Since the blocks are independent identically distributed random matrices, it is sufficient to record when a block has previously appeared in a path. Then stands for any of the blocks , stands for any block, different from , etc. For instance
The analogous evaluation for the Laplacian block matrix is more involved
Each block is the null matrix , with probability or it is a rank-one random matrix , with probability , where is a random vector of length one, chosen with uniform probalibilty in .
Then, for instance,
with probability or with probability .
And with probability or with probability . The expected number of non-zero blocks in each row or column of the Adjacency matrix is , then is the average connectivity of the large graph (or the average degree of the vertices).
Finally the average over the uniform probability of the direction of all the random vectors involves integrals for each of them over the unit sphere in . Let us denote such integrals. For instance
Appendix B The moments of the limiting models
b.1 The simple random graph
For a simple (that is: no multiple edges, no edge with just one vertex) random graph, where the probability of any edge is , the moments of the spectral distribution of the Adjacency matrix and the Laplacian matrix were evaluated in the limit, and fixed average connectivity at every order bau . We report here the first few moments, from Table 1 and 2 of Bauer and Golinelli bau . For the Adjacency matrix we have:
while for the Laplacian matrix we have:
b.2 Effective medium approximation
In the same model, the spectral distribution of the Adjacency matrix in the Effective Medium (EM) approximation, is
It is difficult to evaluate the moments by analytic integration, but the first few moments are easily obtained from the series solution of the cubic
b.3 Marchenko-Pastur distribution
The Marchenko-Pastur distribution reads as
The moments are well known, and are given by
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