Instanton Approach to Large Harish-Chandra-Itzykson-Zuber Integrals
Abstract
We reconsider the large asymptotics of Harish-Chandra-Itzykson-Zuber integrals. We provide, using Dyson’s Brownian motion and the method of instantons, an alternative, transparent derivation of the Matytsin formalism for the unitary case. Our method is easily generalized to the orthogonal and symplectic ensembles. We obtain an explicit solution of Matytsin’s equations in the case of Wigner matrices, as well as a general expansion method in the dilute limit, when the spectrum of eigenvalues spreads over very wide regions.
The ability to perform explicit calculations of sums and integrals is at the heart of much groundbreaking progress in theoretical physics, in particular, in field theory or statistical mechanics. In that respect, the so-called Harish-Chandra-Itzykson-Zuber (HCIZ) integral Harish-Chandra (1957); Itzykson and Zuber (1980) is among the most beautiful results, and has found several applications in many different fields, including Random Matrix Theory, disordered systems or quantum gravity (for a particularly insightful introduction, see Tao ()). The generalized HCIZ integral is defined as:
(1) |
where the integral is over the (flat) Haar measure of the compact group or in dimensions and are arbitrary symmetric (hermitian or symplectic) matrices. The parameter is the usual Dyson “inverse temperature”, with or , respectively for the three groups. In the unitary case and , it turns out that the HCIZ integral can be expressed exactly, for all , as the ratio of determinants that depend on , and additional -dependent prefactors:
(2) |
with , the eigenvalues of and , the Vandermonde determinant of [and, similarly, for ], and .
Although the HCIZ result is fully explicit for , the expression in terms of determinants is highly nontrivial and quite tricky. For example, the expression becomes degenerate () whenever two eigenvalues of (or ) coincide. Also, as is well known, determinants contain terms of alternating signs, which makes their order of magnitude very hard to estimate a priori. This difficulty appears clearly when one is interested in the large asymptotics of HCIZ integrals, for which one would naively expect to have a simplified, explicit expression as a functional of the eigenvalue densities of . [The scaling can be guessed by noting that generically , but of course this is insufficient]. But even this large limit turns out to be highly nontrivial. In a remarkable paper, Matytsin Matytsin (1994) suggested a mapping to a nonlinear hydrodynamical problem in one-dimension, the solution of which gives, in principle, access to . Matytsin’s result for was later shown by Guionnet and Zeitouni Guionnet and Zeitouni (2002) to be mathematically rigorous. Still, neither Matytsin’s, nor Guionnet and Zeitouni’s derivation is very transparent (at least to our eyes). In this Letter, we recover Matytsin’s equations using a rather straightforward instanton approach to the large deviations of the Dyson Brownian motion that describes the (fictitious) dynamics of eigenvalues connecting to . Our approach is easily adapted to arbitrary values of , including the orthogonal case which yields Zuber’s “-rule” when , i.e. Zuber (2008). We then solve exactly Matytsin’s equation in two particular cases (i) both and are Wigner semicircle distributions (of arbitrary widths ); (ii) and are arbitrary, but with diverging widths . We compare our results with the small- expansion obtained in Collins (2003).
Our main idea is to study, using the method of instantons, the large deviations of the Dyson Brownian motion of eigenvalues that brings an initial distribution of eigenvalues to a final distribution (see Fig. 1). This occurs with a probability that is exponentially small, , with a rate that we are able to relate directly to the HCIZ integral – see below. (The idea to use Dyson Brownian motion in that context can also be found, but in a very different language, in Guionnet (2004).) Suppose that one adds to a certain matrix small random Gaussian Hermitian matrices of variance . It is well known that in the limit, the eigenvalues of the time-dependent matrix evolve according to (see Dyson (1962)):
(3) |
where is the standard Brownian motion and we set henceforth , corresponding to unitary matrices. The calculation of can be done using two different (but complementary) languages:
that of particle trajectories and that of densities, using the Dean-Kawasaki formalism. We start with the particle point of view, and sketch the density functional
method later. We introduce the total potential energy
, and the corresponding “force” .
The probability of a given trajectory for the Brownian motions between time and time is given by (see Fig. 1):
(4) | |||||
where is some normalization. The action contains a total derivative equal, in the continuum limit, to:
(5) |
and:
(6) |
The “instanton” trajectory that dominates the probability for large is such that the functional derivative with respect to all is zero (see e.g. Bray and McKane (1989)):
(7) |
which leads, after a few algebraic manipulations, to
(8) |
This can be interpreted as the motion of unit mass particles, accelerated by an attractive force that derives from an effective two-body potential .
The hydrodynamical description of such a fluid is given by the Euler equations for the density and the velocity field
(9) |
and
(10) |
where is the pressure field which reads, from the virial formula in one dimension (Le Bellac et al., 2006, p. 138):
(11) |
because the fluid is at an effective temperature (see below). Now, using the same argument as Matytsin Matytsin (1994), i.e,
writing and , one finally finds
(12) |
and therefore Matytsin’s equations for and . Plugging this back in the action , and going to the continuous limit, one also finds:
(13) |
which is exactly Matytsin’s action Matytsin (1994). Finally, the probability to observe the set of eigenvalues of for a given set of eigenvalues for is proportional to where is obtained by plugging into Eq. (13) the solution of the Euler equations (9, 10), with chosen in such a way that and .
Now, the idea is to interpret the HCIZ integrand in the unitary case, , as a part of the propagator of the diffusion operator in the space of Hermitian matrices.
Indeed, adding to small random Gaussian Hermitian matrices of variance , the probability to end up with matrix in a time is
. Writing with , the change of variables, as is well known, induces a probability measure on alone that includes a Vandermonde determinant
. Since the conditional distribution of is obviously invariant under
where is an arbitrary unitary transformation, we get another expression for
(14) | |||||
Comparing this last expression for with the above calculation, and taking care of the proportionality coefficients, we get as a final expression for :
(15) |
which is, apart from the term which comes from the prefactor in Eq. (2), precisely Matytsin’s result Matytsin (1994). Now, the whole calculation above can be repeated for the (orthogonal group) or (symplectic group) with the final (simple) result . This coincides with the result obtained by Zuber in the orthogonal case Zuber (2008) (see also Guionnet (2004); Collins et al. (2009)).
We now briefly explain how to obtain the same result using the Dean-Kawasaki framework Kawasaki (1973); Dean (1996). As shown by Dean Dean (1996), the density of interacting particles obeying the Langevin equation (3) is found to satisfy the (functional) Langevin equation , with
(16) | |||||
where is the two-body interaction potential, is a normalized Gaussian white noise (in time and in space), and, unlike in Dean (1996), we define . One can again write the weight of histories of using Martin-Siggia-Rose path integrals. This reads
(17) |
Performing the average over gives the following action (and renaming ):
(18) | |||||
with . Taking functional derivatives with respect to and then leads to the following set of equations:
(19) |
and
(20) | |||||
The Euler-Matystin equations are recovered, after a little work, by setting . One can finally check Bun et al. (forthcoming) that the coincides with when using the equation of motion satisfied by and . Note that this second method gives rise to additional “diffusion” terms, of order , which lead to a viscosity term in the velocity equation. This second method might therefore be more adapted to search for subleading corrections (in ) to the action.
Somewhat surprisingly, Matytsin’s formalism has not been exploited to find explicit solutions for in some special cases. One fully solvable case is when
and have centered Wigner semicircle spectra
(21) |
and . Note that , as it should be. Injecting these expressions into Eqs. (13), (Instanton Approach to Large Harish-Chandra-Itzykson-Zuber Integrals) finally leads to (with )
(22) |
For arbitrary matrices , the narrow spectra limit (corresponding to ) has been worked out by Collins Collins (2003). Specializing his general result to the case of Wigner matrices, one finds:
(23) |
which coincides with the small expansion of Eq. (22). In the opposite limit , we find from Eq. (22):
(24) |
Note that Eq. (22) has a singularity (in the complex plane) for . The general analytical properties of have attracted a lot of attention recently, see Goulden et al. (2011) and references therein.
The limit can be called the dilute limit and can be studied in full generality, since the solution of the Euler-Matytsin equations can be constructed as
a power series of , where we define (we choose here, without loss of
generality
(25) | |||||
which is identical to Eq. (24) when is a Wigner semicircle, but holds more generally. Note that terms appear in order of importance in the above formula.
The general expression for is cumbersome and will be given in a longer version of this work Bun et al. (forthcoming). To order , the result reads:
(26) | |||||
where is such that . For , one recovers Eq. (25) by changing variables back from to , with the Jacobian . The leading term in the above expansion is in fact and is easy to interpret: it comes from the fact that in the limit , HCIZ integrals Eq. (1) are dominated by the matrix that diagonalizes in the diagonal base of (and the corresponding eigenvalues are ordered).
The main achievements of this work are twofold: we first rederived the large asymptotics of HCIZ integrals, first obtained by Matytsin, using Dyson’s Brownian motion and the method of instantons. We also provided an exact, explicit solution for the case of Wigner matrices, as well as a general expansion method in the dilute limit, when the eigenvalue spectra spread over very wide regions. Beyond providing a relatively straightforward and transparent interpretation of Matytsin’s method, our work could provide a valuable starting point to obtain new results, such as the generalization to other ensembles (orthogonal, symplectic, Wishart), as in Guionnet (2004), but also to understand the structure of subleading (in ) corrections. Our explicit results in the dilute limit should also be useful for applications, such as, for example, the Bayesian estimate of large correlation matrices using empirical data Bun et al. (in preparation).
We thank R. Allez, J. Bonart, R. Chicheportiche, A. Guionnet, and J. B. Zuber for useful comments and suggestions. S.N.M acknowledges support from ANR Grant No. 2011-BS04-013-01 WALKMAT.
Footnotes
- We neglect the Jacobian which is small in the large (small temperature) limit, as usual.
- Actually, the Dean-Kawasaki formalism allows one to see that a viscosity term, of order , is in fact also present – see below.
- This also implicitely assumes that assuming that the density vanishes at the edges of the spectrum.
- A more rigorous derivation in the unitary case , that includes all prefactors, uses Johansson’s formula Johansson (2001).
- The case where can be easily treated by noting that the Euler-Matytsin equations are invariant under Galilean transformations, which allows one to recover the trivial result from shifting in the HCIZ integral, Eq. 1.
- see previous footnote.
References
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