{'fraction_non_alphanumeric': 0.029817624273724985, 'fraction_numerical': 0.011660748870238864, 'mean_word_length': 4.579693831607384, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Eigenvalue Dynamics and the Matrix Chain

We introduce a general method for transforming the equations of motion following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into a boundary value problem in one-dimensional quantum mechanics. For the particular case of a one-dimensional chain of interacting NxN Hermitean matrices, the corresponding large N boundary value problem is mapped into a linear Fredholm equation with Hilbert-Schmidt type kernel. The equivalence of this kernel, in special cases, to a second order differential operator allows us recover all previously known explicit solutions for the matrix eigenvalues. In the general case, the distribution of eigenvalues is formally derived through a series of saddle-point approximations. The critical behaviour of the system, including a previously observed Kosterlitz-Thouless transition, is interpreted in terms of the stationary points. In particular we show that a previously conjectured infinite series of sub-leading critical points are due to expansion about unstable stationary points and consequently not realized.


Introduction
The utility of studying the statistical mechanics of systems which can be encoded in terms of matrix variables has been evident over the past twenty years. From the generalization of Wigner and Dyson's work by Brézin, Itzykson, Parisi and Zuber [1] to the use of matrix models in low dimensional gravity (see [2] for reviews), to modern implementations of matrix strings [3], matrix models have proven to be a convenient framework for calculation. Remarkably, throughout this period the dynamics of matrix models have been considered only in the 'stationary' case where solutions of the equations of motion are assumed to be time-independent.
There arise circumstances though where the time evolution of a system governed by an action over matrix variables is relevant, especially if there are definite initial and final conditions on the state of the system. Examples include two-dimensional Yang-Mills theory on the cylinder and sphere [4,5], matrix model approaches to the continuous Hirota equation [6] and lattice QCD [7] and, the case we will focus on here, linear chains of interacting Hermitean matrices.
It is our goal to give a coherent framework in which in to study this class of matrix model problems.
When an action over matrix variables can effectively be reduced to the eigenvalues, the treatment of the eigenvalues as a single, collective coordinate [8] provides a convenient viewpoint for determining the Hamiltonian of the system. For the matrix actions we will consider here the particular form is known as the Das-Jevicki-Sakita Hamiltonian [9]. The associated equations of motion are the non-linear Euler equations of fluid dynamics in one dimension and with the addition of appropriate boundary conditions present a difficult mathematical problem. Here we will show how these boundary value problems can be interpreted in the familiar framework of the semi-classical limit of one-dimensional quantum mechanics. While in general solutions are still difficult to obtain, we will focus our attention on the case of a one-dimensional chain of interacting N × N Hermitean matrices which was shown to be a Das-Jevicki-Sakita-type system by Matytsin [10] in the limit of large N. In this situation a restatement of the equations of motion in terms of quantum mechanics allows one to consider a linear problem which is most conveniently stated as a Fredholm integral equation. Formal solution of this equation in terms of iterations of the integral kernel, which can be evaluated by saddle-point methods, will provide a framework for detailing the nature and position of critical points which arise in the system.
An interesting system from a statistical mechanical point of view, the one-dimensional matrix chain was discussed previously as a model for string theory with a discrete target space [12]. In addition to this direct application, the matrix chain is related by duality transformation to string theory on a circle with radius R = 1/a [13,14], string theory at a finite temperature [15] and to O(2) sigma models coupled to two dimensional quantum gravity [16,17]. The model itself consists of Hermitean matrices φ x on the sites {x} of a one-dimensional lattice interacting with nearest neighbours. The action of the system is where a is the lattice spacing and V is the potential at each site. Assuming the site potentials V (φ x ) to be analytic in φ x , the action can be reduced to the eigenvalues of the matrix variables by SU(N) gauge rotations Λ and the known result for the correlator [11] Consequently, in the limit of large N, the calculation of the partition function reduces to determining solutions of equations of motion for the eigenvalues. Despite its relative simplicity and importance in low-dimensional string theory and quantum gravity, explicit solutions of the matrix chain are only known for a few particular choices of potential V . In Section Four we will show these cases arise when the equations of motion are equivalent to certain Sturm-Liouville eigenvalue problems.
This model exhibits a remarkable range of behaviour as one adjusts the lattice spacing, a.
For instance, in the limit of vanishing a one recovers the canonical example of a Das-Jevicki-Sakita system, matrix quantum mechanics (restricted to a singlet sector) In fact, the methods we will introduce in the next Section and use to study the matrix chain find their most natural form when applied to such matrix mechanical systems on a finite More interestingly, fixing the potential V to satisfy V ′′ (0) = −4, at critical lattice spacing a = 1, the matrix chain was argued to undergo a Kosterlitz-Thouless phase transition [13].
The physics of this phase transition are most transparent in the dual picture of string theory on the circle. Here strings wrapping around the non-contractible target space play the role of vortices which are liberated when the radius R ∼ 1/a becomes sufficiently small. In terms of a discrete lattice, at sufficiently small lattice spacing a < 1 the universal behaviour of the system is that of the continuous model (1.3) due to communication between lattice sites.
When the spacing is larger than unity this communication is broken and the system can be expected to factorize into a product of single D = 0 matrix models, which in fact it does as a → ∞. The main objective of studying the matrix chain is to understand precisely how this change of character from D = 1 matrix quantum mechanics to isolated D = 0 matrix models takes place at a = 1.
In addition to a = 0 and a = 1, it was argued in [16] and [17] that at a discrete infinity of lattice spacings non-analytic behaviour arises in the distribution of eigenvalues of the matrix variables φ x . Considering iterative approximation of a functional solution to the equations of motion, it was argued these critical points arise when a = sin (pπ/2q) where p ≤ q are each positive integers. In Section Five, where we develop the general solution for the distribution of eigenvalues as a series of saddle-point integrations, we will show that this behaviour is not realized in the matrix chain. In fact it will be shown that such critical behaviour arises when unstable stationary points are mistakenly considered relevant in saddle-point approximations.
to be determined as a temporal boundary problem in one-dimensional quantum mechanics. In where, for convenience, we have defined the 'velocity' field Explicitly we see that the equations of motion are of the form of the Euler equations for a one-dimensional inviscid fluid experiencing a time-dependent force derived from V ∂ρ ∂t As in all evolution problems we need to specify boundary conditions to supplement the equations of motion. In the case of matrix quantum mechanics the classical treatment of [1] assumed free boundary conditions for the evolution of the system over infinite time. In this case one is led to a study of the ground state of the system and the equations of motion (2.4) are solved for the stationary case of v = 0. In general though we are interested in arbitrary boundary conditions for the evolution of the system through some time T . To be definite we will specify the generic conditions on the fields v and ρ at time t = 0 and t = T We would now like to rewrite the equations of motion in terms of a single quasi-linear first order differential equation. Introducing the complex function it is easy to see that (2.4) leads to For vanishing potential and real f this is the Hopf equation which is the prototypical one dimensional model of an equation which admits wave solutions complete with shocks and other inherently non-linear behaviour [18]. We would like to solve this equation with the boundary conditions given in (2.5). While the solution of the (complex) Hopf equation is known explicitly [10] in terms of initial data, typically we are interested in cases where boundary data is supplied. In these situations the explicit solution leads to an ill-posed, inverse-type problem for the boundary conditions. Consequently we would like to find a more natural presentation of the problem. To this end we will consider a generalized version of (2.7) and introduce a dispersive term [18] which will serve to smooth out the solutions and allow one to calculate the details of the solution f in terms of boundary conditions. Adding such a term leads to a complex version of the forced Burgers' equation Obviously, in the limit ash → 0 the forced Hopf equation (2.7) is recovered. The usefulness of this addition to the problem is that there exists a well-known (to mathematicians) change of variable which will reduce (2.8) to a second order, linear differential equation. This transformation is known as the Cole-Hopf transform and is given As is readily verified, this transformation leads (up to an irrelevant additive constant of integration) to the Schrödinger equation Consequently we see that the time evolution of the collective field Hamiltonian (2.1) is given by the semi-classical (h → 0) limit of one-dimensional quantum mechanics with the same potential, up to a factor.
Restricting to time independent potentials 1 V(x, t) = V(x), the boundary conditions (2.5) enter into the picture as the start and end points of evolution by H through time T where the wavefunction is defined in terms of the eigenvalue density and velocity Given the Green's function G(x, u|T ) for the Schrödinger problem it may be convenient to cast (2.11) as an integral equation In applications the implementation of boundary conditions and Green's function varies depending on the situation. For example, in investigations of the continuous Hirota equation [6] one is interested in the case of time-dependent potentials V(x, t) leading to non-trivial Green's functions. Here the initial and final velocities are specified and the initial and final eigenvalue distributions are identified (ρ 0 = ρ T ) so that (2.13) takes on the form of a non-linear integral equation for ρ 0 .
In this case the coordinates (x, u) in (2.

An application: the matrix chain
An application of the preceding formalism, which we will be investigating for the remainder of this Paper is that of the one-dimensional matrix chain. Instead of (1.1), it will be convenient to consider a different form of the action where we have rescaled the matrix field φ x and the new site potential U is related to V by The evolution of eigenvalues in this system, in the limit of large N, was shown by Matytsin [10] to be given by the Das-Jevicki-Sakita Hamiltonian (2.1) with vanishing potential V. Here the boundary conditions (2.5), which play a crucial role, are given by The objective is to solve for the initial and final eigenvalue density ρ 0 in terms of the potential U.
With V = 0, and ǫ = ih, the Schrödinger equation ( where and we have introduced the constant λ which does not affect the asymptotic solution but is convenient for later discussion. In this form (3.4) has the natural interpretation of an eigen-problem with eigenvector χ and associated eigenvalue λ. It is interesting to note that the same integral equation arose in the context of matrix chains previously [12,13]. In these instances the authors were seeking the lowest lying eigenvalues λ in order to calculate the partition function. Here we are most concerned with the asymptotic (ǫ → 0) form of the eigenfunctions χ in order to recover ρ 0 .
Hence we have re-stated the differential form of the boundary value problem (eqns.
Likewise, at this stationary point ρ 0 can be formally evaluated using its relation to χ from Labeling x by G + in equation (3.6) and u s by G − in equation (3.7), we see the saddle-point evaluation of the integral equation (3.4) leads to Matytsin's solution [10] of the boundary value problem This form of the solution of the problem is well-suited to detailed local analysis as demonstrated in [16,17]. However, for a global analysis which can accommodate the existence of multiple stationary points, we will take the (well-studied) integral form (3.4) over the functional equation (3.8). Before considering the particular characteristics of solutions though, we will utilize the interpretation of integral operators as self-adjoint operators to recover all previously known explicit solutions ρ 0 of the matrix chain.

Integral and differential operators
An analysis of an integral equation centers on the properties of the kernel K. In our case, the kernel of (3.4) is This positive definite kernel is real, symmetric and hence Hermitean. In the following we will assume the kernel to have many convenient properties which follow from the boundedness of all n-fold iterations Consequently we will only consider potentials U(x) which increase at infinity faster than x 2 .
Under such conditions K defines a compact operator and represents a self-adjoint operator on the space of square-integrable functions. From classical operator theory [20] there is a countable infinity of orthogonal eigenfunctions {χ n (x)} which form a complete set on this space. It is our goal to obtain, as a function of the site potential U, the asymptotic form of these eigenfunctions which determine the large N solution of the eigenvalue distribution ρ 0 through (3.5).
There are a number of cases in which the kernel K is equivalent to familiar self-adjoint operators in physics. In fact only in these instances are explicit solutions of the large N matrix chain known. In each case the eigenvalue distribution ρ 0 is the solution of a quadratic equation. Here we will show that such situations arise by considering the ǫ → 0 limit of second order self-adjoint differential operators L for which L and K share a complete set of eigenfunctions and define the same action on the Hilbert space.
In order to determine the conditions that L must satisfy to be equivalent to the integral kernel we act on χ n as defined by the integral equation (3.4) Likewise, acting under the integral sign and using the self-adjoint property of L E n χ n (x) = du K(x, u)L u χ n (u) = du χ n (u)L u K(x, u) It follows from the symmetry of the kernel that the action of L generates symmetric functions of x and u which will cancel out in (4.5), where h is an arbitrary function of two variables with Taylor expansion coefficients {a mn }.
This construction is valid for any differential operator but for simplicity we will consider only second order differential operators of Sturm-Liouville form Now it is a simple matter to find the potentials U and associated coefficients a 2 and a 0 which are consistent with the symmetry condition (4.6). Since a second order L can only produce quadratic polynomials in u, (4.6) leads to the equation Having determined the coefficients a 2 and a 0 , we can solve for the explicit form of the associated eigenvalue distribution. The differential form of the integral equation (3.4) is where the eigenvalue E is related to that of the integral form by λ = e E/ǫ . In the limit of vanishing ǫ, the solution of this differential equation is given simply by the WKB approximation which implies the replacement of derivatives with 'momenta' With this substitution, (4.10) reduces to a quadratic equation in the eigenvalue distribution which is solved by Since ρ 0 is to be interpreted as a probability distribution describing the eigenvalues of the matrix variables, it must be properly normalized. This is carried out by integrating ρ 0 over the positive support of (a 0 − E)/a 2 and fixing the undetermined constant E to satisfy dxρ 0 (x) = 1 (4.13) Integrating (4.9), we find the most general potential of the matrix chain which is consistent with a second order, self-adjoint differential operator first considered by Matytsin [10]. Also the asymmetric double Penner considered in connection with the Kazakov-Migdal model of induced QCD [21] can be recovered (see [22] for recent work). In each of these cases the eigenfunctions χ are related to prolate spheroidal wave functions. Additionally, with a 20 = 1, the limit a 02 → 0 recovers the quadratic potential case solved by Gross [23] and Makeenko [24]. With U ′ = (a 01 + 2)x ≡ 2m 2 x, L is the Schrödinger operator for the simple harmonic oscillator and the general expression for the distribution of eigenvalues (4.12), properly normalized, reduces to In each of these cases the analytic structure of the eigenvalue distribution . This suggests that for a generic potential the solution of the matrix chain problem follows from the semi-classical approximation to a pseudo-differential operator as suggested by naively extending the results for finite chains of matrices [25]. We will not pursue this line of reasoning here but return in the next Section to the integral form (3.4) and give a method for obtaining the general solution and describe its unique features.

General solution and classification of critical points
A phase transition in a system is characterized by a change in the analytic structure of observables as functions of external parameters. In standard matrix models this is commonly taken to include a change in analytic structure of the distribution of eigenvalues 2 . In the present case we have shown that the distribution of eigenvalues for a linear chain of Hermitean matrices is determined completely by a linear integral equation (3.4) involving the kernel K (4.1). In this Section we will go further and solve for the asymptotic eigenfunctions χ as a series of iterations of K. Given that iterations of K are calculable by saddle-point methods and the relation (3.5), we can effectively solve for the distribution of eigenvalues ρ 0 . In particular we will be able to determine the critical structure of the matrix chain by analyzing changes in the analytic structure of the kernel K and its iterations.
We begin by slightly generalizing the problem of finding the eigenfunctions χ in order to make use of well-known techniques [20]. Instead of the integral equation (3.4) let us consider for a moment an inhomogeneous version with c some real constant Since there are no constant eigenfunctions of the kernel K, the unique solution of (5.1) is given by the Neumann series which builds the solution up through an iterative procedure Taking the limit of this process generates the solution where R is commonly referred to as the resolvent kernel and K n is given by the convolution of n kernels The convergence of the resolvent kernel is guaranteed by the boundedness of K n and the Hilbert-Schmidt theory of integral equations [20]. In order to recover the homogeneous solution we should set c = 0 which leads to a trivial solution for χ. Fortunately for us only the c independent logarithmic derivative of χ is required to recover the eigenvalue distribution ρ 0 of the matrix chain. From the definition of χ, (3.5) and (5.3) we have the result Hence we can extract the eigenvalue distribution from the resolvent kernel R, which is expressible solely in terms of iterations of the kernel K as defined in (5.3). This is an important observation since the iterated kernels are calculable in the limit of vanishing ǫ. Explicitly, the n th iterated kernel can be written for n ≥ 2 where the 'action' S is given by Evaluating the integral over z = {z 1 , . . . , z n−1 } is a straightforward application of saddlepoint methods in the limit of vanishing ǫ. In this limit the dominant contributions to the integral come from the vicinity of the stationary points of the function S ∂S ∂z i z α = 0 (5.8) In general there are many such stationary points, which we will label by α. Expanding S up to quadratic order about the stationary points, the resulting Gaussian integrations give the approximation to the iterated kernel where λ n ( z α ) is the number of negative eigenvalues of the n × n Hessian H n of second derivatives of S evaluated at the stationary point While we have counted contributions from all stationary points of S in (5.9), the only relevant ones are those with minimum 'action'. The contributions of other, irrelevant, stationary points are exponentially suppressed in the limit of vanishing ǫ.
Unfortunately it is difficult to calculate the iterated kernels and the resolvent in closed form to obtain explicit results for the eigenvalue distribution. It is possible though to extract enough qualitative information from (5.9) to outline a one-dimensional phase diagram of the matrix chain. To be definite we will consider a quartic potential Consequently we see that when evaluating arbitrarily high iterations of the kernel, K n , the Hessian at the origin will being to develop negative eigenvalues at m 2 = 1 and this stationary point becomes unstable. Since S is bounded from below for our choice of potential, there are other stationary points where S has a lower magnitude and it is these which will dominate the integral (5.6) for K n .
This degeneration at m 2 = 1 signals a phase transition in the matrix chain and the physical meaning of it can be found by returning to the original form of the matrix chain action (1.1).
The potential there, V is related to the current U by the one dimensional lattice spacing a It is a straightforward calculation to show that for vanishing ǫ, and to leading order in vanishing a, (5.15) is solved by which is the well-known solution for the ground state of matrix quantum mechanics [1].
As In addition to this standard behaviour it was argued in [16] that there exist a countable infinity of additional points in this phase where the eigenvalue distribution ρ 0 exhibits subleading non-analyticities. In fact we can reproduce these results if we assume, incorrectly, that the origin z = 0 is a relevant stationary point in the evaluation of ρ 0 . Considering the vanishing of Hessian eigenvalues (5.12), we see that the naive saddle-point approximation to K s+1 (5.9) will break down when m 2 = cos rπ s + 1 (5.19) It is well known that at degenerate stationary points there is a change in the analytic structure of a saddle-point approximated integral so it is natural to assume that when (5.19) is satisfied, K s+1 and the eigenvalue distribution ρ 0 will exhibit non-analyticities. In terms of lattice spacing the degeneracy condition (5.19) can be found by comparing with (5.14). The result a = 1 − cos rπ s+1 2 = sin rπ 2(s + 1) (5.20) with r and s ranging over positive integers is exactly the condition for subleading critical behaviour as found in [16].  It would seem that the same mechanism which produced the critical exponents of matrix quantum mechanics for −1 < m 2 < 1 produces different exponents here. This may be because of the total degeneracy of the origin but explicit calculations of the iterated kernels where the eigenvalue density ρ 0 is common to both models.
The one drawback of this model of induced QCD is that the action (6.1) has a Z Z N symmetry under the center of the gauge group which leads to super-confinement of gauge degrees of freedom, even on the lattice scale [27]. Obviously this not a feature that one would like to have in the continuum limit of the model and the easiest way to avoid it is for the system not to realize this symmetry faithfully in its solution. Unfortunately for the potentials U of logarithmic form, for which we have ρ 0 explicitly (see Section Four), this symmetry persists [21,28]. It would be interesting to see if solutions of the matrix chain for more general potentials might lead to non-trivial propagation of gauge fields in the continuum limit of the Kazakov-Migdal model. This can be checked in principle using the solution of the matrix chain for the eigenvalue density ρ 0 and the formalisms of [29] and [30] which express the correlation of gauge fields in the model in terms of ρ 0 .