Quenched free energy in random matrix model

We compute the quenched free energy in the Gaussian random matrix model by directly evaluating the matrix integral without using the replica trick. We find that the quenched free energy is a monotonic function of the temperature and the entropy approaches $\log N$ at high temperature and vanishes at zero temperature.

energy log Z(β) in the matrix model, where Z(β) = Tr e −βH is the partition function with the inverse temperature β = T −1 and the expectation value is defined by the integral over the N × N hermitian matrix H (1.1) One can compute the quenched free energy by the replica method 3 log Z(β) = lim n→0 Z(β) n − 1 n . (1.2) In the high temperature regime the n-point correlator Z(β) n is approximated by the disconnected correlator Z(β) n ≈ Z(β) n , (1.3) and the n → 0 limit in (1.2) gives rise to log Z(β) ≈ lim n→0 Z(β) n − 1 n = log Z(β) . (1.4) The right hand side of this equation is known as the annealed free energy. On the other hand, in the low temperature regime it is not clear how to define the analytic continuation of Z(β) n to n < 1. This is the origin of the difficulty found in [11]. It turns out that we can avoid this difficulty of analytic continuation by directly evaluating the quenched free energy by the matrix integral In order for the entropy to be positive, the free energy F should be a monotonically decreasing function of T . In the replica computation of the quenched free energy of JT gravity [11], a pathological non-monotonic behavior of F is found under a certain prescription of the analytic continuation in n. We find that the direct computation of the quenched free energy in the Gaussian matrix model (1.5) gives rise to a well-defined monotonic behavior of the free energy F . This paper is organized as follows. In section 2, we find the explicit integral representation of the quenched free energy (1.5) and study its behavior in the high and low temperature regimes. In section 3, we study the exact free energy and entropy for N = 2, 3 as examples. We find that the free energy exhibits a well-defined monotonic behavior as a function of T . In section 4, we comment on the computation using the replica method. We propose a necessary condition for the analytic continuation of Z(β) n to satisfy. Finally, we conclude in section 5 with some discussions on the interesting future problems.

Quenched free energy in Gaussian matrix model
In this paper we will analyze the quenched free energy in Gaussian matrix model (1.5) directly without using the replica trick. From the standard argument, the matrix integral in (1.5) is written as an integral over the N eigenvalues Here the normalization factor Z is given by where G 2 (N + 1) denotes the Barnes G-function. Using this expression (2.1), in subsection 2.1 and 2.2 we will study the behavior of quenched free energy in the high temperature and the low temperature regimes, respectively.

High temperature regime
In the high temperature regime, the quenched free energy is approximated by the annealed free energy (1.4). The one-point function Z(β) in the Gaussian matrix model happens to be the same as the computation of the 1/2 BPS Wilson loop in N = 4 super Yang-Mills theory (SYM), and the exact result at finite N is found in [22] in terms of the Laguerre polynomial The large N behavior of the one-point function Z(β) can be computed from the genuszero eigenvalue density known as the Wigner semi-circle distribution. Then in the large N limit the one-point function Z(β) becomes where I 1 (2β) is the modified Bessel function of the first kind. From this expression one can easily find the expansion of the free energy and the entropy in the high temperature regime (T 1) In particular, the high temperature limit of entropy is log N This is expected since N is the dimension of the Hilbert space and log N is the maximal entropy of the system.

Low temperature regime
Next let us consider the low temperature regime (T 1). In the low temperature limit β → ∞, one can see that log Tr e −βH becomes Thus we find that the low temperature limit of the quenched free energy is determined by the expectation value E 0 = min{E i } of the smallest eigenvalue Note that E 0 is explicitly written as the eigenvalue integral We do not know the closed form of this integral for general N , but it is possible to evaluate this integral for small N . For instance, for N = 2, 3 we find It is known [23] that in the large N limit E 0 converges to the edge of the Wigner semi-circle It turns out that one can systematically compute the small T corrections to the leading term in (2.9). In the eigenvalue integral (2.1), one can choose E N as the smallest eigenvalue without loss of generality. Then the range of other eigenvalues E i (i = 1, · · · , N − 1) is restricted to E i > E N . With this remark in mind, the quenched free energy is written as This is further simplified by shifting (2.14) This is our master formula. The small T behavior of (2.14) is found by rescaling one of the integration variables E i → T E i . In this way we find that the quenched free energy at low temperature (T 1) behaves as 3 Numerics for N = 2, 3 In this section we study numerically the integral representation of the quenched free energy (2.14) for N = 2, 3. For N = 2 the integral (2.14) becomes and for N = 3 we find One can easily evaluate these integrals numerically. In Fig. 1 we show the plot of free energy as a function of temperature. At high temperature, the quenched free energy approaches the annealed free energy F ann = −T log Z(β) (orange dashed curve) as expected. In Fig. 2 we show the plot of entropy S. One can see that S approaches log N at high temperature and vanishes at zero temperature.  Let us take a closer look at the low temperature regime for N = 2. The small T expansion of the integral (3.1) is obtained by rescaling the integration variable E → T E and expanding the Gaussian factor in (3.1) Note that the first small T correction is of order T 4 which is consistent with the general result in (2.15). In Fig. 3 we show the plot of quenched free energy for N = 2 and its small T expansion up to T 6 in (3.3). One can see that the exact quenched free energy is a monotonic function of T even in the low temperature regime and F becomes E 0 at zero temperature. A pathological non-monotonic behavior found in [11] using the replica trick does not occur in the exact result of quenched free energy.

Comment on the replica method
Let us compare our direct calculation of quenched free energy with the replica method (1.2). As we mentioned in section 1, one can easily apply the replica method in the high temperature regime and obtain the result (1.4). In particular, in the high temperature limit β → 0, the partition function Z(β) = Tr e −βH reduces to the dimension of the Hilbert space lim β→0 Z(β) = Tr 1 = N. Thus the quenched free energy approaches the maximal entropy of the system in the limit On the other hand, the application of the replica trick in the low temperature regime is rather subtle. Under a certain prescription of the analytic continuation in the number of replicas n, it is found that the free energy exhibits a non-monotonic behavior as a function of temperature [11].
Our direct computation of the quenched free energy puts a certain constraint on the possible form of the analytic continuation in n. At low temperature, the smallest eigenvalue We can regard (4.3) as a condition for the possible analytic continuation of Z(β) n to satisfy. Then we can apply the replica method in the low temperature regime which reproduces the correct behavior of the quenched free energy (2.9). Note that there is no log N entropy term in (4.4) since only a single eigenvalue (the lowest energy state) contributes to Z(β) n in the low temperature limit. This explains the vanishing of entropy at zero temperature (2.17).
We would like to understand the role of replica symmetry breaking in a possible large N phase transition. When n is a positive integer, the n-replica correlator Z(β) n is expanded in terms of the connected correlators (4.5) Here [p νp ] = [1 ν 1 2 ν 2 · · · n νn ] denotes a partition of n. In the high temperature regime the disconnected part Z(β) n corresponding to the partition [1 n ] is dominant, while at low temperature the totally connected part Z(β) n conn corresponding to the partition [n 1 ] is dominant [26]. Then one might naively think that the quenched free energy in the low temperature regime is given by the totally connected correlator Z(β) n conn log Z(β) = lim n→0 Z(β) n conn − 1 n . (4.6) One can try to compute Z(β) n conn for integer n and analytically continue it to n = 0. However, this analytic continuation is very subtle since Z(β) n conn scales as N 2−n in the large N limit and the naive n → 0 limit of Z(β) n conn is not 1 and the limit (4.6) does not exist. It is not clear how to define the analytic continuation of Z(β) n conn which satisfies the condition (4.3).
A similar problem has appeared in the so-called random energy model [27]. 4 In [27] this problem is circumvented by promoting (p, ν p ) in (4.5) as a continuous variable and the correct low temperature behavior is obtained by plugging ν p = n p and extremizing the term ( Z(β) p conn ) n/p in (4.5) with respect to p. The n-point function Z(β) n obtained with this prescription indeed satisfies the necessary condition (4.3) and we can safely take the n → 0 limit [27]. It would be interesting to see if the same prescription works in the present case of random matrix model. We leave this as an interesting future problem.

Discussion
In this paper we have analyzed the quenched free energy in Gaussian matrix model directly without using the replica method. We find an integral representation of the exact quenched free energy (2.14). The exact quenched free energy is a monotonic function of temperature as expected, and the entropy computed from this free energy approaches log N at high temperature and vanishes at zero temperature.
There are many interesting open questions. It is very interesting to see if there is a phase transition in the large N limit. In the case of random energy model, it is known that there is a phase transition associated with the replica symmetry breaking and the low temperature phase corresponds to a spin glass [29]. Since the random matrix model considered in this paper can be thought of as a generalization of the random energy model, it is tempting to speculate that the quenched free energy of the random matrix model also exhibits a phase transition. To settle this issue it is important to understand the analytic continuation of Z(β) n to n < 1. We proposed a simple condition (4.3) for the analytic continuation of Z(β) n to satisfy.
It would be very interesting to generalize our analysis to the JT gravity matrix model and see if the spin glass phase is realized at low temperature [11]. In [11] the quenched free energy is computed by a certain prescription of the analytic continuation of Z(β) n and it leads to a pathological behavior at low temperature. It is argued in [12] that this problem is resolved by including the non-perturbative effect. It would be very interesting to complete the program of replica computation of the free energy in JT gravity.
Our analysis suggests that at low temperature the smallest eigenvalue (or the lowest energy state) gives a dominant contribution to the quenched free energy. This reminds us of the "eigenbrane" introduced in [30]. Perhaps the spacetime picture of the low temperature phase is described by an eigenbrane with one of the eigenvalues pinned to the edge of the spectral density. It would be interesting to investigate this picture further.