Random Energy Models. Indian Statistical Institute, Bangalore. November 23-27, 2007

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1 Random Energy Models Indian Statistical Institute, Bangalore ovember 23-27, 2007 B.V.Rao The aim of these lectures is to introduce random energy models, to understand some of the problems, their solutions and techniques. either are the results presented most general, nor are the paths geodesic.. How did we end up here: The origins of the subject go back to our attempts to understand magnetism. While it was felt that the collective behaviour of atoms more precisely the alignment of the spins is the cause, the reasons for this collective behaviour were not clear. The discovery by Curie in 895 of the Curie point a certain temperature at which the magnetic property disappears pressed for models to explain the phenomenon. As Bovier [7] put it, Wilhelm Lenz ( ) had a beautiful and simple idea. Imagine atoms are at the sites of Z d, the integer lattice. Assume the simplest spin variables ±. Assume that only neighbours interact and the interaction favours neighbouring atoms to take the same value. There is an external magnetic field which favours globally + or globally spin. Bring in ideas from statistical mechanics and solve the problem. This was the problem he gave to his student Ernest Ising and ever since this is known as Ising model, sometimes called Lenz-Ising model (The Lenz Law in induced magnetism is due to Heinrich Lenz ( )). Mathematical formulation of the model is as follows. Since spins live on the integer lattice Z d and take only ± values, the configuration space is {, +} Zd. If σ = (σ i ; i Z d ) is a configuration, then the energy of the system in this configuration, called Hamiltonian, is H(σ) = Jσ i σ j h σ i. Here the numbers J and i j = i h are respectively the interaction between spins and strength of the external field. Of course i j = means that the two points i and j of Z d differ by exactly one coordinate, that is, they are neighbours. Eventhough the sum is over neighbouring pairs, it is an infinite sum and does not converge. The best way to interpret is to consider particle system and take limit as. To do this we first need some notation. For a set Λ Z d, put Ω Λ = {, +} Λ. For any finite set Λ and σ Ω Λ and η Ω Λ c put H Λ (σ η) = Λ Jσ iσ j h ı Λ σ i where the first sum Λ is over all pairs i and j with i j = and at least one

2 of them is in Λ. In such a case, either both i and j are in Λ or one of them is in Λ and the other one in Λ c ; a boundary point of Λ. When this happens and j Λ c, the corresponding spin σ i is to be taken as η i. This is the energy of the configuration σ in the finite volume Λ when the outside configuration is η. Of course, since only neighbours interact, this depends only on sites in Λ and their neighbours. The key principle of statistical mechanics is the following (Feynman s [30]). If a system in equilibrium can be in one of states then the probability of the system having energy E n is Q e En/κT where Q = e En/κT. Here κ is n the Boltzman constant, T is temperature and Q is called the partition function. The expected value of an observable quantity f is Q f(i)e Ei/κT. This fundamental law (formulated in the quantum set up) is the summit of statistical mechanics and the entire subject is either slide down from the summit as the principle is applied to various cases, or the climb up where the fundamental law is derived and the concepts of thermal equilibrium and temperature T clarified. Returning to the Lenz-Ising model, given a finite set Λ Z d we have the Hamiltonian H Λ (σ η) defined above for σ Ω Λ and η Ω Λ c. The partition function is Z η Λ = e βhλ(σ η). Here β is the inverse temperature and κ is σ Ω Λ taken as one. Let F Λ denote the σ-field on Ω generated by the coordinate maps in Λ. When Λ = Z d, we denote this σ-field by F. For a finite set Λ we can think of the Gibbs distribution exp{ βh Z η Λ (σ η)} as the conditional probability on Λ F Λ given F Λ c. To describe the equilibrium state of the system, one looks for a probability µ on (Ω, F) such that under µ the conditional probability on F Λ given F Λ c are as above for every finite subset Λ Z d. Whether there is any such probability at all, if so how many exist, if more than one exist then the nature of the set of all such probabilities and its extreme points etc are all classical. Moreover one considers general interaction functions (See Ruelle [48], Sinai [53], Preston [47], Georgii [32]). The present story starts with efforts to understand magnetic properties of alloys. In the sixties there was much interest in the behaviour of isolated magnetic impurities in non-magnetic hosts (like Mn in Cu or Fe in Au). The magnetic moments are not arranged in a lattice. The interaction between the magnetic moments is not always of the same sign. This calls for a change from the classical set up. One does see spins exhibiting a collective behaviour, a random freezing which exhibits several peculiarities. To explain the observed phenomena a model was proposed by Edwards and Anderson in 975. i 2

3 As Sherrington [52] says Edwards and Anderson produced a paper in 975 that at one fell swoop recognized the importance of the combination of frustration and quenched disorder as the fundamental ingredients, introduced a more convenient model, a new and novel method of analysis, new types of order parameters, a new mean field theory, new approximation techniques, and the prediction of a new type of phase transition apparently explaining the observed (magnetic) susceptibility cusp. Edwards and Anderson s new approach was beautifully minimal, fascinating and attractive, their analysis was highly novel and sophisticated, involving radically new concepts and methods but also unusual and unproven ansatz, as well as several different approaches. However, this model (see the Hamiltonian later in section 2) was not exactly solvable. Sherrington and Kirckpatrik proposed a mean-field model. The adjective mean-field refers to the fact that each spin interacts with all the others. In 98 and later in 985, Bernard Derrida proposed two models called Random Energy Model and Generalized Random Energy model. In its bare essentials then, for an -particle system we have a configuration space Σ and for each σ in this space we have a random variable H (σ), the Hamiltonian in this configuration. The partition function is Z = e βh (σ) σ where β is the inverse temperature. There are three main issues which lead to several problems. Firstly, does log Z have a limit and if so is it nonrandom. This is the limiting energy, or just energy. It is important to have this limit to be non-random. Secondly, we have random Gibbs measures G (σ) = e βh (σ) /Z on the space Σ. This is random because H is so. Can we say anything about their limit over. Of course this needs careful formulation, since these Gibbs measures live on different spaces. Thirdly, the ground state in any system is the configuration having minimum energy and the ground state energy is the minimum possible energy. Thus we are looking at min H (σ) and σ the states σ where it is attained. Can we say anything about their asymptotics? The first issue, loosely put, says that a random quantity (like log Z ) which depends smoothly on a large number of independent variables, but not too much on any one of them is essentially constant. Formulated this way it leads to an extremely rich and powerful theory [52]. The third issue formulated as an optimization problem with random quantities, again belongs to a bigger picture, known as, combinatorial optimization theory [54]. 2. Examples of Hamiltonians: The particle configuration space is denoted by Σ. The Hamiltonian is 3

4 defined for σ in this space. We start with classical non-random Hamiltonians. Curie-Weiss Ising Model: We have two real numbers J and h, called coupling constant and strength of the external field, respectively. ( Σ = {, +} ; H (σ) = J ) 2 σ i 2 + h σ i When J > 0, this model is called ferro-magnetic. Curie-Weiss Potts model: We have a real number J, an integer q >, S = {, 2,, q}, and h is a map from S to R. Σ = S ; H (σ) = J 2 I σi=σ j + i,j= h(σ i ) i= Here I σi=σ j is one or zero according as σ i = σ j or not. Curie-Weiss Clock model: The numbers J and q are as above. ow S is the set of q-th roots of unity, h is a function from S to R. Σ = S ; H (σ) = J 2 σ i σ j + i,j= h(σ i ) i= Heisenberg Model: As earlier J is a real number. ow S is the surface of the (r +)-dimensional unit ball; h is a nice map from S to R. Σ = S ; H (σ) = J 2 σ i σ j + i,j= h(σ i ) i= (what if r = 0?) Here σ i σ j is their inner product. Derrida s Random Energy Model: Here Σ = {, +} and H (σ) are independent centered Gaussian random variables with variance. This is also called Gaussian REM. Derrida s Generalized Random Energy Model: We fix an integer n, called the level of the GREM. We have numbers a i for i n, called weights. For each, we have positive integers k(i, ) for i n adding upto. Clearly, Σ = {, +} = i {, +}k(i,). With such a representation, σ Σ is thought of as (σ σ 2 σ n ). It is assumed 4

5 that for each i the limit, lim k(i, )/ exists and is strictly positive. For σ {, +} k(,) we have a random variable ξ σ, for each σ σ 2 {, +} k(,) {, +} k(2,) we have a random variable ξ σσ 2, and so on, finally we have ξ σσ 2 σ n. All these are centered Gaussian with variance. More important is that these are all independent. Σ = {, +} = i {, +} k(i,) ; H (σ σ 2 σ n ) = n a i ξ σσ 2 σ i This is also called Gaussian GREM. SK Model: SK stands for Sherrington and Kirkpatrick. For i < j we have centered Gaussian variable J ij with variance. These are all independent. Σ = {, +} ; H (σ) = i<j i= J ij σ i σ j. This is also called Gaussian SK model. Sometimes it is convenient to express H = ξ ij σ i σ j where the ξ are independent standard normals. i<j Hopfield Model: We have an integer p, we have independent random variables ξ µ i p µ p and i. We put for i j, J ij = ξ µ i ξµ j. µ= Σ = {, +} ; H (σ) = J ij σ i σ j 2 Viana-Bray Model: We have a number α > 0, have a random variable ξ which is Poisson with parameter α, have a sequence (J µ ) of i.i.d. symmetric random variables, have sequences i µ and j µ i.i.d random variables uniformly distributed over {, 2,, }. ALL these random variables are independent. i j for Σ = {, +} ; ξ H (σ) = J µ σ iµ σ jµ. µ= Curie-Weiss Spin glass model: This is just Hopfield model with p =. There are several other models. Bethe lattice Model (for the particle system) has the spins living on a random graph with connectivity constraints. 5

6 For example one can take the set of all graphs on vertices with z edges equipped with uniform probability; or graph on vertices where i and j are connected with probability z/( ). Here z is an appropriate parameter. The idea is that as the distribution of the coordination number is Poisson(z), that is for large, the total number of links is z. Hamiltonian is J ij σ i σ j where the J are independent centered Gaussian with variance z /2 and the sum is over all edges in the graph. Thus only spins connected by an edge interact. In Edwards-Anderson Model the spins live on d-dimensional integer lattice, only nearest neighbours interact and the interaction variables are centered Gaussian with variance d /2. In Large Range Edwards-Anderson Model the spins live on d-dimensional integer lattice, only spins at distance less than R interact and the interaction variables are centered Gaussian with variance R d/2. Remark: See Derrida [23, 24, 25], Ligget etal [42], Parisi [45], Guerra etal [35]. 3. Free Energy: 3. REM Definition: A sequence of probabilities (µ n ) on a polish space X is said to satisfy Large Deviation Principle with rate funcion I if (i) I : X [0, ] is not identically infinity and is such that for each a <, the set (x : I(x) a) is a compact set ; (ii) for every closed set F, lim sup n log µ n(f ) inf I(x) and for every x F open set G, lim inf n log µ n(g) inf I(x). x G There are several reasons why this notion is important. Here is one. Theorem (Varadhan s Lemma): If (µ n ) LDP (I) and h is a bounded continuous function on X, then n log e nh(x) dµ n (x) inf{h(x) + I(x)}. Here is one way to get the rate function. Theorem 2 (Getting rates): Let (µ n ) be a sequence of probabilities on a polish space X, all supported on a fixed compact set. Let A be a countable open base for the topology of X. Assume that for each A A the limit, lim n log µ n(a) exists and equals, say L(A), where 0 L(A). Set, I(x) = sup{l(a) : x A A}. Then (µ n ) LDP (I). Why are we interested in this? ow let us return to REM. For each, we have 2 many i.i.d random variables H (σ) Gaussian with mean zero and 6

7 variance and Z = e βh (σ). Just to recap, here the summation is σ over all σ 2, space of sequences of + and - of length, which is the configuration space of the -particle system. We are interested in showing the existence of the limit, lim log Z and calculating it. For fixed ω define the map σ H (σ)(ω) from Σ to R. Let µ (ω) be the induced probability on R when Σ has uniform probability. Put Ψ = {x R : x 2 2 log 2}. Define I(x) = x 2 /2 for x Ψ and for x not in Ψ. Theorem 3 (REM large deviation rate): For almost every sample point ω, the sequence of probabilities (µ (ω)) satisfies LDP(I). This combined with Varadhan s lemma leads immediately to the following. Theorem 4 (REM energy): Consider REM where for the -particle system the Hamiltonians in different configurations are independent centered gaussian with variance. Let Z (β) be the partition function. Then lim log Z (β) exists almost surely and equals log β2 for 0 β 2 log 2 and equals β 2 log 2 for β > 2 log 2. In situation as above, one says that there is phase transition at the critical value β c = 2 log 2. ote that for small β the limit is quadratic in β where as for large β it is linear. If p(β) denotes the above limit, then a uniform integrability argument leads to the following. Theorem 5 (REM Energy, L p -convergence): For any β, log Z (β) converges almost surely as well as in L P ( p < ) to the nonrandom value p(β) above. Proof of Theorem : It is convenient to use the notation I(A) for infimum of I over the set A. First we claim that lim inf n log e nh dµ sup{ h(x) I(x)}. So fix x 0 and ɛ > 0. Take an open ball G around x 0 in which h(x) < h(x 0 ) + ɛ. Since X G, the required liminf is at least h(x 0) ɛ I(G) h(x 0 ) I(x 0 ) ɛ. True for every ɛ > 0 and every x 0, we are done. ow we claim that lim sup n log e nh dµ sup{ h(x) I(x)} = Λ, say. o loss to assume that Λ <. Fix M > 0 so that h M. Fix a > 0 so that M a < Λ. Fix ɛ > 0. Define the compact set K = {x : I(x) a}. Recall that h is continuous and for each b, the set (I > b) is an open set. For each x K, fix an open ball G x centered at x, through out which h is at least h(x) ɛ and I is at least I(x) ɛ. Let B x be the open ball centered at x and radius half that 7

8 of G x. These balls cover K. Take a finite subcover, say (B i ), with B i centered at x i. Set F i to be closure of B i and F to be the complement of B i. For any i, lim sup n log F i e nh dµ n h(x i ) + ɛ I(F i ) h(x i ) I(x i ) + 2ɛ Λ + 2ɛ This is true for each F i. Moreover F being a subset of K c, I(F ) a and hence lim sup n log e nh dµ n M I(F ) M a Λ F Observe that the integrand being non-negative, X F i +. ow to complete the proof, use the fact that lim sup F n log a in max lim sup n i i n log a in. n This last fact can be proved as follows. Let M n = max n log i a in n [log k+log M ]. Since lim sup n a in. Then i k n log M n n log M n = max lim sup i n log a in, n we are done. Proof of Theorem 2: Clearly I maps X to [0, ]. If I(x 0 ) > a, then there is A A such that x 0 A and L(A) > a. In particular, at all points of A, I value is larger than a. So the set (I a) is a closed set. If K is the compact set on which all the probabilities are supported, then clearly, (I < ) K. This shows that for any a <, the set (I a) is a compact set. Let G be any open set. eed to show that lim inf n log µ n(g) I(G). Pick x G and A A with x A G. lim inf n log µ n(g) lim inf n log µ n(a) = L(A) I(x). This being true for every x G we have lim inf n log µ n(g) sup{ I(x)} = I(G). x G Let F be any closed set. Shall show lim sup n log µ n(f ) I(F ). If F does not intersect K, the compact set on which all the probabilities are supported, then both sides are. So need to consider the case when F has points of K. In this case clearly, the left side of the inequality remains unaltered when F is replaced by F K. Right side also remains unaltered because I being infinity outside K, the infimum of I over F is same as that over F K. Thus safely assume that F is a compact set. Temporarily assume that for all x F, I(x) <. Fix ɛ > 0. For each x F, pick A(x) A such that x A(x) and I(x) L(A(x)) + ɛ. These sets cover F, get a finite subcover (A(x i )). ow lim sup n log µ n(f ) lim sup n log µ n (A(x i )) max i { I(x)} + ɛ giving the desired result. sup x F { L(A(x i ))} max{ I(x i )} + ɛ i 8

9 To remove the condition that I is finite on F, proceed as follows. Fix M > 0 and set I M as minimum of I and M so that for any x, I(x) I M (x). Proceed as above with I replacing I M to conclude that lim sup n log µ n(f ) I M (F ). This is true for every M > 0 and as M increases, I M (F ) increases to I(F ) completing the proof. Proof of Theorem 3: Since, all the H (σ) have the same distribution, we let H stand for a random variable with this distribution. Fix an interval J = (a, b) and put q = P ( H J). Let m = inf x and M = sup x. Clearly J [ M, m] J J [m, M]. If M = make these intervals open at M. q = b a 2π e x2 /2 dx 2 2π m e x2 /2 dx m e m2 /2. If m = 0, take this bound as one. If 0 < δ < M m, then q (m+δ) m 2π e x2 /2 dx 2π δe (m+δ) 2 /2. Claim : If [a, b] Ψ =, then for almost every sample point ω we have eventually µ (a, b) = 0. Indeed if m and M are the min and max (in modulus) of the interval [a, b] then hypothesis and the fact that Ψ is a symmetric interval imply that m 2 > 2 log 2. In particular m 0. The probability that one of the points H / is in (a, b) is at most 2 m e m2 /2 = m e 2 (m2 2 log 2) which is summable over and Borel-Cantelli completes the proof. Claim 2: If (a, b) contains a point x with x 2 < 2 log 2, then for almost every sample point ω we have eventually the following: for any ɛ > 0; ( ɛ)q µ (a, b) ( + ɛ)q. Pick x as in the hypothesis and m be as earlier. Since x and x are in Ψ and m x, we conclude that m 2 < 2 log 2. Fix δ such that (m + δ) 2 < 2 log 2 and 0 < δ < M m. Recall that µ (a, b) = I (a,b) ( H (σ) ). Denoting by E expectation w.r.t. ω, we have 2 σ V ar{µ (a, b)} = E 2 2 E{µ (a, b)} = q σ,τ I (a,b) ( H (σ) )I (a,b)( H (τ) ) q2 q 2 9

10 where the last inequality is arrived at after cancelling the σ τ terms of the first sum with the second term. As a consequence, chebyshev gives P [ µ (a, b) q > ɛq ] ɛ 2 2 q ɛ 2 2 δ (m+δ) 2 2 e [log 2 ]. Since the last expression is summable over, Borel-Cantelli completes the proof of claim 2. To complete the proof of the theorem, let A denote the collection of all open intervals (a, b) such that a and b are either rational or ±, but different from the two points 2 log 2 and + 2 log 2. This family forms a base for the topology of the real line. If A A and closure of A is disjoint with Ψ then claim shows that lim log µ (A) exists and equals. If x > 2 log 2, then we clearly have sets A as above containing the point x and so for such points x, we have sup L(A) =. ow consider a point x with x 2 2 log 2. Take any x A A set A A with x A. Then hypothesis of claim 2 holds for the interval A and hence, for any fixed ɛ > 0, its conclusion holds. The bounds obtained for q at the beginning of the proof show that lim log q exists and equals m 2 /2. As a consequence, lim log µ (A) exists and equals m 2 /2. It is now easy to see that, sup L(A) = x 2 /2. This is so for all points x with x 2 2 log 2. x A A All this happens for almost every sample point ω (Remember µ is random) because A is a countable family. Theorem 2 applies completing the proof. Proof of Theorem 4: Use Varadhan s lemma for the sequence (µ n (ω)) and the function βx along with the above theorem to conclude that for almost every sample point the required limit exists and equals log 2 + sup{βx 2 x2 : x 2 2 log 2}. We add log 2 because Z is the sum of all the Gibbs factors, not their average. The required sup is same as sup of βx 2 x2 over 0 x 2 log 2. If β 2 log 2 then this sup is attained at x = β and the value of the sup is 2 β2. However if β 2 log 2, then this sup is attained at 2 log 2 with value β 2 log 2 log 2. Proof of Theorem 5: We show that the family ( log Z ) is L p bounded. First observe that if ξ σ for σ 2 are i.i.d. standard normal, then the Hamiltonian H (σ) = ξ(σ). If M denotes the max of the 2 standard normals ξ σ, then clearly βm / log Z log 2 + βm /. As a consequence we need to show that the sequence (M / ) is L p bounded. Fix p <. [( ) p ] ( ) M M E I M >0 = P > x /p dx 0 0

11 P (M > x /p ) 2 x /p 2π e t2 /2 dt C x /p e [ 2 x2/p log 2] Let a = (4 log 2) p/2 so that if x > a, then 2 x2/p log 2 > 4 x2/p. Thus [( ) p ] M E I M >0 a + C which is a finite number not depending on. Also 0 which is at most P ( M < x /p) dx a 0 x /p e 4 x2/p dx + 2π x /p e 2 x2/p dx P (ξ > x /p )dx [( ) p ] showing that E M IM <0 is also bounded in. This completes the proof of uniform integrability. Remarks:. In the definition of rate function, usually one only demands that for each a, the set {x : I(x) a} is a closed set, in other words, that I is lower semicontinuous. Rate functions for which these sets are compact are called good rate functions. 2. If in the definition of LDP, the inequality required for all closed sets is demanded only for compact sets (the inequality for open sets is, of course, demanded for all open sets) then one says weak LDP holds. 3. In the definition of LDP, there are several indexings used, like n, ɛ, t, and normalizations n, a n or a ɛ etc. We have used n and /n. It all depends on how our probabilities are indexed and the normalization needed. 4. There is no need to use the same function h for all n in Varadhan s lemma, we can have h n but with some extra conditions. 5. There is no need to assume that all the µ n are supported on a fixed compact set in Theorem 2. One can assume exponential tightness etc. 6. Discuss Theorem 2, when µ n has mass /2 at n and /2 at zero. 7. We discussed REM only with Gaussian distributions, one could use other distributions too. 8. For LDP see Varadhan [60, 6] and Dembo etal [22]. Its use in statistical mechanics was systematically explored by Ellis [29], Eisele [28]. For the energy of REM and its fluctuations see Olivieri etal [43], Dorlas etal [27], Jana [36, 37], Galves etal [3], Talagrand [59], Derrida [23] and Ben Arous etal [2].

12 3.2 GREM: In REM, the Hamiltonians in distinct configurations are independent. The idea in generalized random energy model (GREM) is to bring an amount of dependence in the structure of the Hamiltonians. Of course, very little can be achieved by assuming an arbitrary covariance matrix. An n-level tree structure was suggested by Derrida, where the branches of the tree are in correspondence with the configuration space. We first recall GREM. Fix an integer n. This will be called the level of the GREM. Let n be the number of particles, each of which can have two states/spins, +; so that the configuration space is {, +}, denoted as 2. Consider integers k(i, ) for i n such that each k(i, ) and k(i, ) =. The i configuration space 2, naturally splits into product, 2 k(i,) and σ 2 can be written as σ σ 2 σ n with σ i 2 k(i,). An obvious tree structure can be brought in the configuration space. Imagine an n-level rooted tree. There are 2 k(,) nodes at the first level. These will be denoted as σ, for σ 2 k(,). Below each of the first level nodes there are 2 k(2,) nodes at the second level. The second level nodes below σ of the first level will be denoted by σ σ 2 for σ 2 2 k(2,). In general, below a node σ σ 2 σ i of the (i )-th level, there are 2 k(i,) nodes at the i-th level denoted by σ σ 2 σ i σ i for σ i 2 k(i,). Thus a typical branch of the tree reads like σ σ 2 σ n. Obviously the branches are in one-to-one correspondence with 2, the configuration space. At the edge ending with node σ σ i, we place a random variables ξ σ σ i. We assume that all these random variables are i.i.d. centered Gaussian with variance. We associate one weight for each level, say weight a i > 0 for the i-th level. It is assumed tht a 2 i =. These are not random. In a configuration σ = σ σ n i the Hamiltonian is H (σ) = For β > 0 the partition function is n a i ξ σ σ i. i= Z (β) = σ e βh (σ). Since ξ s are random variables both H and Z are random variables. We suppress the parameter ω. As usual log Z (β) is the free energy of the - particle system. As in the case of REM, the random probabilities µ are defined on R n by 2

13 transporting the uniform distribution of 2 = 2 k(,) 2 k(n,) to R n via the map ( ξσ (ω) σ, ξ σ σ 2 (ω),, ξ ) σ σ n (ω). It is easy to see that µ δ 0 a.s. as. Here δ 0 is the point mass at the zero vector of R n. We assume from now on that k(i,) p i > 0 for i n. Let Ψ = { x R n : k x 2 i i= k 2p i log 2, k n}. i= We define the map I : R n R, by I( x) = 2 n x 2 i i= if x Ψ and = otherwise. Theorem 6 (GREM large deviation rate): For almost every sample point, the (random) sequence {µ } satisfies LDP with rate function I. Theorem 7 (GREM Energy): There are numbers 0 < β < < β K < = β K+ and numbers 0 < r < < r K = n such that almost surely, lim log Z (β) = log 2 + β2 n 2 i= a2 i if β < β j l= (β l β) 2 r l r l + a2 i if β j β < β j+. = log 2 + β2 2 n i= a2 i 2 The exact computation of the numbers β j is in the proof of the Theorem. Two simple cases are worth mentioning. The numbers β j mentioned below are same as the above, in these particular cases. Theorem 8 (GREM Energy, Special Cases): a.s. i) Let 0 < p a 2 < p2 a 2 2 < < pn a. Put β 2 j = n 2pj log 2 a j for j =,, n. Then lim log Z (β) = log 2 + β2 2 if β < β, = n p i log 2 + β j a i 2pi log 2 + β2 2 j+ n a 2 i j+ if β j β < β j+ for j < n, = β n a i 2pi log 2 if β β n. 3

14 ii) Let p = p2 = = pn a 2 a 2 a > 0. Then a.s. 2 2 n lim log Z (β) = log 2 + β2 2 if β < 2 log 2 = β 2 log 2 if β 2 log 2. ote that in case i) above all the n levels of the GREM showed up in the formula for the energy where as in case ii) all the levels reduced to just one level, leading to REM situation. Let us denote by p(β) the non-random limit in Theorem 7. Theorem 9 (GREM Energy, L p -convergence): For any β, log Z (β) converges almost surely as well as in L P ( p < ) to the nonrandom value p(β) above. Proof of Theorem 6: The proof is similar to that of Theorem 3. Here are the main steps. Let J = J J n, where each J i is an interval of R. Put m i = inf x Ji x, M i = sup x Ji x, and q i = P ( ξ J i) we have q i 2 2π M i mi e x 2 2 dx < e x 2 mi 2 dx e m 2 i mi with the understanding that when m i = 0, the last expression is 2 and q i 2π M i mi e x 2 2 dx > 2 (m i+δ) mi 2, e x 2 δ 2 dx > 2 e 2 (mi+δ) 2, for any 0 < δ < M i m i. Firstly, one shows that if J Ψ =, then a.s. eventually µ (J) = 0. Moreover, the sequence {µ } is supported on a compact set. Secondly, if ( J Ψ) 0, then for any ɛ > 0 a.s. eventually ( ɛ)q q n µ (J) ( + ɛ)q q n. Finally, one shows that almost surely lim log µ (J) = n 2 i= m2 i if ( J Ψ) 0 φ = if J Ψ = φ. This completes a sketch of a proof of the theorem. Proof of theorem 7: 4

15 The free energy is given by lim log Z (β) = log 2 + β2 2 n a 2 i 2 i= inf x Ψ + n (x i βa i ) 2 where Ψ + consists of points of Ψ with all coordinates non-negative. Here is the general idea. Let c, c 2,, c n 0 with c > 0. Let α = (α,, α n ) R n with each α i > 0. Let S R n be the set of all points x = (x,, x n ) R n with nonnegative coordinates and i x2 j i c j for n i =, 2,, n. Here then is the formula for l = inf x S (x i α i ) 2. i) If c + +c i α 2 + +α2 i ii) Let γ = min i Assume that c k++ +c i α 2 k+ + +α2 i i= for all i then clearly α S and l = 0. c + +c i. Let k be the largest index such that γ = c+ +c α 2 k. + +α2 α i 2 + +α2 k, for i > k. Put α = (α,, αn) where αi = γα i for i k = α i for i > k. Clearly α S. Moreover the infimum, l = k (α i α i) 2 = ( γ) 2 k To see this, consider any x S. By Cauchy-Schwarz, k α i x i k α i α2 i. 2 and hence k α i (α i x i) 0. Since γ <, k α i (α i x i) k α i(αi x i). A simple algebra shows k (x i α i ) 2 k (αi α i ) 2 k (x i αi ) 2 0. ( ) iii) Let γ and k be as above. Suppose c k++ +c i η = min i>k α 2 k+ + +α2 i <, for some i > k. Put c k+ + +c i, so that η <. Let m be the largest index when this ratio α 2 k+ + +α2 i, for i > m. Put equals η. Clearly m > k. Assume that cm++ +ci α 2 m+ + +α2 i α i = γα i for i k = ηα i for k + i m = α i for i > m. Clearly α S. Further, the infimum, l = m (α i α i) 2 = ( γ) 2 k α2 i + ( η) 2 m k+ α2 i. To see this, consider any point x S. It is enough to show ( ) with k replaced by m. As earlier k α i (α i x i) 0 and m α i (α i x i ) 0. Using γ < η <, we have m α i (α i x i) m η α i (α i x i) m η α i (α i x i)+( γ η ) k α i (α i x i). In other words, m α i(αi x i) m α i (α i x i). A simple algebra completes proof of ( ) with k replaced by m. 5

16 We shall not continue with the generalities, instead we explain this in our situation, namely, S = Ψ +, α i = βa i and c i = p i log 2. Following the above analysis, let us put B j,k = j k n. Set and in general (p j+ +p k )2 log 2 a 2 j + +a2 k β = min k B,k r = max{i : B,i = β } β 2 = min k>r B r+,k r 2 = max{i > r : B r+,i = β 2 } β m+ = min k>r m B rm+,k r m = max{i > r m : B rm+,i = β m+ }. for Clearly, for some K with K n, we have r K = n. Put r 0 = β 0 = 0 and β K+ =. ote that β 0 < β < β 2 < β K < β K+ =. Fix j K and let β (β j, β j+ ]. Define x Ψ + as follows: x i = β l a i if i {r l +,, r l } for some l, l j = βa i if i > r j +. Then inf x Ψ n i= (x i βa i ) 2 occurs at x. This proves the theorem. Proof of Theorem 8: For the first part one only needs to see that the hypothesis implies that the constants β as defined above coincide with the values given in the statement of the theorem. The formula itself, for the energy, is just a special case of the one in the earlier theorem. For the second part one only needs to realize that in this case there is only one β i. Proof of Theorem 9: The argument given for REM applies to show uniform integrability in this set-up as well. Remarks:. It is possible to take, as in the case of REM, distributions other than Gaussian. In fact it is possible to take different distributions at different levels of the tree. However, explicit formulae for the energy appear to be difficult. 2. One can give a general tree formulation of the GREM and solve the resulting model. This allows one to consider random trees as well. For example, toss an n-faced die times to decide what should be the numbers k(i, ). However, no interesting trees that exhibit any behaviour other than expected have been found. May be there is some universality w.r.t. the trees. 6

17 3. An enirely new formulation in terms of rate functions of the driving distributions can be given to these models. 4. There are other models akin to GREM that are considered in the literature. 5. For the energy of GREM see Derrida [24], Derrida etal [25], Dorlas etal [26], Capocaccia etal [8], Contucci etal [2], Galves etal [3], Jana [37], and Jana etal [38, 39], Bolthausen etal [9]. 3.3 SK Model: For the SK model, to recap, we have H (σ) = i<j J ij σ i σ j ; Z = σ e βh (σ) To show the existence of the limit, lim log Z the method used is called the smart path method. Roughly speaking, if you have two quantities a and b computed in two systems and want to show that a b, one way is to find a path ϕ(t) for 0 t, (between the two systems that produced the values a and b respectively) such that ϕ is increasing; ϕ(0) = a and ϕ() = b. Such a path is definitely a smart path! We need some preliminaries about Gaussian expectations. Theorem 0 (integration by parts): (i) Let X be centered Gaussian and F : R R be a C function such that both F and its derivative F have exponential growth, that is, for some constants c and d, F (x) ce d x and similarly for F. Then E(X F (X)) = E(X 2 ) E(F (X)). (ii) Let X, Y, Y 2,, Y n be jointly Gaussian and centered. F : R n R be a C function with exponential growth, that is, F (y) ce dσ yi and similarly for its first partial derivatives. Then denoting by F i the derivative of F w.r.t. the i-th coordinate, and by Y the vector (Y,, Y n ), E(X F (Y )) = i E(X Y i )E(F i (Y )). Theorem (Slepian s Lemma): Let F : R M R be a C 2 function of exponential growth along with its first two derivatives. Assume that for i j, 2 x i x j F 0. Let U = (U,, U M ) 7

18 and V = (V,, V M ) be centered Gaussian with E(Ui 2) = E(V i 2 ) for all i, and E(U i U j ) E(V i V j ) for i j. Then E(F (U)) E(F (V )). Theorem 2 (Gaussian concentration inequality): Let F : R M R be a C 2 function such that for some number A > 0, we have for all x and y; F (x) F (y) A d(x, y). Let X = (X,, X M ) where (X i ) are independent standard normal. Then for any t > 0, P { F (X) E[F (X)] t} 2e t2 /4A 2. Here d is the usual Euclidean distance. Results which assert that a random variable, with high probability, takes values near a number (mean, median) are called concentration inequalities. Theorem 3 (Concentration inequality for SK model): Consider the SK model with inverse temperature β. Let us denote p (β) = E log Z (β). Then for any t > 0, P { log Z p t} 2e t2 /2β 2. Inequalities like the above are very useful, they allow us to deduce the almost sure convergence of the sequence log Z from the convergence of their means. Theorem 4 (SK model, Energy): The sequence (E[log Z ]) is a superadditive sequence of numbers. E[log Z ] converges to a finite limit. The sequence log Z converges almost surely. either the convergence nor the value of the limit seem to depend on the Gaussian nature of the environment (a term used to describe the randomness that enters the Hamiltonian). There is a universal behaviour. Anything reasonable seems to do. Theorem 5 (Universality of Energy): Consider any probability µ on R with mean zero, variance one and finite third moment. Consider the SK Hamiltonian and partition function, H (σ) = i<j J ij σ i σ j ; Z = σ e β H (σ) where J ij are i.i.d. having distribution µ. Then log Z converges almost surely, E(log Z ) converges. Moreover this limit is same as that for the Gaussian environment. Proof of Theorem 0: 8

19 Condition on F says required integrals below exist. First part follows by observing that if EX 2 = σ 2, then integration by parts gives E(X F (X)) = xe x2 /2σ 2 F (x)dx = σ 2 e x2 /2σ 2 F (x)dx. 2πσ 2πσ For the second part put Z i = Y i c i X where c i = E(Y i X)/E(X 2 ). See that Z = (Z,, Z n ) is independent of X. Apply earlier part to X and the function ϕ(x) = F ( z i + c i x ) for fixed numbers (z i ). We get (remember chain rule for derivative) E(Xϕ(X)) = E(X 2 )E(ϕ (X)) = E(Y i X)E X (F i ( z i + c i X )). Since Z is independent of X, now taking expectation w.r.t. Z we get E(XF (Y )) = E(Y i X) E(F i (Y )). Proof of Theorem : There is no loss to assume that the families U and V are independent. Put W (t) = tu+ tv and ϕ(t) = E[F (W (t))]. Thus W (t) = (W (t),, W M (t)) where W i (t) = tu i + tv i. Under the assumptions of the theorem, we can interchange expectation with differentiation, yielding ϕ (t) = E[ W i (t)f i(w (t))]. Here F i is the derivative of F w.r.t. its i-th coordinate and W i is the derivative w.r.t. t of the i-th coordinate of W. Thus W i (t) = 2 U t i 2 V t i. ote that by independence of the Us and V s, we have E[W i (t)w j(t)] = 2 [E(U iu j ) E(V i V j )]. By previous theorem E[W i (t) F i (W (t))] = j E[W i (t) W j (t)] E[F ij (W (t))]. As a consequence ϕ (t) = 2 [E(Ui U j ) E(V i V j )]E[F ij (W (t))] By hypothesis, this quantity is positive. Thus E(F (V )) = ϕ(0) ϕ() = E(F (U)). Proof of Theorem 2: Consider R 2M and think of a point y = (y,, y 2M ) as (y, y 2 ) with y = (y,, y M ) and y 2 = (y M+,, y 2M ). Define G on R 2M by G(y) = e s[f (y ) F (y 2 )] for a fixed number s R. Let (U i ) i 2M be independent standard Gaussian. Fix independent standard gaussian (V i ) i M independent of the U family and also put V i = V i M for M + i 2M. ote that E(U i U j ) E(V i V j ) is zero unless i j = M, in which case it is. Set W (t) = tu + tv where U and V are the 2M dimensional vectors (U i ) 9

20 and (V i ) respectively. Set ϕ(t) = E[G(W (t))] so that as in the above theorem, we have ϕ (t) = [E(U i U j ) E(V i V j )] E[F ij (W (t))] = E G i,i+m (W (t)) 2 i,j where, as usual, the suffixes for F and G denote the corresponding partial derivatives. Observe that G i,i+m (y) = s 2 F i (y )F i (y 2 )G(y) so that ϕ (t) = s 2 E[ F i (W (t))f i (W 2 (t))g(w (t)] i M But for all x R M our hypothesis tells [F i (x)] 2 A 2 So applying Cauchy- Schwarz for the F terms above, we get i M ϕ (t) s 2 A 2 ϕ(t). This combined with ϕ(0) = yields ϕ() e s2 A 2. In other words, we have proved that for every s R, Ee s[f (U ) F (U 2 )] e s2 A 2. Writing this expectation E as E U E U2 and performing E U2 with the help of Jensen, we get so that Ee s[f (U) E(F (U))] e s2 A 2 P (F (U) E[F (U)] t) e s2 A 2 /e st. Taking s = t/(2a 2 ) we see that this last quantity is e t2 /(4A 2). Apply this to F to complete the proof. Proof of Theorem 3: Take M = ( )/2. For σ Σ, let a(σ) be the vector of R M given by ( β σ i σ j : i < j ). ote that if J is a M dimensional standard normal vector, then the SK Hamiltonian is nothing but H (σ) = J a(σ). If we consider the function F (x) = log e a(σ) x then F (J) = log Z. If we σ show that F (x) F (y) β 2 d(x, y), then the previous theorem completes the proof. But this is clear because, a(σ) β /2 and hence a(σ) x a(σ) y β /2d(x, y). Thus a(σ) x a(σ).y + β /2 d(x, y) Take exponentials, add over σ and take log to get F (x) F (y) + d(x, y) β/ 2. Interchange x and y to complete the proof. Proof of Theorem 4: Almost sure convergence of log Z follows from the convergence of the numbers E(log Z ) by an application of Theorem 3. But the convergence i M 20

21 of these numbers, in turn, follows from the super-additivity of the sequence of numbers E(log Z ). This will be observed first. Let (a n ) be a super-additive sequence of numbers, that is, a m+n a m + a n for every m and n. Set s = sup(a n /n). Clearly, lim sup(a n /n) s. We shall n show that lim inf(a n /n) s, then it follows that (a n /n) converges to s. First consider the case s <. Fix ɛ > 0. Fix k such that (a k /k) > s ɛ/2. Fix > k such that for r k, (a r /) > ɛ/2. ow take any n >. Let n = kd + r with r < k. Then a n = a kd+r da k + a r dk(s ɛ 2 ) + a r by super additivity and choice of k. Thus a n n dk ( s ɛ dk + r 2 ) + a r n (s ɛ 2 ) ( r ) ɛ n 2, showing that lim inf(a n /n) s ɛ. If s =, we fix M > 0 and need to show that lim inf(a n /n) > M. Fix k as above but with (a k /k) > 2M and proceed. Of course, in our case s <. Indeed E(log Z ) log β2. This is because by concavity of log function, Jensen yields E(log Z ) log E(Z ) = log[2 E(e βx/ )] where X is centered Gaussian with variance ( )/2. We now proceed to show that the sequence of numbers {E[log Z ]} is indeed super-additive. Fix integers = + 2. Take independent standard standard normals (J ij ) for i < j ; (J ij ) for i < j ; and (J ij ) for < i < j. Set for σ {, +}, H 2 (σ) = and for 0 t H (σ) = i<j i<j J ijσ i σ j + J ij σ i σ j 2 + i<j H(t)(σ) = t H (σ) + t H 2 (σ). J ijσ i σ j We use the notation of Slepian s lemma. We take M = 2 ; The family (U σ ) is the family (H (σ)); (V σ ) is the family (H 2 (σ)); and the funcion F (x) = log e βxσ. Then (W σ ) is precisely the family (H(t)(σ)). As a result we have, σ denoting ϕ(t) = E[F (W (t))], ϕ (t) = [E(U σ U η ) E(V σ V η )] E[F ση (W (t))]. 2 ση 2

22 Here F ση is the derivative of F w.r.t. these variables. In the present case, E(U σ U η ) = σ i σ j η i η j = 2 [R2 ση ] i<j where R ση = ( σ i η i )/, called the overlap between σ and η. ote that this i quantity always lies between and +; it equals one if and only if the two configurations σ and η are the same. Similarly, keeping the independence of the families (J ij ) and (J ij ), we get E(V σ V η ) = 2 [ (R ση) 2 ] + 2 [ 2(R ση) 2 ] where Rση = ( σ i η i )/ and R ση = ( σ i η i )/ 2. i <i Since F (x) = log e βxτ we have, for its derivatives τ F σ (x) = βe βxσ /(Σ τ e βxτ ) = βg(σ, x) where G(σ, x) is the Gibbs measure of σ corresponding to the vector x, namely, e βxσ /Z where Z = e βxτ. τ F ση (x) = β2 e βxσ e βxη ( e βxτ ) 2 = β 2 G(σ, x)g(η, x) for σ η, τ F σσ (x) = β2 e 2βxσ ( e βxτ ) 2 τ + β2 e βxσ = β 2 G(σ, x)g(σ, x) + β 2 G(σ, x) e βxτ τ substituting these value in the expression for ϕ above, we get 4 β ϕ equals 2 [Rση 2 Rση 2 2 Rση 2 + ]E[ G(σ, W t )G(η, W t )] + E[G(σ, W t )] ση σ { = [ E G(σ, W t )G(η, W t ) Rση 2 R 2 ση ] } 2 R 2 ση ση ote that R ση = R ση + 2 R ση The funcion x x 2 being convex function, we have Rση 2 R 2 ση 2 R 2 ση 0 22

23 The Gs being positive we conclude from the above expression concerning ϕ that ϕ 0. Thus ϕ(0) ϕ(). In other words E[log Z 2 ] E[log Z ] where the Z i are the partition functions corresponding to the Hamiltonians H i. It is not difficult to see that E[log Z 2 ] = E[log Z ] + E[log Z 2 ] and E[log Z ] = E[log Z ]. i<j This completes the proof the super-additivity of the sequence of numbers E[log Z n ]. Proof of Theorem 5: Let H (σ) = J ij σ i σ j where (J ij ) are standard normal. Let H 2 (σ) = i<j J ij σ iσ j where (J ij ) are i.i.d. mean zero, variance one, finite third moment variables. Set Z = e βh (σ)/ and Z 2 = e βh 2 (σ)/. We σ σ need to show that E[ (log Z log Z 2 )] 0. Some notational issues are to be addressed first. We set M = ( )/2. A vector x R M is indexed by (ij) with i < j. The indices (ij) are also arranged in some order l M when needed. For σ {, +}, we let σ denote also the vector of R M whose (ij)-th coordinate is σ i σ j. There will be no confusion with this dual notation for σ, it will be obvious from the context as to what is meant. Let J be the M-dimensional random vector whose (ij)-th coordinate is J ij. Similarly J. With this notation, H = J σ and H 2 = J σ where the dot denotes the inner product. For 0 l M let X l be the M-dimensional random vector which has J ij variables upto and including the coordinate l and J ij variables after the l-th coordinate. Thus XM is just J and X 0 is J. Let F : R M R be defined by F (x) = log e βx σ/ σ where the dot in the exponent is the inner product. With all this notation, what we need to show amounts to E[F (X M ) F (X 0 )] = M E[F (X l ) F (X l )] 0 Set Y l to be the M-dimensional random vector which is X l in all its coordinates except the l-th coordinate which is set as zero. Thus replacing the l-th coordinate of Y l by the l-th coordinate of J we get X l, while replacing the l-th coorinate of Y l by the l-th coordinate of J we get X l. Since our function F is smooth, we have F (X l ) F (X l ) = [F (X l ) F (Y l )] [F (X l ) F (Y l )] 23

24 which equals, by Taylor expansion F l (Y l )[J l J l] + 2 F ll(y l )[J 2 l J 2 l ] + 6 [F lll(?)j 3 l F lll (??)J 3 l ] Here the suffixes for F denote the partial derivative w.r.t. that coordinate. The question marks denote that F lll is evaluated at an appropriate point in R M, the actual point is not relevant (enough to note that we have random variables). Since Y l, J l, J l are independent and also that J l and J l have zero means and unit second moments we get, on taking expectations of the above equation, E[F (X l ) F (X l )] = 6 E[F lll(?)j 3 l F lll (??)J 3 l ] ow let us calculate the required derivatives. Let the coordinate l be (ij). Keep in mind that the partition function corresponding to x is Z(x) = e βx σ/, σ the Gibbs measure corresponding to x on the σ-space is G(σ, x) = e βx σ/ Z(x) and for any function f on the σ-space its expectation w.r.t. this probability is denoted by f x. F l (x) = β 3/2 Z(x) ( F ll (x) = β2 σ 2 F (x) = log σ σ e βx σ/ e βx.σ/ σ i σ j = β 3/2 σ iσ j x e βx.σ/ σ i σ j ) 2 e βx.σ/ σi 2σ2 j σ Z 2 (x) Z(x) F lll (x) = β2 5/2 2 σ = β2 2 [( σ iσ j x ) 2 ] e βx σ/ σ i σ j. σ Z 2 (x) e βx σ/ 2( σ e βx σ/ σ i σ j ) 3 Z 3 (x) = 2β2 5/2 [ σi σ j 3 x σ i σ j x ] C 5/2 As a consequence E[F (X l ) F (X l )] C 5/2 and finally E[F (X M ) F (X 0 )] MC 5/2 Since M = ( )/2 this last expression converges to zero as n, completing the proof. 24

25 Remarks:. The overlap that appeared in the proof of theorem 4 plays a prominent role in understanding SK model. 2. See Talagrand [59] for SK model. Guerra and Toninelli [34] proved the existence of the limiting energy for the SK model. The identification of the limit with a conjecture of Parisi was achieved by Guerra [33] and Talagrand [58]. 3. The universality was noted in Carmona etal [9] and Chatterjee [20], see also Talagrand [56]. The proof is an imitation of the Lindeberg proof of the Central Limit Theorem as given, for example, in Billingsley [7]. 4. For β <, the limiting energy is log β2. See Talagrand [59]. 5. We have not included the external field term, but that can also be included in proving the existence of the limit. 4. Gibbs Distribution: Let us now turn our attention to another problem, namely, the fate of the Gibbs distributions. For each, we have a random probability on {, +}, namely the Gibbs distribution, which is given by G(σ) e βh (σ). More precisely, it is the probability which puts mass e βh (σ) /Z at the point σ. This is a random probability, because H are random. The question is to find out if there is any limiting object for these. Of course, one has to formulate carefully, because the space on which these probabilities live changes with. Theorem 6 (REM High temperature Gibbs measures): Consider the Gaussian REM with β < 2 log 2. Then almost surely the Gibbs measures converge to the uniform distribution on the space{, +} in the following sense. Fix any k. For every k and ω, the Gibbs distribution on {, +} be denoted by G (ω,.). Let its marginal on the first k coordinate space be denoted by G k (ω,.). Then for almost every ω, as n, these converge to the uniform distribution on {, +} k. To discuss the fate of Gibbs measures at low temperatures, we need some notation. For 0 < m <, let P m be the Poisson Point Process on (0, ) with intensity x m dx. This means the following. Say that a subset D (0, ) is locally finite if for any two numbers a and b with 0 < a < b <, the set D [a, b] is a finite set. Clearly, a locally finite set is countable. Let Ω be the collection of all locally finite sets. For a Borel set B (0, ) let B be the number map defined on Ω by D B D, where A is the number of points in the set A. This map takes non-negative integer values, including possibly infinity. Let F be the smallest σ-field on Ω making all these maps B as B 25

26 varies over Borel sets measurable. There is a unique probability P m on F such that for each Borel B (0, ) the P m distribution of B is Poisson with parameter ν(b) = x m dx and moreover for disjoint Borel sets B i (0, ) B the random variables Bi ; i k are independent. Poisson distribution with parameter means mass one at infinity and parameter 0 means mass one at zero. Since for each a > 0 we have x m dx <, we conclude that almost a every D (under P m ) has only finitely many points larger than a. Thus we can arrange D as a decreasing sequence. Since x m dx = we conclude that D has infinitely many points and hence, when arranged as decreasing sequence, it converges to zero. Since x.x m dx < the sum of points of D which are 0 smaller than one is finite. But since there are only finitely many points of D larger than one, we deduce that the sequence D is summable. Let S [0, ] be the set of all seqeunces (x, x 2, ) which are decreasing with sum at most one. Clearly S is a compact set (usual topology, as a subset of the product space). We define a map from Ω to S as D ( d d, d2 d, ) where (d i) is decreasing enumeration of D and d is sum of points of D. Let Λ m be the probability induced on S by P m via this map. This is called the Poisson-Dirichlet distribution with parameter m. Let us now consider REM with β > 2 log 2. Put m = β/ 2 log 2, so that 0 < m <. Arrange the 2 numbers H (σ)/z in decreasing order and continue with zeros to get a random point of S. Let its distribution be denoted by µ. Thus µ is the distribution of the Gibbs measures. Theorem 7 (REM Low temperature Gibbs measures): Consider the Gaussian REM with β > 2 log 2. Denote m = β/ 2 log 2 and µ be the distribution of Gibbs measures. Then µ Λ m weakly on S. Proof of Theorem 6: If β 2 < log 2 then the Gibbs distributions converge almost surely to the uniform probability on {, +} hereafter denoted 2 ω. Let us put Z n = e β ξ σ where ξ σ are independent standard normal. Then σ var(z ) = E σ,η α n = EZ = 2 e β2 /2 0 e β ξ σ e β ξ η σ,η E[e β ξ σ ]E[e β ξ η ]. 26

27 Cancelling the σ η terms and ignoring the remaining positive terms from the second sum we get var(z ) σ E[e 2β ξ σ ] = 2 e 2β2 As a consequence ( P Z ) > ɛ α ɛ 2 var(z /α ) ɛ 2 e (log 2 β2 ) Since the last terms form a convergent series, Borel-Cantelli shows that Z α almost surely. Fix τ 2 k. Then for > k the projection of the Gibbs measure G on 2 k puts mass G k (τ) at τ which is described as follows. For η 2 k, let us denote the concatenation of τ followed by η as τη 2. Put Z = e β ξ τη. Then G k (τ) = Z /Z. One can repeat the above η argument to show that E[ Z ] = 2 k α and the ratio ( Z )/(2 k α ) converges to one almost surely. Combining this result with the earlier one for Z one gets that, G k (τ) = Z /Z 2 k almost surely to complete the proof. ow let us assume that log 2 β 2 < 2 log 2. The proof is similar to the above, just that on needs to truncate the Hamiltonians. Recall that H (σ) = ξσ where ξs are independent standard normal. We shall fix a δ such that 2 log 2 < δ < 2β. Put Z = σ eβh (σ) I H (σ) δ. Then P (Z Z ) 2 P (H > δ) 2 e δ2/2 e [ δ2 2 log 2] 2π δ δ which is summable, so that almost surely eventually, Z and Z are equal. Further E(Z ) = 2 δ 2π e 2 (x β)2 dx e β2 /2 > 2 2 e β2 /2 where we used δ > β. Regarding variance, note that var(z ) equals E[ σ,η e βh (σ) I (H (σ) δ)e βh (η) I (H (η) δ)] σ,η E[e βh (σ) I (H (σ) δ)] E[e βh (η) I (H (η) δ)]. Cancel the σ η terms and ignore the others from the second expression, to get var(z ) is at most 2 δ 2π e (x 2β)2 /(2) dx e 2β2 = 2 e 2β2 (2β δ) 2π e y2 /2 dy 27