Local Energy Statistics in Directed Polymers

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. We show that this conjecture holds true as well for directed polymers in random environment. We also show that, under certain conditions, this conjecture holds for directed polymers even if energy levels that grow moderately with the volume of the system are considered.


Introduction and results
Recently, Bauke and Mertens have proposed in [2] a new and original look at disordered spin systems.This point of view consists of studying the micro-canonical scenario, contrary to the canonical formalism, that has become the favorite tool to treat models of statistical mechanics.More precisely, they analyze the statistics of spin configurations whose energy is very close to a given value.In discrete spin systems, for a given system size, the Hamiltonian will take on a finite number of random values, and generally (at least, if the disorder is continuous) a given value E is attained with probability 0. One may, however, ask : How close to E the best approximant is when the system size grows and, more generally, what the distribution of the energies that come closest to E is ?Finally, how the values of the corresponding configurations are distributed in configuration space ?
The original motivation for this viewpoint came from a reformulation of a problem in combinatorial optimization, the number partitioning problem (this is the problem of partitioning N (random) numbers into two subsets such that their sums in these subsets are as close as possible) in terms of a spin system Hamiltonian [1,16,17].Mertens conjecture stated in these papers has been proven to be correct in [4] (see also [7]), and generalized in [8] for the partitioning into k > 2 subsets.Some time later, Bauke and Mertens generalized this conjecture in the following sense : let (H N (σ)) σ∈ΣN be the Hamiltonian of any disordered spin system with discrete spins (Σ N being the configuration space) and continuously distributed couplings, let E be any given number, then the distribution of the close to optimal approximants of the level √ N E is asymptotically (when the volume of the system N grows to infinity) the same as if the energies H N (σ) are replaced by independent Gaussian random variables with the same mean and variance as H N (σ) (that is the same as for Derrida's Random Energy spin glass Model [12], that is why it is called the REM conjecture).
What this distribution for independent Gaussian random variables is ?Let X be a standard Gaussian random variable, let δ N → 0 as N → ∞, E ∈ R, b > 0. Then it is easy to compute that Let now (X σ ) s∈ΣN be |Σ N | independent standard Gaussian random variables.Since they are independent, the number of them that are in the interval [ converges, as N → ∞, to the Poisson point process in R + whose intensity measure is the Lebesgue measure.In other words, the best approximant to , where W is an exponential random variable of mean 1.More generally, the kth best approximant to , where W 1 , . . ., W k are independent exponential random variables of mean 1, k = 1, 2 . . .It appears rather surprising that such a result holds in great generality.Indeed, it is well known that the correlations of the random variables are strong enough to modify e.g. the maxima of the Hamiltonian.This conjecture has been proven in [9] for a rather large class of disordered spin systems including short range lattice spin systems as well as mean-field spin glasses, like p-spin Sherringthon-Kirkpatrick (SK) models with Hamiltonian ..,ip≤N where J i1,...,ip are independent standard Gaussian random variables, p ≥ 1. See also [5] for the detailed study of the case p = 1.
Two questions naturally pose themselves.(i) Consider instead of E, N -dependent energy levels, say, E N = constN α .How fast can we allow E N to grow with N → ∞ for the same behaviour (i.e.convergence to the standard Poisson point process under a suitable normalization) to hold ?(ii) What type of behaviour can we expect once E N grows faster than this value ?
The second question (ii), that is the local behaviour beyond the critical value of α, where Bauke and Mertens conjecture fails, has been investigated for Derrida's Generalized Random Energy Models ( [13]) in [10].
Finally, the paper [3] introduces a new REM conjecture, where the range of energies involved is not reduced to a small window.The authors prove that for large class of random Hamiltonians the point process of properly normalized energies restricted to a sparse enough random subset of spin configuration space converges to the same point process as for the Random Energy Model, i.e.Poisson point process with intensity measure π −1/2 e −t √ 2 ln 2 dt.In this paper we study Bauke and Merten's conjecture on the local behaviour of energies not for disordered spin systems but for directed polymers in random environment.These models have received enough of attention of mathematical community over past fifteen years, see e.g.[11] for a survey of the main results and references therein.Let ({w n } n≥0 , P ) is a simple random walk on the d-dimensional lattice Z d .More precisely, we let Ω be the path space Ω = {ω = (ω n ) n≥0 ; ω n ∈ Z d , n ≥ 0}, F be the cylindrical σ-field on Ω and for all n ≥ 0, ω n : ω → ω n be the projection map.We consider the unique probability measure P on (Ω, F ) such that ω 1 − ω 0 , . . ., ω n − ω n−1 are independent and where δ j = (δ kj ) d k=1 is the jth vector of the canonical basis of Z d .We will denote by ) the space of paths of length N .We define the energy of the path where {η(n, x) : n ∈ N, x ∈ Z d } is a sequence of independent identically distributed random variables on a probability space (H, G, P).We assume that they have mean zero and variance 1.
Our first theorem extends Bauke and Merens conjecture for directed polymers.
Theorem 1 Let η(n, x), {η(n, x) : n ∈ N, x ∈ Z d }, be the i.i.d.random variables of the third moment finite and with the Fourier transform φ(t) such that |φ(t Then the point process converges weakly as N ↑ ∞ to the Poisson point process P on R + whose intensity measure is the Lebesgue measure.Moreover, for any ǫ > 0 and any b ∈ R + P(∀N 0 ∃N ≥ N 0 , ∃ω N,1 , ω N,2 : cov (η(ω N,1 ), η(ω N,2 )) > ǫ : The decay assumption on the Fourier transform is not optimal, we believe that it can be weaken but we did not try to optimize it.Nevertheless, some condition of this type is needed, the result can not be extended for discrete distributions where the number of possible values the Hamiltonian takes on would be finite.
The next two theorems prove Bauke and Mertens conjecture for directed polymers in Gaussian environment for growing levels E N = cN α .We are able to prove that this conjecture holds true for α < 1/4 for polymers in dimension d = 1 et and α < 1/2 in dimension d ≥ 2. We leave this investigation open for non-Gaussian environments.
The values α = 1/4 for d = 1 and α = 1/2 for d ≥ 2 are likely to be the true critical values.Note that these are the same as for Gaussian SK-spin glass models for p = 1 and p = 2 respectively according to [6], and likely for p ≥ 3 as well.
Then the point process converges weakly as N ↑ ∞ to the Poisson point process P on R + whose intensity measure is the Lebesgue measure.Moreover, for any ǫ > 0 and any b ∈ R + P(∀N 0 ∃N ≥ N 0 , ∃ω N,1 , ω N,2 : cov (η(ω N,1 ), η(ω N,2 )) > ǫ : Then the point process converges weakly as N ↑ ∞ to the Poisson point process P on R + whose intensity measure is the Lebesgue measure.Moreover, for any ǫ > 0 and any Acknowledgements.The author thanks Francis Comets for introducing him to the area of directed polymers.He also thanks Stephan Mertens and Anton Bovier for attracting his attention to the local behavior of disordered spin systems and interesting discussions.
2 Proofs of the theorems.
Our approach is based on the following sufficient condition of convergence to the Poisson point process.It has been proven in a somewhat more general form in [8].
Theorem 4 Let V i,M ≥ 0, i ∈ N, be a family of non-negative random variables satisfying the following assumptions : for any l ∈ N and all sets of constants b j > 0, j = 1, . . ., l where the sum is taken over all possible sequences of different indices (i 1 , . . ., i l ).Then the point process δ {Vi,M } on R + converges weakly in distribution as M → ∞ to the Poisson point process P on R + whose intensity measure is the Lebesgue measure.
Hence, in all our proofs, we just have to verify the hypothesis of Theorem 4 for V i,M given by where the sum is taken over all sets of different paths (ω N,1 , . . ., ω N,l ).
Informal proof of Theorem 1.Before proceeding with rigorous proofs let us give some informal arguments supporting Theorem 1.
The number of such sets is exponentially smaller than (2d) N l .Here two possibilities should be considered differently.
The first one is when the covariance matrix is non-degenerate.Then, invoking again the Central Limit Theorem, the probabilities P(•) in this case are not greater than From the definition of the covariances of η(ω N,i ), det B N (ω N,1 , . . ., ω N,l ) is a finite polynomial in the variables 1/N .Therefore the probabilities P(•) are bounded by (2d) −N l up to a polynomial term, while the number of sets (ω N,1 , . . ., ω N,l ) such that b i,j (N ) = o(1) some i = j, i, j = 1, . . ., l, is exponentially smaller than (2d) N l .Therefore the sum (11) over such sets (ω N,1 , . . ., ω N,l ) converges to zero exponentially fast.
We now give rigorous proofs.We start with proofs of Theorems 2 and 3 in Gaussian environment and give the proof of Theorem 1 after that.
Proof of Theorem 2. For η ∈]0, 1/2[ let us denote by Step 1.As a first preparatory step, we need to estimate the capacity of R η N,l in (14).Let us first note that for any two paths ω e. the number of moments of time within the period [0, N ] when the trajectories ω N,1 and ω N,2 are at the same point of the space Z equals s.But due to the symmetry of the simple random walk Taking into account the fact that the random walk starting from 0 can not visit 0 at odd moments of time, we obtain that (13) equals This last number is well-known for the simple random walk on Z : it equals 2 2N 2 s−2(2N ) 2N 2(2N )−s (see e.g.[15]) which is, by Stirling's formula, when /4 as N → ∞.Finally, we obtain that for all N ≥ 0 the number (13) it is not greater than 2 2N e −hN 2η with some constant h > 0. It follows that for all N > 0 where C > 0, h > 0 are some constants.
Let us now treat the sum over S ⊗l N \ R ηn N,l .Let us first study the sum over (ω N,1 , . . ., ω N,l ) such that the matrix B N (ω N,1 , . . ., ω N,l ) is non-degenerate.By (18) each term in this sum is bounded by 2 −N l e c 2 lN 2α /2 N k(l) for some k(l) > 0. The number of terms in this sum is bounded by CN 2 N l exp(−hN 2ηn ) by (14).Since α < η n by (23), this sum converges to zero exponentially fast.
Let us finally turn to the sum over (ω N,1 , . . ., ω N,l ) such that the matrix B(ω N,1 , . . ., ω N,l ) is degenerate of the rank r < l.By (18) each term in this sum is bounded by for some k(r) > 0.
The number of such sets of r different paths is exponentially smaller than 2 N r : there exists p > 0 such that is does not exceed 2 N r e −pN .(In fact, consider r independent simple random walks on Z that at a given moment of time occupy any k < r different points of Z. Then with probability not less than (1/2) r , at the next moment of time, they occupy at least k + 1 different points.Then with probability not less than ((1/2) r ) r at least once during r next moments of time they will occupy r different points.So, the number of sets of different r paths that at each moment of time during [0, N ] occupy at most r − 1 different points is not greater than 2 N r (1 Given any set of r paths with η(ω N,1 ), . . ., η(ω N,r ) linearly independent, there is an N -independent number of possibilities to complete it by linear combinations η(ω N,r+1 ), . . .η(ω N,l ).To see this, first consider the equation λ 1 η(ω N,1 ) + • • • + λ r η(ω N,r ) = 0 with unknown λ 1 , . . ., λ r .For any moment of time m ∈ Let us construct a matrix A with r columns and at least N and at most rN rows in the following way.For any m > 0, according to given ω 1 m , . . ., ω r m , let us add to A n(m) rows : each equation λ i1 + • • • + λ i k = 0 gives a row with 1 at places i 1 , . . ., i k and 0 at all other places.Then the equation λ 1 η(ω N,1 ) + • • • + λ r η(ω N,i ) = 0 is equivalent A λ = 0 with λ = (λ 1 , . . ., λ r ).Since this equation has only a trivial solution λ = 0, then the rank of A equals r.The matrix A contains at most 2 r different rows.There is less than (2 r ) r possibilities to choose r linearly independent of them.Let A r×r be an r ×r matrix consisting of r linearly independent rows of A. The fact that η(ω N,r+1 ) is a linear combination ) can be written as A r×r µ = b where the vector b contains only 1 and 0 : if a given row t of the matrix A r×r corresponds to the mth step of the random walks and has 1 at places i 1 , . . ., i k and 0 elsewhere, then we put b Thus, given ω N,1 , . . ., ω N,r , there is an N independent number of possibilities to write the system A r×r µ = b with non degenerate matrix A r×r which determines uniquely linear coefficients µ 1 , . . ., µ r and consequently η(ω N,r+1 ) and the path ω N,r+1 itself through these linear coefficients.Hence, there is not more possibilities to choose ω N,r+1 than the number of non-degenerate matrices A r×r multiplied by the number of vectors b, that is roughly not more than 2 r 2 +r .
These observations lead to the fact that the sum (11) with the covariance matrix B N (ω N,1 , . . ., ω N,l ) of the rank r contains at most (2 r 2 +r ) l−r 2 N r e −pN different terms with some constant p > 0.Then, taking into account the estimate (24) of each term with 2α < 1, we deduce that it converges to zero exponentially fast.This finishes the proof of (6).
To show (7), we have been already noticed that the sum of terms P(∀ 2 i=1 : |η(ω N,i )−E N | < b i δ N ) over all pairs of different paths ω N,1 , ω N,2 in S ⊗l N \ R η0 N,l converges to zero exponentially fast.Then (7) follows from the Borel-Cantelli lemma.
Proof of Theorem 3. We have again to verify the hypothesis of Theorem 4 for V i,M given by δ −1 N |η(ω N,i ) − E N |, i.e. we must show (11).
For β ∈]0, 1[ let us denote by Step 1.In this step we estimate the capacity of the complementary set to K β N,l in (26) and (27).We have : It has been shown in the proof of Theorem 2 that the number equals the number of paths of a simple random walk within the period [0, 2N ] that visit the origin at least [N β ] + 1 times.Let W r be the time of the rth return to the origin of a simple random walk (W 1 = 0), R N be the number of returns to the origin in the first N steps.Then for any integer q where E k is the event that exactly k of the variables W s − W s−1 are greater or equal than N , and q − 1 − k are less than N .Then It is shown in [14] that in the case d = 2 with some constant h > 0. Finally for d = 2 and all N > 0 by ( 25) with some constant h 2 > 0.
In the case d ≥ 3 the random walk is transient and with some constant h d > 0.
Step 2. Proceeding exactly as in the proof of Theorem 2, we obtain that uniformly for (ω N,1 , . . ., ω N,l ) ∈ K β N,l , where we denoted by E N the vector (E N , . . ., E N ).Moreover, if the covariance the matrix B N (ω N,1 , . . ., ω N,l ) is of the rank r ≤ l (using the fact that its determinant is a finite polynomial in the variables 1/N ) we get as in the proof of Theorem 2 that for some k(r) > 0.
Step 3. Having (26), ( 27), ( 28) and (29), we are able to carry out the proof of the theorem.For given α ∈]0, 1/2[, let us choose first Next, let us choose then etc.After i − 1 steps we choose Let us take e.g.β i = (i + 1)β 0 .We stop the procedure at n = [2α/β 0 ]th step, that is We know that the function f ω N,1 ,...,ω N,r N ( t) is the product of N generating functions : Moreover, each of these functions is itself a product of (at minimum 1 and at maximum r) generating functions of type φ(( ).More precisely, let us construct the matrix A with r columns and at least N and at most rN rows as in the proof of Theorem 2. Namely, for each step n = 0, 1, 2, . . ., N , we add to the matrix A at least 1 and at most r rows according to the following rule: n for any j ∈ {1, . . ., r} \ {i 1 , . . ., i k }, we add to A a row with 1 at places i 1 , . . ., i k and 0 at other r − k places.Then Since B N (ω N,1 , . . ., ω N,r ) is non-degenerate, the rank of the matrix A equals r.Let us choose in A any r linearly independent rows, and let us denote by A r the r × r matrix constructed by them.
Then by the assumption made on φ with some C ′ > 0. Hence, with some constant C 0 > 0 depending on the function φ and on b 1 , . . ., b r only.Since the matrix A r is non-degenerate, using easy arguments of linear algebra, one can show that for some constant C 1 > 0 depending on the matrix A r only, we have The proof of (42) is given in Appendix.But the right-hand of (42) is finite.This shows that the integrand in (36) is in L 1 (R d ) and the inversion formula (36) is valid.Moreover, the right-hand side of (42) equals C 1 (2((2d) −N + (2d) −N N ln 2d + (2d) −N )) r .Hence, the probabilities above are bounded by the quantity C 0 N r/2 C 1 2 r (2+N ln(2d)) r (2d) −N r with C 0 depending on φ and b 1 , . . ., b r and C 1 depending on the choice of A r To conclude the proof of (35), it remains to remark that there is an N -independent number of possibilities to construct a matrix A r (at most 2 r 2 ), since it contains only 0 or 1.
Proposition 1 There exist constants N 0 , C, ǫ, δ, ζ > 0 such that for all (ω N,1 , . . .ω N,l ) ∈ R η N,l and all N ≥ N 0 the following estimates hold: , for all t ≤ ǫN 1/6 .(48) The proof of this proposition mimics the one of the Berry-Essen inequality and is given in Appendix.
The first part of I 1 N is just the probability that l Gaussian random variables with zero mean and covariance matrix B N (ω N,1 , . . ., ω N,l ) belong to the intervals

Appendix
Proof of (42).It is carried out via trivial arguments of linear algebra.Let m = 1, 2, . . ., r + 1, D m−1 be a non-degenerate r × r matrix with the first m − 1 rows having 1 on the diagonal and 0 outside of the diagonal.(Clearly, D 0 is just a non-degenerate matrix and D r is the diagonal matrix with 1 everywhere on the diagonal.)Let us introduce the integral Now, observe that the left-hand side of (42) is J 0 (A r ).By (53) it is bounded by J 1 (A r 1 ) + a −1 1,1 J 1 (A r B −1 A r ).Again by (53) each of these two terms can be estimated by a sum of two terms of type J 2 (•) etc.After 2 r applications of (53) J 0 (A r ) is bounded by a sum of 2 r terms of type J r (D r ) multiplied by some constants depending only on the initial matrix A r .But all these 2 r terms J r (D r ) are the same as in the right-hand side of (42).

{δ − 1 N N − 1
E − δ N b, E + δ N b] has a Binomial distribution with parameters (2δ N b) 1/(2π)e −E 2 /2 (1 + o(1)) and |Σ N |.If we put δ N = |Σ N | −1 √ 2π(1/2)e E 2 /2 , by a well known theorem of the course of elementary Probability, this random number converges in law to the Poisson distribution with parameter b as N → ∞.More generally, the point process σ∈ΣN δ N → ∞, to the Poisson point process in R + whose intensity measure is the Lebesgue measure.So, Bauke and Mertens conjecture states that for the Hamiltonian (H N (σ)) σ∈ΣN of any disordered spin system and for a suitable normalization C(N, E) the sequence of point processes σ∈ΣN δ {C(N,E)|HN (σ)− √ N E|}

J m− 1 (
m−1 t) k | d t.Sice D m−1 is non-degenerate, there exists i ∈ {m, . . ., r} such that d m,i = 0 and the matrix D m which is obtained from the matrix D m−1 by replacing its mth row by the one with 1 at the place (m, i) and 0 at all places (m, j) for j = i is non-degenerate.Without loss of generality we may assume that i = m (otherwise juste permute the mth with the ith column in D m−1 and t i with t m in the integralJ m−1 (D m−1 ) above).Since either |t m−1 | < |(D m−1 t) m−1 | or |t m−1 | ≥ |(D m−1 t) m−1 |, we can estimate J m−1 (D m−1 )roughly by the sum of the following two terms :J m−1 (D m−1 )here is just J m (D m ).Let us make a change of variables in the second one : let z = B Dm−1 t, where the matrix B Dm−1 is chosen such that z 1 = t 1 , . . ., z m−1 = t m−1 , z m = (D m−1 t) m , z m+1 = t m+1 , . . ., z r = t r .(Clearly, its mth row is the same as in the matrix D m−1 , and it has 1 on the diagonal in all other r − 1 rows and 0 outside of it.)Since d m,m = 0, the matrix B is non-degenerate.Then D m−1 t = D m−1 B −1 Dm−1 z, where the matrix D m−1 B −1 Dm−1 is non-degenerate, and, moreover, it has the first m rows with 1 on the diagonal and 0 outside of it, as we have (D m−1 t) 1 = t 1 = z 1 , . . ., (D m−1 t) m−1 = t m−1 = z m−1 , (D m−1 t) m = z m .Then (52) can be written as J m−1 (D m−1 ) ≤ J m (D m ) + d −1 m,m J m (D m−1 B −1 Dm−1 ).