Finite-size corrections to disordered systems on Erdös-Rényi random graphs

Ferrari, Ulisse; Lucibello, Carlo; Morone, Flaviano; Parisi, G.; Ricci-Tersenghi, Federico; Rizzo, Tommaso

doi:10.1103/physrevb.88.184201

Cited by 23 publications

(39 citation statements)

References 25 publications

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“…It has been recently found that short chains have an important role in the finite size corrections to disordered models on diluted graphs 18 and in perturbative expansions around the Bethe approximation on Euclidean systems 27 . Therefore the analytical tools we have developed also apply to these contexts.…”

Section: Discussionmentioning

confidence: 99%

“…The only replica expression whose derivation is left as an open problem is the free energy of closed chains. We recall that closed chains are rather important objects that appears in perturbative computations developed around the tree approximation 18 . We conclude this introduction by briefly discussing the connection between our results and the extensive literature on disordered Ising chains.…”

Section: Introductionmentioning

confidence: 99%

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One-dimensional disordered Ising models by replica and cavity methods

2014

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Using a formalism based on the spectral decomposition of the replicated transfer matrix for disordered Ising models, we obtain several results that apply both to isolated one-dimensional systems and to locally tree-like graph and factor graph (p-spin) ensembles. We present exact analytical expressions, which can be efficiently approximated numerically, for many types of correlation functions and for the average free energies of open and closed finite chains. All the results achieved, with the exception of those involving closed chains, are then rigorously derived without replicas, using a probabilistic approach with the same flavour of cavity method.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

One-dimensional disordered Ising models by replica and cavity methods

2014

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“…Further, Claim 5.8 yields 32 Therefore, since SYM provides that the distribution P is invariant under permutations,…”

Section: Proof Of Lemma 54mentioning

confidence: 95%

Charting the Replica Symmetric Phase

Coja-Oghlan

Efthymiou

Jaafari

et al. 2018

Commun. Math. Phys.

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Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous 'cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics [Krzakala et al.: PNAS 2007]. In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the k-XORSAT model and the diluted k-spin model for even k. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention [Decelle et al.: Phys. Rev. E 2011].1 k h=1 ρ π h (σ h ) . 2 and d cond (k, β) = inf{d > 0 : sup π∈P 2 * ({1,−1}) B k−spin (d, β, π) > ln 2}. Then 0 < d cond (k, β) < ∞ and (k, β).From now on we assume that k ≥ 4 is even. The regime d < d cond (k, β) is called the replica symmetric phase. According to the cavity method, its key feature is that with probability tending to 1 in the limit n → ∞, two independent samples σ 1 , σ 2 ('replicas') chosen from the Gibbs measure µ H,J ,β are "essentially perpendicular". To formalize this define for σ, τ : V n → {±1} the overlap as ̺ σ,τ = x∈V n σ(x)τ(x)/n. We write 〈 · 〉 H,J ,β for the average on σ 1 , σ 2 chosen independently from µ H,J ,β and denote the expectation over the choice of H and J by E [ · ]. Theorem 1.2. For all β > 0 and k ≥ 4 even we have d cond (k, β) The corresponding statement for k = 2 was proved by Guerra and Toninelli, but as they point out their argument does not extend to larger k [37]. Theorem 1.2 implies the absence of extensive long-range correlations in the replica symmetric phase. Indeed, for two vertices x, y ∈ V n and s, t ∈ {+1, −1} letbe the joint distribution of the spins assigned to x, y. Further, letρ be the uniform distribution on {±1} × {±1}. Then the total variation distance µ H,J ,β,x,y −ρ TV is a measure of how correlated the spins of x, y are. Indeed, in the case that k is even for every x ∈ V n the Gibbs marginals satisfy µ H,J ,β,x (±1) = 〈1{σ 1 (x) = ±1}〉 H,J ,β = 1/2 because µ H,J ,β (σ) = µ H,J ,β (−σ) for every σ ∈ {−1, +1} n . Therefore, if the spins at x, y were independent, then µ H,J ,β,x,y = µ H,J ,β,x ⊗ µ H,J ,β,y =ρ. Furthermore, it is well known (e.g., [13, Section 2]) thatThus, Theorem 1.2 implies that for d < d cond (k, β), with probability tending to 1, the spins assigned to two random vertices x, y of H are asymptotically independent. By contrast, Theorem 1.2 and (1.2) show that extensive longrange dependencies occur beyond but arbitrarily close to d cond (k, β).1.3. The Potts antiferromagnet. Let q ≥ 2 be an integer, let Ω = {1, . . . , q} be a set...

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“…The factor F(n) accounts for the fact that s (φ) is not normalized for arbitrary n, as can be noted from Eq. (16). Plugging Eq.…”

Section: The Distribution Of Eigenvalues In the Replica Symmetricmentioning

confidence: 99%

Finite-size corrections to the spectrum of regular random graphs: An analytical solution

2014

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We develop a thorough analytical study of the O(1/N) correction to the spectrum of regular random graphs with N→∞ nodes. The finite-size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the O(1/N) correction to the continuous part of the spectrum. The comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement, confirming the correctness of our expression.

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Finite-size corrections to disordered systems on Erdös-Rényi random graphs

Cited by 23 publications

References 25 publications

One-dimensional disordered Ising models by replica and cavity methods

One-dimensional disordered Ising models by replica and cavity methods

Charting the Replica Symmetric Phase

Finite-size corrections to the spectrum of regular random graphs: An analytical solution

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