2014/03/31 by Cécile Monthus, Cecile Monthus
Economics, Econometrics and Finance · Physics and Astronomy · #Complex Network Analysis Techniques #Complex Systems and Time Series Analysis #Theoretical and Computational Physics #cond-mat.dis-nn
paper · pdf · doi:10.1088/1742-5468/2014/14/p06015
published as J. Stat. Mech. (2014) P06015 · v2=revised version (17 pages) with new section VII concerning the Dyson hierarchical Spin-Glass model
arxiv created 2014/05/15 · openalex publication_date 2014/06/30 · arxiv updated 2014/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J ( r ) ∼ r − σ and distributed with the Lévy symmetric stable distribution of index 1 < μ ≤ 2 (including the usual Gaussian case μ = 2), we consider the region σ > 1/ μ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings J L ∝ L θ μ ( σ ) is found to be whenever the long-ranged couplings are relevant θ μ ( σ ) ≥ −1. For the statistics of the ground state energy over disordered samples, we obtain that the droplet exponent θ μ ( σ ) governs the leading correction to extensivity of the averaged value . The characteristic scale of the fluctuations around this average is of order , and the rescaled variable is Gaussian distributed for μ = 2, or displays the negative power-law tail in 1/(− u ) 1+ μ for u → − ∞ in the Lévy case 1 < μ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.