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Heejune Kim

Publications and source records attributed to Heejune Kim.

5 recordsLinked to original sources
arXiv2026

Color symmetry in the Potts spin glass at high temperature

We show that color symmetry is preserved at high temperatures in the Potts spin glass model with $κ\ge 3$ colors. Our proof employs the second moment method applied to the balanced model with a suitable centering of the Hamiltonian, while incorporating results from the non-disordered Potts model Ellis--Wang (1990), https://doi.org/10.1016/0304-4149(90)90122-9. For $κ= 2$, we exploit the model's gauge symmetry to show that unbalanced configurations occur with exponentially small probability at all temperatures $β\in [0, \infty]$.

math.PR↗
arXiv2025

On the de Almeida--Thouless Transition Surface in the Multi-Species SK Model with Centered Gaussian External Field

We study the phase transition of the Parisi formula for the free energy in the multi-species Sherrington--Kirkpatrick model with a centered Gaussian external field and a positive-semidefinite variance profile matrix. We show that in terms of the strength of the external field and the variance profile, the de Almeida--Thouless surface delineates the boundary between replica symmetric solutions and replica symmetry breaking solutions.

math.PR↗
arXiv2025

Disorder Chaos in Short-Range, Diluted, and Lévy Spin Glasses

In a recent breakthrough [arXiv:2301.04112], Chatterjee proved site disorder chaos in the Edwards-Anderson (EA) short-range spin glass model utilizing the Hermite spectral method. In this paper, we demonstrate the further usefulness of this Hermite spectral approach by extending the validity of site disorder chaos in three related spin glass models. The first, called the mixed even $p$-spin short-range model, is a generalization of the EA model where the underlying graph is a deterministic bounded degree hypergraph consisting of hyperedges with even number of vertices. The second model is the diluted mixed $p$-spin model, which is allowed to have hyperedges with both odd and even number of vertices. For both models, our results hold under general symmetric disorder distributions. The main novelty of our argument is played by an elementary algebraic equation for the Fourier-Hermite series coefficients for the two-spin correlation functions. It allows us to deduce necessary geometric conditions to determine the contributing coefficients in the overlap function, which in spirit is the same as the crucial Lemma 1 in [arXiv:2301.04112]. Finally, we also establish disorder chaos in the Lévy model with stable index $α\in (1, 2)$.

math.PR↗
arXiv2024

Some Rigorous Results on the Lévy Spin Glass Model

We study the Lévy spin glass model, a fully connected model on $N$ vertices with heavy-tailed interactions governed by a power law distribution of order $0<α<2.$ Our investigation is divided into three cases $0<α<1$, $α=1$, and $1<α<2.$ When $1<α<2,$ we identify a high temperature regime, in which the limit and fluctuation of the free energy are explicitly obtained and the site and bond overlaps are shown to exhibit concentration, interestingly, while the former is concentrated around zero, the latter obeys a positivity behavior. At any temperature, we further establish the existence of the limiting free energy and derive a variational formula analogous to Panchenko's framework in the setting of the Poissonian Viana-Bray model. For $α=1$, the free energy scales super-linearly and converges to a constant proportional to $β$ in probability at any temperature. In the case of $0<α<1$, the scaling for the free energy is again super-linear, however, it converges weakly to the sum of a Poisson Point Process at any temperature. Additionally, we show that the Gibbs measure puts most of its mass on the configurations that align with signs of the polynomially many heaviest edge weights.

math.PR↗