bibkey: bianchicolletmagnanini2022ghs
authors: Alessandra Bianchi; Francesca Collet; Elena Magnanini
year: 2022
title: "The GHS and other correlation inequalities for the two-star model"
doi: 10.30757/ALEA.v19-64
url: https://arxiv.org/abs/2107.08889v2
claim: "The paper proves the FKG, GKS and GHS inequalities for the two-star exponential random graph model with nonnegative parameters, extends FKG and GKS to general exponential random graphs with nonnegative non-edge coefficients, and conjectures that in this general setting the GHS inequality holds exactly when every edge marginal is at least one half."
strata_touched:
  - D5/S3/StatisticalMechanics/ErgmGhsMarginalConjectureRefutation
license: citation-only
triage: anchor

Bianchi, Collet and Magnanini, the GHS and other correlation inequalities for the two-star model

A. Bianchi, F. Collet and E. Magnanini, arXiv:2107.08889v2 [math.PR] (6 December 2021; v1 19 July 2021); ALEA Lat. Am. J. Probab. Math. Stat. 19 (2022) 1679–1695. Quotations are from the v2 TeX source, with the macros \cG, \cE, \b, \E written as $\mathcal{G}$, $\mathcal{E}$, $\beta$, $\mathbb{E}$ and cross-references omitted.

The homomorphism density, for $G\in {\mathcal{G}}_n$ and a finite simple graph $H$:

$t(H,G):=\frac{|\text{hom}(H,G)|}{|V(G){|}^{|V(H)|}}$,

and the exponential random graph with pre-chosen finite simple graphs $H_1,\dots ,H_k$ and real parameters ${\beta }_1,\dots ,{\beta }_k$:

${\mu }_{n;\mathbf{\beta }}(G)=\frac{\exp \left(H_{n,\mathbf{\beta }}(G)\right)}{Z_{n;\mathbf{\beta }}}\text{for }G\in {\mathcal{G}}_n$, with $H_{n;\mathbf{\beta }}(G)=n^2{\sum }_{j=1}^k{\beta }_{j}t(H_j,G)$.

The third Ursell function of the edge indicators $x_i$:

$u_3(i,j,k)=\mathbb{E}(x_i{x}_j{x}_k)-\mathbb{E}(x_i)\mathbb{E}(x_j{x}_k)-\mathbb{E}(x_j)\mathbb{E}(x_i{x}_k)-\mathbb{E}(x_k)\mathbb{E}(x_i{x}_j)+2\mathbb{E}(x_i)\mathbb{E}(x_j)\mathbb{E}(x_k)$,

and the GHS inequality: for all $i,j,k\in {\mathcal{E}}_n$, $u_3(i,j,k)\le 0$.

The general setting (Section 4) takes $H_1$ to be an edge; for ${\beta }_1\in \mathbb{R}$ and ${\beta }_2,\dots ,{\beta }_k\ge 0$ the FKG lattice condition and the GKS inequality hold. The closing conjecture:

Since ${\text{Cov}}_{n;\mathbf{\beta }}(x_i,x_j)\ge 0$, due to the GKS inequality, we can conjecture that the necessary and sufficient condition for the GHS inequality to hold in the present general setting is again only the requirement ${\mathbb{E}}_{n;\mathbf{\beta }}(x_i)\ge 1/2$.

Verified locator

Source: https://arxiv.org/abs/2107.08889v2

• DOI: https://doi.org/10.30757/ALEA.v19-64 (ALEA 19 (2022) 1679–1695; author list, journal, volume and first page from Crossref).
• URL: https://arxiv.org/abs/2107.08889v2 (the e-print is a gzipped tar archive; member GHS_arXiv-replacement.tex, 1428 lines, md5 b353bfa014348d3c29f4b9c509b5665d):
  • the homomorphism density, the Gibbs density and the Hamiltonian, l. 286–312;
  • the edge set ${\mathcal{E}}_n$ and the edge indicators, l. 322–331;
  • the Ursell function $u_3$, l. 577–589;
  • the GHS inequality, Definition 3.4, l. 617–636;
  • the general setting, Lemma 4.1 and the GKS inequality, l. 1243–1287;
  • the conjecture, l. 1322.
• The author list and the arXiv dates are from the arXiv metadata.