In this paper, we derive quenched scaling limits for linear functionals and the empirical spin field of Ising models on inhomogeneous random graphs generated by a graphon (encompassing both dense and sparse graphs), in the high-temperature regime. We first prove a joint central limit theorem (CLT) for finite collections of linear statistics of the spin configurations, where the limiting covariance is characterized by the resolvent of the associated graphon integral operator. Building on this result, we establish functional CLTs for the average magnetization and for the spin field indexed by suitable classes of regular test functions. We further prove convergence of the full empirical spin field, viewed as a random generalized function in negative Sobolev spaces. These scaling limits provide applications to both Bayesian neural networks and causal inference. Specifically, for the former, we derive infinite-width Gaussian-process limits for two-layer Bayesian neural networks with Ising-dependent output-layer signs, while for the latter, we establish the asymptotic normality of H\'{a}jek estimators for average treatment effects under network interference.
We design randomized approximation schemes for the partition function of antiferromagnetic Ising models with uniform external field on random regular bipartite graphs. Our algorithm generalizes the approach of Kocurek, Oveis Gharan and Tjowasi (arXiv, 2026) for hard-core models on the same random graph model beyond the uniqueness threshold. We show that, as long as $\lambda$ is upper bounded by a constant and $\lambda(1 - \beta) \lesssim \Delta^{-1/2}$, an efficient randomized algorithm approximates the partition function with high probability. The algorithm first truncates configurations that are large on either side of the bipartition and then samples from Gibbs distributions conditioned on fixed sizes on one or both sides. To choose an optimal truncation bound, we establish concentration properties of the Gibbs distribution on random regular bipartite graphs. Then we apply high-dimensional expansion and prove trickle-down theorems to obtain fast samplers for the conditioned distributions.
Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include many network topologies common in applications. We derive the mean-field limit for the resulting model and obtain exact results for phase transitions in such systems. Specifically, we show that the critical temperatures of the Ising model on graphons are determined by the eigenvalues of the Hilbert-Schmidt operator associated with the graph limit. For many important network topologies, these eigenvalues can be computed explicitly. We illustrate our results with three representative random network models: Erd\H{o}s-R\'{e}nyi, small-world, and power-law. In the small-world case, we demonstrate phase transitions to both ferromagnetic and antiferromagnetic phases, as well as coexistence of local minima of the free energy. The latter gives rise to multistability, as confirmed by Monte Carlo simulations. The results of this work demonstrate that the Ising model on graphons combines the analytical tractability of exactly solvable mean-field models with the ability to accommodate a broad range of network topologies. We expect that the use of graphons in spin models will lead to new insights into the statistical physics of interacting systems on complex networks.
We prove recurrence criteria for inhomogeneous long-range percolation in dimensions one and two. In dimension one, recurrence follows from a purely geometric scarcity condition: long edges eventually disappear on exponential scales. This applies to weight-dependent random connection models and related one-dimensional spatial scale-free graphs whenever the standard strong-decay long-edge estimate holds. In dimension two, we combine the linear chemical-distance estimate of L\"uchtrath with an area-order bound on the degree measure. Graph-distance layers in exponentially separated bands then give the required Nash-Williams cutsets for planar random geometric graphs satisfying the polynomial mixing and long-edge estimates [J. Theoret. Probab. 39 (2026), Paper No. 12]. As a concrete consequence, every connected component of the two-dimensional weight-dependent random connection model with interpolation kernel is recurrent throughout the strong-decay region $\delta>2$, $\gamma<1-\frac{1}{\delta}$, and $\alpha<1-\gamma$.
Johannes Bäumler, Lukas Lüchtrath, Christian Mönch· 0 citations
We study Poisson Galton--Watson trees on a standard Borel type space when the offspring kernel is multiplied by a scalar parameter. On finite trees, we identify the Radon--Nikodym derivative between two parameter values and show that it remains measurable after projection to the total progeny measure. Under a uniform bound on the offspring intensities, differentiation yields exact differential and integral equations for the projected laws without irreducibility, reversibility, or a positive eigenfunction. With an additional positive eigenfunction bounded above and away from zero, we relate these equations to an infinite spinal tree, uniform pruning, the Doob transform, and the Aldous--Pitman ascension process. For a uniformly bounded offspring kernel, we also prove uniform exponential integrability of the total progeny throughout the spectrally subcritical regime. As an application, under the graphical-kernel assumptions of Bollobas, Janson and Riordan, the number $K_n$ of connected components satisfies $K_n/n \to {\mathbb E}_{\pi}[1/T_u]$ in probability and in $L^1$, where $T_u$ is the total progeny of the associated branching process and $1/\infty=0$. If $q_u(x)$ is its extinction probability from type $x$, re-rooting and extinction duality give the explicit limit $$ \int_S q_u(x)\,\pi(dx) - {u\over 2}\int_{S\times S}\kappa(x,y)q_u(x)q_u(y)\,\pi(dx)\pi(dy). $$ This extends the finite-type and compact-continuous formulas to the full BJR graphical-kernel setting, allowing separable noncompact type spaces and kernels that may be unbounded or reducible.
Using the properties of GL(2,R), the general linear group of invertible 2×2 real matrices, we investigate random fields of spin variables on finite one-dimensional rings with a unit cell of p∈N sites. The interaction parameters are assumed to be periodic with period p. The cases p=1 and p=2 recover the one-dimensional and alternating Ising models, respectively. The couplings between adjacent spins may be ferromagnetic (positive) or antiferromagnetic (negative). Utilising the recurrence relations of the Chebyshev polynomials and a bijection between the number of spins and the polynomial index, we derive explicit formulae well suited to the finite-size analysis of the partition functions, free energy, and specific heat of both models. We show that, for the (p=2) case, the specific heat exhibits a double-Schottky anomaly whenever the characteristic exchange energy scales are sufficiently separated. We prove that this double-peak structure originates from the coexistence of distinct energy scales induced by the periodic modulation of the coupling signs and characterise its dependence on the model parameters. We demonstrate that the universality hypothesis in critical Casimir force theory remarkably holds without requiring small fields or large interaction parameters, suggesting a form of “hyper-universal” behaviour valid for arbitrary model parameters.
Nicholay S. Tonchev, D. Dantchev· Axioms· 0 citations
We study the local eigenvalue statistics (LES) associated with one-dimensional lattice models of random polymers. We consider models constructed from two polymers. Each polymer is a finite interval of lattice points with a finite potential. These polymers are distributed along
\mathbb{Z}
according to a Bernoulli distribution. The deterministic spectrum for these models is dense pure point, and is known to contain finitely-many critical energies. In this paper, we prove that the LES centered at these critical energies is described by a uniform clock process, and that the LES for the unfolded eigenvalues, centered at any other energy in the deterministic spectrum, is a Poisson point process. These results add to our understanding of these models that exhibit dynamical localization in any energy interval avoiding the critical energies (D. Damanik, R. Sims, and G. Stolz (2004) and S. De Bièvre and F. Germinet (2000)), and non-trivial transport for wave packets with initial states supported at an integer point (S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz (2003)). We show that the projection of these initial states onto spectral subspaces associated with any energy interval that contains all of the critical energies exhibit non-trivial transport, refining the connection between non-trivial transport and the critical energies. Finally, we also prove that the transition in the unfolded LES is sharp at the critical energies.
P. Hislop, F. Nakano· Journal of Spectral Theory· 0 citations