Explore the vast range of titles available.

We examine the Ising model at its critical temperature with an external magnetic field ha158 on aZ2 for a,h>0 . A new proof of exponential decay of the truncated two-point correlation functions is presented. It is proven that the mass (inverse correlation length) is of the order of h815 in the limit h→0 . This was previously proven with CLE-methods in Camia et al. in (Commun Pure Appl Math 73(7):1371–405, 2020). Our new proof uses instead the random current representation of the Ising model and its backbone exploration. The method further relies on recent couplings to the random cluster model (Aizenman et al. in Invent Math 216:661–743, 2018) as well as a near-critical RSW-result for the random cluster model (Duminil-Copin and Manolescu in Planar Random-Cluster Model: Scaling Relations, 2020).

The truncated two-point function of the ferromagnetic Ising model on Z d ( d ≥ 3 ) in its pure phases is proven to decay exponentially fast throughout the ordered regime ( β > β c and h = 0 ). Together with the previously known results, this implies that the exponential clustering property holds throughout the model’s phase diagram except for the critical point: ( β , h ) = ( β c , 0 ) .

In this article, we prove that the height function associated with the square-ice model (i.e. the six-vertex model with a = b = c = 1 on the square lattice), or, equivalently, of the uniform random homomorphisms from Z 2 to Z , has logarithmic variance. This establishes a strong form of roughness of this height function.

We prove that at any inverse temperature β and on any transitive amenable graph, the automorphism-invariant Gibbs states of the ferromagnetic Ising model are convex combinations of the plus and minus states. The theorem is equivalent with the differentiability of the free energy with respect to the temperature at any temperature. This is obtained for a general class of interactions, that is automorphism-invariant and irreducible coupling constants. The proof uses the random current representation of the Ising model. The result is novel when the graph is not ℤd , or when the graph is ℤd but endowed with infinite-range interactions, or even ℤ² with finite-range interactions. Among the other corollaries of this result, we can list continuity of the magnetization at any noncritical temperature and the uniqueness of FK-Ising infinite-volume measures at any temperature.

We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that For the Potts model on transitive graphs, correlations decay exponentially fast for β < β c . For the random-cluster model with cluster weight q ≥ 1 on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime. For the random-cluster models with cluster weight q ≥ 1 on planar quasi-transitive graphs 𝔾, p c ( 𝔾 ) p c ( 𝔾 * ) ( 1 - p c ( 𝔾 ) ) ( 1 - p c ( 𝔾 * ) ) = q As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices. (This provides a short proof of a result of Beffara and the first author dating from 2012.) These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian free field, and the random-cluster and Potts models with infinite range interactions.

We prove that for Voronoi percolation on \\[ R^d\\]\\[(d 2)\\], there exists \\[p_c=p_c(d)ın (0,1)\\] such thatfor \\[p0\\] such that \\[ P_p[0 connected to distance n] (-c_pn)\\],there exists \\[c>0\\] such that for \\[p>p_c, P_p[0 connected to infinity] c(p-p_c)\\]. For dimension 2, this result offers a new way of showing that \\[p_c(2)=1/2\\]. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition; see [10, 11].

What does a discrete diffusion model learn: a denoiser, a score ratio, or a bridge plug-in predictor? At the level of jump rates, these are one object in different coordinates, and reading a neural network in the wrong coordinate changes the process being trained and sampled. Starting with a rigorous derivation of the continuous-time Markov chain (CTMC) ELBO for any noising process, boundary terms included, we prove the Oracle Distance theorem: the negative ELBO is exactly equal to the data entropy plus the path KL from the oracle reverse process to the learned one, not merely a bound. Its unique optimizer is therefore the conditional expectation of the true reverse jump rate given the current noisy state, and its irreducible cost is the rate at which the forward process \\(Z_t\\) destroys information about the clean data \\(Z_0\\), \\(-ddtI(Z_0; Z_t)\\), so every noising process shares the same best achievable negative ELBO: the data entropy. For sequences with token-factorizing noise, the oracle projection yields three exact coordinates for the optimizer: denoiser, cavity (bridge plug-in), and score, with closed-form conversions among them. This framework identifies which law each loss in the literature actually optimizes, recovering MDM, UDM, SEDD, and GIDD as special cases; explains why denoiser and cavity coincide for masked diffusion but not for uniform diffusion; proves that a denoiser parameterization makes the uniform ELBO diverge at initialization while the bridge plug-in stays finite; and calibrates ELBO implementations exactly at initialization. Every identity is verified numerically, without approximation, on an exactly solvable model.

We prove that at any inverse temperature \\(\\) and on any transitive amenable graph, the automorphism-invariant Gibbs states of the ferromagnetic Ising model are convex combinations of the plus and minus states. This is obtained for a general class of interactions, that is automorphism-invariant and irreducible coupling constants. The proof uses the random current representation of the Ising model. The result is novel when the graph is not \\(Z^d\\), or when the graph is \\(Z^d\\) but endowed with infinite-range interactions, or even \\(Z^2\\) with finite-range interactions. Among the corollaries of this result, we can list continuity of the magnetization at any non-critical temperature, the differentiability of the free energy, and the uniqueness of FK-Ising infinite-volume measures.

The truncated two-point function of the ferromagnetic Ising model on \\( Z^d\\) (\\(d3\\)) in its pure phases is proven to decay exponentially fast throughout the ordered regime (\\(>_c\\) and \\(h=0\\)). Together with the previously known results, this implies that the exponential clustering property holds throughout the model's phase diagram except for the critical point: \\(( h) = (_c,0)\\).

We examine the Ising model at its critical temperature with an external magnetic field \\(h a^158\\) on \\(aZ^2\\) for \\(a,h >0\\). A new proof of exponential decay of the truncated two-point correlation functions is presented. It is proven that the mass (inverse correlation length) is of the order of \\(h^815\\) in the limit \\(h 0\\). This was previously proven with CLE-methods in \\( 1 \\). Our new proof uses instead the random current representation of the Ising model and its backbone exploration. The method further relies on recent couplings to the random cluster model \\( 2 \\) as well as a near-critical RSW-result for the random cluster model \\( 3 \\).