Continuity, critical-current geometry, and Pfaffian universality for two-dimensional finite-range Ising models

The 2019 'emergent planarity' paper of Aizenman, Duminil-Copin, Tassion and Warzel proved that at criticality the Pfaffian (fermionic) structure of correlation functions, which is exact for planar nearest-neighbor Ising models, emerges asymptotically for Ising models on the square lattice with finite-range ferromagnetic interactions. Its discussion section lists four problems left open, which together complete the universality picture for this model class: (i) prove that the spontaneous magnetization vanishes at the critical temperature, i.e. that the phase transition is continuous; (ii) prove polynomial mixing of the critical sourceless random current; (iii) prove Russo–Seymour–Welsh-type crossing estimates for the percolation of positive currents; and (iv) strengthen the asymptotic Pfaffian relation for bulk order–disorder correlations so that the error is relative to the order–disorder correlation itself.

Background

The Ising model is the archetypal lattice model of a phase transition. Spins $\sigma_x\in\{-1,+1\}$ sit on the vertices $x$ of the square lattice $\mathbb{Z}^2$ and interact through a Hamiltonian $H(\sigma)=-\sum_{x,y}J_{x,y}\sigma_x\sigma_y$, where the coupling constants are ferromagnetic ($J_{x,y}=J_{y,x}\ge0$), translation-invariant ($J_{x,y}=J_{x-y}$), and finite-range ($J_{x,y}=0$ whenever $|x-y|$ exceeds some fixed radius), and the graph of nonzero couplings is connected. At inverse temperature $\beta$, configurations are weighted by $e^{-\beta H}$. The model has a critical point $\beta_c\in(0,\infty)$ separating a disordered phase from an ordered phase in which the spontaneous magnetization $M(\beta)$ — the limiting expectation of $\sigma_0$ as a positive external field is sent to zero and the finite volume grows to $\mathbb{Z}^2$ — is strictly positive. The transition is called continuous if $M(\beta_c)=0$ and discontinuous if the order parameter jumps at $\beta_c$.

For the nearest-neighbor model the two-dimensional Ising model is exactly solvable: Onsager and Kaufman computed the free energy, and Yang proved in 1952 that $M(\beta_c)=0$. Whether continuity persists for general finite-range interactions — where exact solvability is lost — is a basic universality question: it asks whether the nature of the critical point depends only on dimensionality and symmetry class rather than on microscopic details of the interaction. In the plane, continuity is known for nearest-neighbor models on the square lattice (Yang 1952), for planar random-cluster and Potts models with cluster weight $1\le q\le4$ in the nearest-neighbor case (Duminil-Copin–Sidoravicius–Tassion 2017), and for the Ising model on arbitrary planar biperiodic graphs (Duminil-Copin–Lis 2019); in dimensions $d\ge4$ and $d=3$ it is known for nearest-neighbor models on $\mathbb{Z}^d$ (Aizenman–Fernández 1986; Aizenman–Duminil-Copin–Sidoravicius 2015). Beyond the nearest-neighbor planar case the question is open, and it is genuinely subtle: on quasi-planar 'book' graphs $\mathbb{Z}^2\times\{1,\dots,k\}$ with $k$ large, the critical Ising model in fact exhibits long-range order (Duminil-Copin–Garban–Tassion 2023), so planarity of the interaction graph is essential.

The random current representation rewrites the Ising partition function as a sum over integer-valued 'currents' $n=(n_e)_{e\text{ edge}}\in\mathbb{N}^{E}$, with spin correlation functions expressible through the connectivity properties of the set $\hat{\mathbf{n}}$ of edges carrying positive current. The sourceless measure $\mathbf{P}^{\emptyset}_{\mathbb{Z}^2,\beta_c}$ is the infinite-volume law of a current configuration with no sources (even total current at every vertex) at criticality; its stochastic geometry at $\beta_c$ is the central object below. Russo–Seymour–Welsh (RSW) estimates are uniform lower bounds on the probability that a long thin rectangle is crossed by an open path; in planar percolation theory they are the workhorse behind criticality, universality, and scaling-limit results, and no analogue is currently available for the current percolation of finite-range models.

For strictly planar nearest-neighbor models, boundary spin correlations and, more generally, order–disorder correlation functions have an exact Pfaffian structure: the $2n$-point function equals the Pfaffian of the matrix of two-point functions. The Pfaffian of an antisymmetric $2n\times2n$ matrix $A=(a_{ij})$ is the signed sum of products of entries over all pairings of indices, $\mathrm{Pf}(A)=\sum_{\text{pairings}}\pm\prod_{k}a_{i_kj_k}$, the polynomial whose square is $\det A$; it is the signature of fermionic (Majorana) Gaussian states, in the same way that the Wick rule governs bosonic Gaussian fields. Order–disorder (Kadanoff–Ceva) operators pair spin variables $\sigma_x$ with disorder variables $\tau$ associated with dual-graph paths along which the couplings are flipped; their correlations probe both the magnetic and the dual (disorder) sectors of the model.

Aizenman, Duminil-Copin, Tassion and Warzel (2019) proved that this Pfaffian structure 'emerges' at criticality even when the interaction is not planar: for finite-range, translation-invariant, reflection-invariant ferromagnetic couplings on the half-plane, boundary spin correlations satisfy the Pfaffian relation with relative error $1+o(1)$ at large separations (Theorem 1.2), and the analogous statement holds for order–disorder correlations, with an error term relative to the corresponding spin correlation (Corollary 6.5). These theorems are unconditional — the authors handle both a continuous and a discontinuous critical point. Section 7 of that paper lists what remains: the four problems stated below, whose common objective is to complete the universality picture for the finite-range planar Ising class — continuity of the transition and universality of the fermionic correlation structure — through the stochastic geometry (mixing and crossing estimates) of the critical random current.

Problem Statement

Settle the four questions posed in Section 7 of Aizenman–Duminil-Copin–Tassion–Warzel (2019) for Ising models on $\mathbb{Z}^2$ whose couplings are ferromagnetic ($J_{x,y}\ge0$), translation-invariant ($J_{x,y}=J_{x-y}$), reflection-invariant (invariant under the reflections of the lattice $\mathbb{Z}^2$), and finite-range with connected interaction graph. The common objective uniting the components is to extend the stochastic-geometric picture of the critical random current — and with it continuity of the phase transition and universality of the fermionic correlation structure — from the exactly solvable nearest-neighbor model to the entire finite-range class:

(i) Continuity of the order parameter (Question 7.1): prove that the spontaneous magnetization vanishes at the critical temperature, $M(\beta_c)=0$, for every model in the class (or refute by exhibiting a model in the class whose order parameter jumps at $\beta_c$).

(ii) Polynomial mixing of random currents (Question 7.2): prove that there exist constants $c,C>0$ such that for any two events $A$ (depending on edges inside a box of size $n$) and $B$ (depending on edges outside a concentric box of size $N\ge n$),

$$\bigl|\mathbf{P}^{\emptyset}_{\mathbb{Z}^2,\beta_c}[A\cap B]-\mathbf{P}^{\emptyset}_{\mathbb{Z}^2,\beta_c}[A]\,\mathbf{P}^{\emptyset}_{\mathbb{Z}^2,\beta_c}[B]\bigr|\le C\,(n/N)^{c}\,\mathbf{P}^{\emptyset}_{\mathbb{Z}^2,\beta_c}[A]\,\mathbf{P}^{\emptyset}_{\mathbb{Z}^2,\beta_c}[B],$$

where $\mathbf{P}^{\emptyset}_{\mathbb{Z}^2,\beta_c}$ is the sourceless single random-current measure in the full plane at criticality.

(iii) Crossing estimates for the current percolation (Question 7.3): prove that there exists $c>0$ such that for every $n$, the probability, under $\mathbf{P}^{\emptyset}_{\mathbb{Z}^2,\beta_c}$, that an $n\times2n$ box is crossed in the hard direction (that is, by a path of positive current spanning the side of length $2n$) is at least $c$; the source article establishes this only for infinitely many values of $n$.

(iv) Full asymptotic Pfaffian relation for order–disorder correlations (Question 7.4): strengthen Corollary 6.5 of the source so that the error term becomes $o(1)\,|\langle\tau_1\cdots\tau_{2n}\rangle_{\mathbb{Z}^2,\beta_c}|$ in place of $o(1)\,|\langle\sigma_{x_1}\cdots\sigma_{x_{2n}}\rangle_{\mathbb{Z}^2,\beta_c}|$, thereby establishing the fermionic Pfaffian structure for bulk order–disorder fields in finite-range models.

A complete answer settles each of (i)–(iv); the components are the stated remaining steps of one program, (ii) and (iii) being the stochastic-geometric inputs expected to drive the continuity statement (i) and the Pfaffian strengthening (iv).

References

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