Uniqueness Phase Transition in Statistical Models

• Uniqueness phase transition is the shift from a single equilibrium state to multiple macroscopic states when critical system parameters are crossed.
• Critical thresholds are characterized by mathematical criteria such as fixed-point, spectral, or branching process methods in models like random graphs, percolation, and spin systems.
• The transition has practical implications, affecting sampling complexity and computational tractability by marking the boundary between efficient algorithms and #P-hard regimes.

A uniqueness phase transition, or uniqueness/non-uniqueness phase transition, refers to the abrupt change in the qualitative structure of equilibrium states, macroscopic clusters, or Gibbs measures of a random or statistical-mechanical model as one or more system parameters cross a critical threshold. In the uniqueness regime, there is exactly one equilibrium (or giant component, or infinite cluster), while beyond the transition, either multiple distinct macroscopic states or the loss of a unique "giant" occurs. This phenomenon is central to random graph theory, Gibbs measures of lattice models, percolation theory, stochastic processes, and spin glass models, and governs both physical properties and computational hardness in these systems.

1. Prototype Models and Definitions

In random graphs, percolation, and statistical mechanics, uniqueness phase transitions describe points at which uniqueness of the relevant infinite-volume object (Gibbs state, infinite cluster, giant component) is lost:

• Random Intersection Graphs: For inhomogeneous random intersection graphs with $n$ vertices, $m$ attributes, and incidence probabilities $\mathbf{p}=(p_1,\ldots,p_m)$, the critical parameter is $\Lambda_n = n \sum_{i=1}^m p_i^2$. At $\Lambda_n=1$, a phase transition occurs between a regime with all components small and a regime with a unique giant component (“uniqueness of the giant component”) (Bradonjić et al., 2013).
• Spin Systems on Trees and Graphs: For models such as the hardcore model or anti-ferromagnetic Ising, the uniqueness phase transition is defined by a critical parameter (e.g., fugacity $\lambda_c(d)$ or inverse temperature $\beta_c(d)$) corresponding to decay or persistence of boundary influence on the root on the infinite $d$-regular tree. Uniqueness of the Gibbs measure holds below the critical threshold; above, multiple extremal measures emerge (Sly, 2010, Li et al., 2011, Cai et al., 2012).
• Continuum Percolation and Level Sets: For random fields built from Poisson clouds (e.g., level set percolation of attenuation fields, occupied sets of Brownian paths), percolation of the upper level set exhibits a unique infinite component in the supercritical regime, establishing uniqueness in the sense of cluster structure (Broman et al., 2016, Erhard et al., 2013).
• Stochastic Chains (g-Measures): In regular attractive stochastic chains defined by an infinite-memory kernel, the uniqueness phase transition separates a regime where there is a unique compatible chain ("g-measure") from one with at least two extremal stationary chains (Gallo et al., 2011).
• Random Walk Vacant Set Phase Transition: For random interlacements and the vacant set of high-dimensional random walks, a sharp threshold $u_*$ separates a regime with a unique giant vacant component from a regime where only small clusters exist (Duminil-Copin et al., 2023).

2. Mathematical Criteria and Criticality

The analytic characterization of the uniqueness phase transition is model-dependent, but commonly takes the form of a fixed-point or spectral criterion:

• Branching Process/Galton–Watson Criteria: In inhomogeneous random intersection graphs, the extinction probability $\rho$ is solved by a multi-type Galton–Watson equation; the survival of the process corresponds to a unique giant component, and the criticality is exactly $\Lambda_n=1$ (Bradonjić et al., 2013).
• Spectral or Fixed-Point Criteria for Spin Systems:
  • For two-spin (Ising, hardcore) systems on $d$-regular trees, the uniqueness threshold is given by conditions such as $|f_d'(\zeta)| < 1$, where $f_d$ parameterizes the recursive marginal ratio (Cai et al., 2012, Li et al., 2011).
  • For the hardcore model, $\lambda_c(d) = (d-1)^{d-1} / (d-2)^d$ is the unique solution where the decay of correlations is lost on the infinite $d$-regular tree (Sly, 2010).
  • For the Ising model, the uniqueness threshold is at $(d-1)\tanh \beta_c = 1$.
• Percolation Thresholds: The critical parameter is often defined via the existence of a non-trivial solution for the infinite cluster probability, e.g., for continuum percolation models, existence and uniqueness are proved for $h < h_c$, the critical level where the percolation probability transitions to zero (Broman et al., 2016, Erhard et al., 2013).
• Stochastic Chains: The equivalence theorem asserts that uniqueness of the stationary chain is equivalent to the maximal chain being a finitary coding of an i.i.d. process and to exponential concentration, with the phase transition controlled by the decay rate of the memory kernel (Gallo et al., 2011).

3. Uniqueness, Size Jump, and Critical Behavior

The structural consequences of the uniqueness phase transition include the size and uniqueness of giant components or infinite clusters, the character of the computational problem, and the complexity of sampling or optimization:

Model type	Subcritical Regime	Supercritical Regime	Critical Parameter
Inhom. random intersection graphs	All components $O(\max\{np_\mathrm{max}\log n, \log n\})$	Unique giant component of size $(1-\rho)n$	$\Lambda_n = 1$ (Bradonjić et al., 2013)
Hardcore/Ising on tree	Unique (extremal) Gibbs measure, exponential decay	Two distinct extremes (coexistence)	$\lambda_c(d)$, $\beta_c(d)$
Percolation, level sets	No infinite cluster	Unique infinite cluster (in 2d, continuous l)	$h = h_c$ (Broman et al., 2016)
Stochastic chain (g-measure)	Minimal = maximal chain, finitary i.i.d. code	Distinct extremal chains, long memory	Kernel memory, regularity
Random walk vacant set	Only small clusters (sub-exponential)	Giant component with local uniqueness	$u = u_*$ (Duminil-Copin et al., 2023)

Notably, the "size-jump" at the phase transition may be universal (Erdős–Rényi graphs) or depend on inhomogeneity (intersection graphs, where the jump can be $$O(n p_{\mathrm{max}} \log n) \to \Theta(p_\mathrm{max}^{-1})$$) (Bradonjić et al., 2013).

In spin systems, the uniqueness transition marks the point where boundary effects persist macroscopically and coexistence of extremes becomes possible.

4. Proof Techniques and Structural Insights

• Exploration Process and Coupling: In random graph models, uniqueness is established by coupling the exploration of clusters to (multi-type) branching processes and applying concentration inequalities (Chernoff bounds) to show large structures must merge, implying at most one giant component (Bradonjić et al., 2013).
• Planar Percolation and RSW: For random field level set percolation in $\mathbb{R}^2$, uniqueness proofs adapt finite-energy and mixing arguments (FKG, box-decorrelation, RSW theory) in the absence of finite-range dependence due to the infinite support of the attenuation function (Broman et al., 2016).
• Disagreement Percolation/Block Criteria: For Ising-type models, uniqueness can be established by bounding the maximal influence (as in Dobrushin’s criterion), disagreement percolation (relating per-site influences to percolation thresholds), or Dobrushin–Shlosman block updates (Fernández et al., 2019).
• Perfect Simulation and Coding: In stochastic chains, uniqueness is characterized by the success of coupling-from-the-past constructions and equivalence to i.i.d. finitary codings (Gallo et al., 2011).
• Spectral Independence and Mixing Times: At the uniqueness threshold, the mixing time of Glauber dynamics for Gibbs distributions transitions from logarithmic to polynomial, with explicit dependence on the spectral independence constant C($\eta$), which grows sublinearly at criticality (Chen et al., 5 Nov 2024).

5. Computational Complexity Correspondence

A pivotal correspondence exists between physical uniqueness transitions and computational tractability:

• Complexity Transition: In the hardcore and two-spin anti-ferromagnetic models, uniqueness on the infinite regular tree is equivalent to the existence of efficient (FPTAS) algorithms for partition function approximation and to strong spatial mixing for all bounded-degree graphs. Loss of uniqueness signals the onset of #P-hardness and regime of torpid mixing for Markov chains (Sly, 2010, Cai et al., 2012, Li et al., 2011).
• Categorical Dichotomy:
  • Uniqueness regime: Efficient approximation (FPTAS), strong mixing.
  • Non-uniqueness regime: No FPRAS unless NP=RP; exponentially slow sampling.

This computational “phase transition” matches the statistical uniqueness transition precisely, with the threshold parameter (e.g., $\lambda_c(d)$) defining both boundaries (Sly, 2010, Cai et al., 2012, Li et al., 2011, Chen et al., 5 Nov 2024).

6. Influence of Inhomogeneity and Boundary Effects

The sharpness, universality, and nature of the uniqueness phase transition can be significantly modulated by model inhomogeneities or long-range dependencies:

• Random Intersection Graphs: Degree inhomogeneity alters the sharpness and size of the giant component jump at the critical point, with the possibility of sublinear jumps for large $p_\mathrm{max}$ (Bradonjić et al., 2013).
• Level Set Percolation: Infinite-range dependence (attenuation functions with unbounded support) complicates uniqueness proofs, requiring truncation and specialized spatial-mixing arguments; the classical Burton–Keane approach may fail in this context (Broman et al., 2016).
• Stochastic Chains: The necessary conditions for uniqueness (strong non-nullness and continuity) are essential; violation leads to degeneration or pathological nonuniqueness with different ergodic properties (Gallo et al., 2011).

7. Structural and Physical Implications

• Uniqueness guarantees macroscopic determinacy: In regimes where uniqueness holds, thermodynamic limits, macroscopic observables, and sampling procedures yield consistent, predictable results.
• Phase coexistence and multiplicity: Beyond the uniqueness threshold, multiple thermodynamic phases, orderings, or giants emerge, often corresponding to spontaneous symmetry breaking or metastability.
• Criticality and slow dynamics: At the uniqueness threshold, sampling complexity increases polynomially; mixing times become superlinear, reflecting the slow decay of correlations and the presence of long-range dependencies (Chen et al., 5 Nov 2024).
• Universality and Model Dependence: While the phenomenon of uniqueness phase transitions is universal, the exact point, nature of the transition, and size jump depend intricately on system-specific parameters (e.g., inhomogeneity, support of interaction, dimension).

References: (Bradonjić et al., 2013, Broman et al., 2016, Sly, 2010, Cai et al., 2012, Gallo et al., 2011, Erhard et al., 2013, Chen et al., 5 Nov 2024, Li et al., 2011, Fernández et al., 2019, Duminil-Copin et al., 2023)