Random Gibbs Measure Overview

• Random Gibbs measures are probability measures defined on configuration spaces that integrate randomness in interactions or geometry, extending classical Gibbs measures.
• They are rigorously constructed using tools like measurable selection, metastates, and extremal decompositions to handle disordered environments.
• Applications span disordered spin systems, spatial networks, Bayesian nonparametrics, and algorithmic sampling in complex statistical models.

A random Gibbs measure is a probability measure on a space of configurations—such as spin systems, graphs, permutations, partitions, or function fields—where the defining interaction (Hamiltonian or potential) and/or the underlying "geometry" (lattice, graph, or point set) is itself random or depends on an external realization of disorder. The concept generalizes the classical Gibbs measure by incorporating randomness at the level of interactions, geometry, or construction procedures, making it fundamental for the mathematical description of disordered systems, spatial networks, random graph ensembles, random partitions, and statistical models with random external parameters or environments.

1. Foundational Construction of Random Gibbs Measures

The construction of a random Gibbs measure depends on the specific class of configurations and the nature of the randomness:

• Random Interactions on Lattices: Given a lattice such as ℤᵈ, one assigns random coupling constants (e.g., $J_{xy}$) between neighboring sites. The random Gibbs measure is then defined as a measurable selection $J \mapsto \mu_J$, where $J$ indexes the (random) interaction and $\mu_J$ is the corresponding DLR (Dobrushin–Lanford–Ruelle) solution on spin or field configurations. For unbounded spins or couplings, temperedness conditions (e.g., $J \in J_q$ and spins in $S_p$) are imposed to ensure integrability and compactness (Kondratiev et al., 2010).
• Random Geometry and Random Graphs: The random Gibbs measure may be defined over subgraph ensembles, with the vertex set $\Gamma$ itself given by a realization of a point process (e.g., homogeneous Poisson). Here, the set of possible edges $E$ is determined by the random point set, and the Gibbsian weight combines geometric costs (edge lengths) and combinatorial penalties (collisions, monomers). The exact form is:

$$H(\omega) = \sum_{e:\omega(e)=1} L(e) + \sum_{\gamma \in \Gamma} \varphi(\omega(\gamma)),$$

where $\varphi$ encodes vertex-level penalties (Ferrari et al., 2010). The measure is constructed by taking the infinite-volume limit of finite-volume Gibbs distributions.

• Random Partition Models and Bayesian Nonparametrics: In probabilistic combinatorics and Bayesian nonparametrics, Gibbs-type random probability measures (or π-priors) generate exchangeable random partitions. Here, randomness may enter through the random process (e.g., a Poisson–Kingman process with possibly random mixing parameters) and the random sampling of configurations (e.g., species or clusterings). The induced random partition probability structure (EPPF) has product form, with parameters (e.g., $(\sigma, V_{n, j})$) encoding the random environment (Bacallado et al., 2015, Arbel et al., 2017).
• Random Function Fields and Tessellations: For convex (piecewise-linear) functions whose gradients define random tessellations (e.g., Laguerre), the measure is built via Markovian jump processes where the kernel $f(x, p_-, dp_+)$ governing transitions is itself random or sampled from a prescribed distribution (Ouaki et al., 2021).
• Disordered Hamiltonians and Spin Glasses: In spin systems with Gaussian or mixed-random Hamiltonians (e.g., spherical $p$-spin glasses), the random Gibbs measure is the quenched measure associated to a random Hamiltonian $H_N$. The thermodynamic properties are described in terms of the Parisi measure and may require conditioning on specific energies or gradients at random points (Dembo et al., 28 Sep 2024).

A unifying feature is the requirement that the assignment from the underlying randomness (environment, point configuration, random interaction) to the corresponding Gibbs measure is measurable (and often upper semicontinuous) with respect to the σ-algebra of the environment.

2. Mathematical Properties: Measurability, Selection, and Metastates

• Measurable Selection: For a random environment $\xi$ (which may include random couplings, random field, random graph geometry), the assignment $\xi \mapsto \mu_\xi$ giving the random Gibbs measure is required to be measurable. This is established using set-valued analysis and measurable selection theorems (Kondratiev et al., 2010, Daletskii et al., 2013, Cotar et al., 2018).
• Empirical Distributions and Metastates: In disordered systems with strong chaotic dependence on volume or boundary conditions (typical in spin glasses), the finite-volume Gibbs measures may fail to converge. Instead, empirical distributions (Cesàro averages) and metastates are constructed as measurable maps from the environment to probability measures on the space of infinite-volume Gibbs measures. For example, the Aizenman–Wehr metastate is a measurable map $J \mapsto m(J)$ supported on the set of all possible infinite-volume Gibbs measures $G_p(J)$, and the barycenter of $m(J)$ gives a random Gibbs measure (Kondratiev et al., 2010, Cotar et al., 2018).
• Extremal Decomposition: Every random (possibly nonextremal) Gibbs measure admits a unique decomposition into pure (extremal) states, $p_\xi = \int_{\mathrm{ex} G} w_\xi(dv) v$, where $w_\xi$ is measurable in $\xi$. There also exists, for any metastate, a decomposition metastate supported on extremal states with the same barycenter (Cotar et al., 2018).
• Support and Temperedness: In systems with unbounded spins and/or unbounded degree, random Gibbs measures are supported on well-defined Banach spaces of tempered configurations. For example, measures are supported on $l^p_\alpha(\gamma)$, the space of $p$-summable configurations with exponential spatial weights (Daletskii et al., 2013).

3. Percolation, Phase Transitions, and Cluster Structure

• Percolation Properties: In random graph models with Gibbsian interaction (as in (Ferrari et al., 2010)), the competition between energetic penalties (favoring isolated or short edges) and geometric randomness determines percolation thresholds. For a Poisson vertex process of intensity $\lambda$ and inverse temperature $\beta$, there exists a non-percolating regime specified by

$$(\lambda, T) \in F = \left\{ (\lambda, T) : \lambda \leq \frac{1}{2h_0 J(T)} \right\},$$

with $J(T)$ given explicitly. In this regime, the Gibbs measure almost surely does not admit infinite clusters; clusters are finite and often consist of monomers and dimers.

• Ground States and Sparse Structure: The ground states of the random Gibbs graph are exactly those configurations with only monomers and dimers; open edges longer than $2h_0$ cannot appear. This ensures robustness against the creation of extended structures in low-density or low-temperature regimes (Ferrari et al., 2010).
• Cluster Branching Process: The distribution of cluster sizes and existence (or not) of infinite clusters is studied via an associated branching process, leveraging moment estimates (expected number of offsprings) to establish subcriticality and hence non-percolation.
• Phase Transition and Hierarchical Structures: In spatial Gibbs random graphs, tuning the graph bias parameters can yield emergent hierarchical connectivity ("fractal-like" edge layerings) that minimize graph-theoretic diameters without altering local neighborhoods, unless the bias is sufficiently strong to enforce dense connectivity (Endo et al., 2017, Cerqueira et al., 2019).

4. Applications: Disordered Systems, Random Partitions, and Statistical Mechanics

• Disordered Spin Systems (Spin Glasses): The random Gibbs measure framework provides a rigorous description for spin glasses, random field models, and related disordered systems, especially in infinite volume or at low temperature (e.g., in $k$-RSB phases) (Kondratiev et al., 2010, Dembo et al., 28 Sep 2024). Metastability, phase coexistence, and chaotic volume dependence are addressed through metastate theory.
• Random Graphs and Spatial Networks: The formalism extends to random graphs with spatial constraints, providing tools to paper percolation thresholds, ground state structure, and finite-cluster properties in spatially embedded networks or random geometric graphs. Unique infinite-volume measures and perfect sampling algorithms are constructed via graphical Markov processes and "clan of ancestors" techniques (Cerqueira et al., 2019).
• Random Permutations and Bose–Einstein Condensation: Spatial random permutations with quadratic costs on the jumps have unique random Gibbs measures at sufficiently high temperature and low density, exhibiting only finite cycle decompositions; this links to path integral representations of Bose–Einstein condensation (Armendáriz et al., 2019).
• Random Partitions and Bayesian Nonparametrics: Gibbs-type exchangeable random partitions and associated random probability measures provide a flexible class of priors for species sampling, clustering, and nonparametric Bayesian inference. Explicit formulas for predictive probabilities, conditional distributions, and large-$n$ approximations of predictive weights underpin practical algorithms (Bacallado et al., 2015, Arbel et al., 2017).
• Random Weak Gibbs Measures and Multifractal Analysis: Random weak Gibbs measures arise in random dynamical systems and in the paper of the multifractal structure of inverse measures associated with random Cantor sets, requiring precise construction techniques to control lower local dimensions and fluctuations (Yuan, 2017, Barral et al., 2015).

5. Theoretical Tools and Technical Advances

• Coupling and Stochastic Domination: Stochastic domination arguments (e.g., Holley's inequality), comparison with Bernoulli percolation, and coupling to dominating or dominated processes are frequently used to establish phase transition boundaries and existence results (Ferrari et al., 2010, Cerqueira et al., 2019).
• Set-Valued Analysis and Komlós Theorem: The existence of measurable random Gibbs measures is established using set-valued maps and Komlós' theorem for Cesàro means to handle chaotic size dependence and ensure almost sure weak convergence of empirical distributions (Kondratiev et al., 2010, Daletskii et al., 2013).
• Loss Network Representations and Clans of Ancestors: In random permutation and Gibbs random graph models, Markov processes (loss networks, birth-death chains) coupled with clan-of-ancestors constructions allow the exact (perfect) simulation of equilibrium random Gibbs measures in infinite volume under non-percolation regimes (Armendáriz et al., 2019, Cerqueira et al., 2019).
• Gamma-Convergence and Large Deviations: Scaling limits are described by Gamma-convergence of the sequence of free energy or energy functionals, ensuring that minimizers (equilibrium states) and large deviation rate functions can be computed from the macroscopic limiting theory. Both positive and negative temperature regimes are treated, with extensions to singular mean field models (Berman, 2016).

6. Non-Gibbsian and Generalized Gibbs Phenomena

• Generalized and Almost Gibbsian Measures: Under renormalization (e.g., block-spin transformations or majority rules with overlap), the image measure may lose strict quasilocality but can often be classified as "almost Gibbsian"—admitting a specification whose discontinuity set has measure zero (D'Achille et al., 2021). This situates such measures at the interface between random Gibbsian and generalized non-Gibbsian descriptions.

7. Computational Aspects and Algorithmic Implications

• Algorithmic Hardness and Sampling: In models such as the continuous random energy model (CREM), there is a sharp threshold $\beta_G$ for the algorithmic hardness of sampling from the random Gibbs measure: for $\beta \leq \beta_G$, efficient (polynomial-time) recursive sampling is feasible via tree-based algorithms; for $\beta > \beta_G$, hardness emerges, and polynomial-time approximation becomes exponentially unlikely (Ho, 2023, Ho et al., 2021). The phase transition in the computational problem reflects the concentration of the Gibbs measure on rare or hierarchically structured minima, with analogous phase transitions observed in the statistical mechanics theory.

8. Examples and Key Formulas Table

Context / Model	Random/Gibbs Feature	Key Formula / Structure
Random lattice spins	Random $J$ couplings in $W_{xy}(u,v)$	$G_p(J):$ DLR solutions with integrability
Random geometric graphs	Point process $\Gamma$, edge lengths	$$H(\omega) = \sum L(e) + \sum \varphi(\omega(\gamma))$$
Exchangeable random partitions	Species sampling models, random EPPF	$$p_j^{(n)}(n_1,\dots,n_j) = V_{n,j}\prod (1-\sigma)_{(n_i-1)\uparrow1}$$
Spherical $p$-spin glasses	Random $H_N$, Parisi measure	Conditional Gaussian laws at centers $\mathbf{x}_i$
Spatial permutations	Poisson points, cycle weights	$$\mu(\sigma) \propto e^{-\alpha \sum_x V(\sigma(x)-x)}$$
Tessellations	Markov kernel $f(x, p_-, dp_+)$	Gibbs measure on piecewise-linear convex functions, kinetic PDEs for $f$

9. Broader Implications and Open Directions

Random Gibbs measures provide a rigorous and unifying mathematical structure for the paper of disordered systems, random graphs, spatial networks, and modern statistical models—enabling the precise formulation and realization of equilibrium, metastable, and dynamically evolving states in the presence of randomness in both interaction and geometry. Their properties are central to the analysis of phase transitions, percolation, equilibrium uniqueness and non-uniqueness, and computational complexity in high-dimensional and disordered systems. The interplay between random environments and Gibbsian structure continues to be an active domain for theoretical advances and algorithmic exploration.