Generalized p-adic Quasi Gibbs Measures

• Generalized p-adic quasi Gibbs measures are p-adic-valued probability measures on hierarchical trees that extend classical Gibbs frameworks to non-Archimedean settings.
• They are constructed using recursive boundary law equations that connect p-adic dynamical systems with the analysis of fixed point stability and phase transitions.
• The framework rigorously characterizes strong and quasi phase transitions with applications in p-adic quantum mechanics and statistical physics.

A generalized $p$-adic quasi Gibbs measure is a non-Archimedean analogue of classical Gibbs measures designed for spin systems (such as Potts, Ising, SOS, or Hard-Core models) defined on hierarchical structures like Cayley trees. Unlike their real-valued counterparts, these measures take values in the field of $p$-adic numbers $\mathbb{Q}_p$ and are constructed to accommodate the features of $p$-adic probability and ultrametricity. Their construction typically involves recursive compatibility conditions or boundary law equations, leading to deep connections with $p$-adic dynamical systems and enabling a rigorous analysis of phase transitions unique to the non-Archimedean setting.

1. Definition and Formulation

Generalized $p$-adic quasi Gibbs measures (often abbreviated as "quasi Gibbs measures" or "p-adic quasi Gibbs measures") are probability measures on configuration spaces of tree-like graphs, whose values lie in $\mathbb{Q}_p$. For a prototypical $(q+1)$-state Potts model on a Cayley tree of order $k$, the finite-volume measure on the subtree $V_n$ is defined as

$$\mu_h^{(n)}(\sigma) = \frac{1}{Z_n^{(h)}} \, p^{H_n(\sigma)} \prod_{x\in W_n} h_{\sigma(x), x}$$

where:

• $H_n(\sigma)$ is the Hamiltonian, typically of the form $$N\sum_{\langle x, y\rangle\in L_n} \delta_{\sigma(x), \sigma(y)}$$,
• $h_{i,x} \in \mathbb{Q}_p$ are local fields,
• $W_n$ denotes the boundary at the $n$-th level of the Cayley tree,
• $Z_n^{(h)}$ is the $p$-adic partition function ensuring normalization.

A key requirement is the compatibility (consistency) of these finite-volume measures, enforced by the $p$-adic analogue of Kolmogorov's extension theorem: $$\sum_{\omega \in \Omega_{W_n}} \mu_h^{(n)}(\sigma\vee \omega) = \mu_h^{(n-1)}(\sigma)$$ for all boundary configurations $\sigma$ on $V_{n-1}$.

The measure is called "generalized" or "quasi Gibbs" if it is constructed via weighting schemes that may differ from the standard $p$-adic exponential prescription, reflecting the necessity to accommodate $p$-adic analytic subtleties and ultrametric features in the compatibility relation (Mukhamedov, 2010, Mukhamedov et al., 2012).

2. Recursive Relations and the Role of Boundary Laws

The essential mathematical structure underlying these measures is a system of nonlinear, recursive relations for the boundary parameters—frequently termed "boundary laws." For translation-invariant or periodic solutions (i.e., those with $h_x = h$ for all $x$ in a class), the recursion often takes the form: $$\widehat{h}_x = \prod_{y\in S(x)} F(\widehat{h}_y;\theta)$$ with

$$F_i(\widehat{h}; \theta) = \frac{(\theta - 1) \widehat{h}_i + \sum_{j=1}^q \widehat{h}_j + 1}{\sum_{j=1}^q \widehat{h}_j + \theta}, \quad i=1,\dots, q,$$

where $\theta = p^N$ encodes the interaction.

The structure of the Cayley tree, combined with the $p$-adic field’s ultrametricity, enables an explicit reduction to low-dimensional dynamical systems in the presence of high symmetry—most notably in translation-invariant and periodic cases (Mukhamedov, 2010, Khakimov, 2014, Mukhamedov et al., 19 Sep 2025).

3. Dynamical Systems and Fixed Point Classification

The recursive relations naturally define $p$-adic dynamical systems, where fixed points and their stability classify possible infinite-volume measures:

• Translation-invariant solutions correspond to fixed points of the governing dynamical system; for example, for the $(q+1)$-state Potts model on the Cayley tree of order two (Mukhamedov, 2010):

$$x = f(x) = \left(\frac{x+q}{x+\theta+q-1}\right)^2.$$

The existence and type (attractive, neutral, or repelling) of fixed points depend on model parameters (e.g., $p$-divisibility of $q$, $N$ sign).

• In more general settings or for periodic/even weakly periodic solutions, these recurrences may be higher-dimensional or involve periodic cycles, leading to the investigation of the corresponding $p$-adic dynamical system's entire orbit structure (Mukhamedov et al., 2012, Mukhamedov, 2014, Mukhamedov et al., 2017).

The distinct $p$-adic norm behavior of the derivative at a fixed point (i.e., $|f'(x^*)|_p$) determines whether the associated measure is "attractive," "neutral," or "repelling," which in turn is linked to the boundedness properties of the corresponding Gibbs measure and thus to phase coexistence or uniqueness.

4. Phase Transitions and Boundedness

The rich structure of generalized $p$-adic quasi Gibbs measures supports various types of phase transitions:

• Strong phase transition: There exist at least two translation-invariant (or periodic) generalized $p$-adic quasi Gibbs measures, one of which is bounded (i.e., norm stays uniformly finite) and the other unbounded (diverges along a sequence of finite volumes). This can occur when $q$ is divisible by $p$ in ferromagnetic Potts models on the Cayley tree (Mukhamedov, 2010).
• Quasi phase transition: Multiple distinct bounded generalized $p$-adic quasi Gibbs measures coexist (e.g., in antiferromagnetic regimes or when $q$ is not divisible by $p$) (Mukhamedov, 2010, Mukhamedov et al., 2012).

Boundedness is of central importance; only bounded measures yield “physical” $p$-adic probability measures suitable for integrating $p$-adic observable functions (Gandolfo et al., 2011, Khakimov, 2014). The onset of unbounded measures is tied, via the dynamical system structure, to the presence of repelling fixed points and to the arithmetic properties of model parameters.

5. Mathematical Structures and Characteristic Equations

Generalized $p$-adic quasi Gibbs measures are closely associated with the algebraic solutions of parametric polynomial equations arising from the recursive consistency conditions. For the translation-invariant case in the order-$k$ Potts model, the fixed point equation often reduces to a polynomial of degree $k$: $$x = \left(\frac{\theta x + a}{x + \theta a}\right)^k,$$ for suitable choices of $a$ and $\theta$ (Mukhamedov et al., 19 Sep 2025, Khakimov, 2014). The algebraic nature of these equations (quadratic, cubic, or higher) makes their solvability in $\mathbb{Q}_p$ deeply dependent on $p$-adic divisibility and the existence of square or cubic roots in the field.

For periodic measures (e.g., $m$-periodic), the corresponding consistency conditions are encoded in fixed point equations for the $m$-th iterate of the recursion. Their solvability and the abundance of periodic solutions can indicate the presence of chaotic dynamics and a “vastness” of the Gibbs measure set (Mukhamedov et al., 2017, Ahmad et al., 2017).

6. Connections to $p$-adic Probability, Markov Chains, and Boundary Laws

The compatibility and recursion structure of these measures can be reformulated using $p$-adic Markov chain concepts and boundary law approaches. Markov property imposition along the tree leads naturally to boundary law equations whose solution space corresponds to the set of (possibly generalized) $p$-adic quasi Gibbs measures (Ny et al., 2019). The uniqueness (or multiplicity) of solutions is governed by ultrametric estimates for the boundary law recursion and is sensitive to the arithmetic properties of the stochastic matrices and the prime $p$.

These connections tightly couple the theory of $p$-adic Markov processes with that of generalized Gibbs measures, allowing powerful generalization of classical probabilistic and measure-theoretic results into non-Archimedean frameworks.

7. Implications, Applications, and Physical Interpretation

Generalized $p$-adic quasi Gibbs measures form a foundational component in the paper of $p$-adic statistical mechanics and mathematical physics. Their main applications and implications include:

• Providing rigorous models for systems exhibiting ultrametric or hierarchical organization, such as those arising in $p$-adic quantum mechanics, complex systems with replica symmetry breaking, and string theory (Mukhamedov, 2010, Mukhamedov et al., 2012).
• Allowing the systematic paper of phase transitions in non-Archimedean contexts, even for models (such as 1D chains) where classical analogues reveal uniqueness only (Mukhamedov, 2011).
• Enabling a precise characterization of measure-theoretic properties (boundedness, norm divergence) tied to qualitative phase phenomena (strong versus quasi transitions), and their dependence on arithmetic properties of the models (Mukhamedov, 2010, Mukhamedov et al., 2012, Mukhamedov et al., 19 Sep 2025).
• Linking $p$-adic dynamical system theory with statistical mechanics, establishing explicit correspondences between fixed point/stability properties and physical phases, and providing analytic techniques for verifying renormalization predictions (Mukhamedov, 2014, Mukhamedov et al., 19 Sep 2025).

The development of generalized $p$-adic quasi Gibbs measures not only extends the classical theory of Gibbs measures but also opens a route to rigorous mathematical frameworks for models where the conventional real-valued probability paradigm is insufficient, revealing new phenomena tied to the arithmetic and topological properties of $\mathbb{Q}_p$.