p-adic λ-Model on the Cayley Tree

• The p-adic λ-model is a non-Archimedean spin system defined on a Cayley tree with p-adic valued interactions and hierarchical structure.
• It employs recursive renormalization equations to analyze fixed points and classify generalized Gibbs measures, revealing phase transition mechanisms.
• The model integrates p-adic analysis and dynamical systems to distinguish between bounded and unbounded measures, underpinning its theoretical significance.

A $p$-adic $\lambda$-model on the Cayley tree is a non-Archimedean analogue of classical spin systems, defined by associating $p$-adic-valued interactions and measures to statistical mechanics models on tree-like hierarchical graphs. The non-Archimedean geometry, recursive self-similar structure of the Cayley tree, and the dynamical (renormalization) systems framework underpin a distinctly $p$-adic paradigm for phase transitions, measure theory, and dynamical classification. This article presents a detailed overview of the structure, underlying dynamical systems, generalized $p$-adic quasi Gibbs measures, phase transition phenomena, and the analytical results central to the theory of $p$-adic $\lambda$-models on the Cayley tree, drawing primarily from recent developments in the literature, especially (Mukhamedov et al., 19 Sep 2025, Mukhamedov, 2014, Mukhamedov, 2010, Ny et al., 2019).

1. Model Specification and Hamiltonian Structure

In the $p$-adic $\lambda$-model, the configuration space consists of spin variables $\{\sigma(x)\}$ attached to vertices $x$ of a Cayley tree of order $k$ (each vertex has $k+1$ neighbors except the root). The values $\sigma(x)$ are typically elements of a finite set, for example $\Phi = \{-1, 1\}$ (Ising), or $\{1,\ldots, q\}$ (Potts/λ-generalizations). The interaction potential $\lambda(\cdot,\cdot)$ is a symmetric function $\Phi\times\Phi\rightarrow \mathbb{Q}_p$, and the Hamiltonian for a finite volume $V_n$ takes the form

$$H_n(\sigma) = \sum_{\langle x, y \rangle \in L_n} \lambda(\sigma(x), \sigma(y)),$$

where $L_n$ is the set of edges internal to $V_n$.

Finite-volume $p$-adic probability measures are defined by

$$\mu^{(n)}_{h, \rho} (\sigma) = \frac{1}{Z_{n, \rho}^{(h)}} \rho^{H_n(\sigma)} \prod_{x\in W_n} h_{x, \sigma(x)},$$

where $\rho\in\mathbb{Q}_p$ is a parameter controlling the measure's regime (Gibbs/quasi Gibbs), $W_n$ is the boundary of $V_n$, and $h$ specifies local boundary weights. Compatibility (Kolmogorov consistency) under restrictions of measures to increasing volumes is the central condition for constructing infinite-volume Gibbsian objects.

2. Dynamical Systems via Recursion and Renormalization

The key to the $p$-adic $\lambda$-model on the Cayley tree is that measure-theoretic consistency translates to recursion relations (renormalization equations) for the boundary weights $h_{x,\cdot}$. For translation-invariant (or periodic) states, these relations reduce to a finite-dimensional nonlinear functional equation.

For Ising interactions, the recursion becomes (for $k$-regular tree): $$h = \left( \frac{\rho^{\lambda(1,1)} h + \rho^{\lambda(1,-1)}}{\rho^{\lambda(-1,1)} h + \rho^{\lambda(-1,-1)}} \right)^k,$$ and general $\lambda(\cdot, \cdot)$ yield analogous rational recursions.

This equation defines a $p$-adic dynamical system on an appropriate domain (e.g., $$E_p = \{x\in \mathbb{Q}_p : |x|_p=1, |x-1|_p < p^{-1/(p-1)}\}$$ for the Ising/λ-cases). The fixed points, periodic orbits, and dynamical classification (attractive, repelling, neutral) of this map correspond to translation-invariant or periodic generalized $p$-adic quasi Gibbs measures. Phase transition phenomena are thus encoded in the multiplicity and stability structure of these dynamical systems (Mukhamedov et al., 19 Sep 2025, Mukhamedov, 2014, Mukhamedov, 2010).

3. Generalized $p$-adic Quasi Gibbs Measures

A central object is the generalized $p$-adic (quasi) Gibbs measure, constructed from the finite-volume measures via the $p$-adic Kolmogorov extension theorem, provided the boundary recursion admits a solution (fixed point or periodic orbit). The term "quasi" denotes the construction using powers $p^{H_n(\sigma)}$ rather than the $p$-adic exponential, allowing for broader parameter regimes and more general interactions (Mukhamedov, 2010, Mukhamedov, 2014).

In the translation-invariant setting, the measure is determined by a solution $h^*$ of the fixed-point equation: $h^* = F(h^*),$ for the appropriate dynamical function $F$. In the periodic case, $m$-periodic orbits of the dynamical system correspond to $m$-periodic generalized Gibbs measures.

A salient feature in the $p$-adic context is the distinction between bounded and unbounded measures: the ultrametric topology produces the possibility of unbounded probability values (with $|h|_p$ diverging along certain sequences). Boundedness is characterized via the norm of the recursion's parameters and is central both for the probabilistic interpretation and for phase transition analysis (Ny et al., 2019, Mukhamedov, 2010).

4. Phase Transitions: Dynamical and Measure-Theoretic Mechanisms

The existence of more than one (generalized) $p$-adic quasi Gibbs measure signals a phase transition. The type of transition—strong or quasi—depends on both the multiplicity and the boundedness of the resulting measures.

• Strong phase transition: Coexistence of at least one bounded and one unbounded (generalized) $p$-adic quasi Gibbs measure, typically occurring when the dynamical system has both attractive and repelling fixed points—small perturbations can lead orbits to different basins (measures) (Mukhamedov, 2014, Mukhamedov, 2010).
• Quasi phase transition: Multiple bounded (generalized) $p$-adic quasi Gibbs measures, associated with the presence of multiple attractive (or neutral) fixed points and corresponding partitioning of space into distinct basins of attraction.

The dynamical systems formalism provides a precise criterion: the norm (and arithmetic) of parameters such as $|\rho|_p$, values of $\lambda$, and the algebraic nature of solutions of the fixed-point equations (existence in $\mathbb{Q}_p$ determined by quadratic or higher residue conditions) determine the transition regime (Mukhamedov et al., 19 Sep 2025, Mukhamedov, 2014, Mukhamedov et al., 2012). For the Ising embedding (i.e., $\lambda(u,v)=Nuv$), explicit criteria depending on $|\rho|_p$ and $|N|_p$ are available, and complete analysis of fixed point stability can be performed.

5. Analysis of Fixed Points and Basins of Attraction

The classification into attracting, repelling, or neutral fixed points is determined by the $p$-adic norm of the derivative of the recursion map at the fixed point: $$|F'(h^*)|_p<1: \text{attracting}, \qquad |F'(h^*)|_p>1: \text{repelling}, \qquad |F'(h^*)|_p=1: \text{neutral}.$$ The explicit computation of these norms, in terms of the model parameters, underlies the rigorous determination of phase structure. For instance, in the regime where $|\rho|_p=1$, many models (including the Ising and Potts cases) display only bounded measures (possibly multiple), while $|\rho|_p<1$ often yields both bounded and unbounded generalized measures.

Basins of attraction for fixed points can be explicitly described in terms of $p$-adic annuli and spheres, and their structure dictates the stability domains of corresponding measures (Mukhamedov, 2010, Mukhamedov, 2014).

6. Verification of Dynamical Predictions: Rigorous Construction and Applications

Rigorous justification of phase transition predictions entails constructing the generalized measure corresponding to each relevant fixed or periodic point, estimating their (un)boundedness via partition function asymptotics, and demonstrating the dynamical separation (distinctness) of these measures. The pioneering results demonstrate:

• For translation-invariant solutions, the number and nature of fixed points exactly matches the number and type (bounded/unbounded) of generalized measures (Mukhamedov, 2014).
• Presence of periodic orbits in the dynamical system, especially those corresponding to symbolic subshifts, leads to existence of, for example, infinitely many periodic $p$-adic Gibbs measures and even chaotic behavior as in the $p$-adic Ising-Vannimenus model (Mukhamedov et al., 2017, Mukhamedov et al., 19 Sep 2025).
• Non-Archimedean functional analysis tools (e.g., contraction mapping principle in $p$-adic Banach spaces, Hensel's lemma for the existence of roots) are essential for proving existence, uniqueness, and multiplicity claims.

These match and generalize predictions based solely on the dynamics of the recursion, confirming the equivalence between dynamical phase structure and measure-theoretic phase structure in the $p$-adic $\lambda$-model (Mukhamedov et al., 19 Sep 2025, Mukhamedov, 2014, Mukhamedov, 2010). The approach is not transferable to the real case, due to the strong triangle inequality and fundamentally ultrametric nature of $\mathbb{Q}_p$.

7. Extensions, Context, and Comparative Analysis

The $p$-adic $\lambda$-model on the Cayley tree encapsulates and generalizes other models (Ising, Potts, hard-core), each with its specific recursion structure and phase diagram (Mukhamedov, 2010, Gandolfo et al., 2011, Mukhamedov, 2012). Comparative studies show:

• The precise form of recursion distinguishes $\lambda$-models from Potts and hard-core models; for example, Potts fixed point equations are typically quadratic or cubic rational maps, while $\lambda$-models may involve higher-degree fractions depending on the parametrization.
• In $p$-adic boundary law and Markov chain approaches (Ny et al., 2019), the uniqueness/multiplicity of $p$-adic Gibbs measures can be derived from properties of associated stochastic matrices.
• The measure-theoretic and dynamical systems approaches align: contexts with unique boundary law solutions (trivial fixed points) yield unique $p$-adic measures (no phase transition), while non-trivial solutions and cycles yield multiple measures (phase transition or complexity).

Moreover, in the countable-state setting (Potts models with infinitely many spin values), results suggest the existence theory is often independent of $p$ and determined by contractivity and regularity conditions on the weights (Mukhamedov, 2012, Rozikov et al., 2022).

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In conclusion, the $p$-adic $\lambda$-model on the Cayley tree is a paradigm where the structure and multiplicity of generalized $p$-adic quasi Gibbs measures—and hence the occurrence of phase transitions—are governed by the fixed point and dynamical properties of $p$-adic renormalization equations. The method synthesizes renormalization, $p$-adic analysis, and non-Archimedean measure theory, confirming that phase transitions correspond exactly to the qualitative changes (number, boundedness, and stability) of the fixed points and periodic orbits arising in the associated $p$-adic dynamical system (Mukhamedov et al., 19 Sep 2025, Mukhamedov, 2014, Mukhamedov, 2010).