1 Introduction

The notion of a Gibbs measure was first introduced by R.L.Dabrushin as well as Lanford and Ruelle [1, 2], makes use of systems of compatible conditional probabilities with respect to the outside of finite subsets, when the outside is fixed in a boundary condition, to reach thereafter infinite-volume quantities. A central problem in the theory of Gibbs measures is to describe infinite (or limiting) Gibbs measures corresponding to a given Hamiltonian (see [5–8]).

In [3] the Potts model with countable set \(\Phi \) of spin values on \(Z^d\) was considered and it was proved that with respect to Poisson distribution on \(\Phi \) the set of limiting Gibbs measure is not empty. In [4] the Potts model with a nearest neighbor interaction and countable set of spin values on a Cayley tree is studied.

It is well known, the XY model is an example with an uncountable single-spin space \(\Omega =\{x \in R^2: \Vert x\Vert _2=1\}\). In [5–8] several models (Hamiltonians) with-nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree were considered. In these papers translation-invariant Gibbs measures are studied via a non-linear functional (integral) equation.

In the present note we continue the investigation from [5] and consider a model with nearest-neighbor interactions and local state space given by the uncountable set [0, 1] on a Cayley tree. Note that, in [5] it was proved that the considered model has at least two translational-invariant Gibbs measure. Here we prove that our model has exactly three translation-invariant Gibbs measures on a Cayley tree of order two.

Let us give basic definitions.

The Cayley tree \(\Gamma ^k\) of order \(k \ge 1\) is an infinite tree, i.e., a graph without cycles, such that exactly \(k+1\) edges originate from each vertex. Let \(\Gamma ^k=(V,L)\) where V is the set of vertices and L the set of edges. Two vertices x and y are called nearest neighbors if there exists an edge \(l \in L\) connecting them and we denote \(l=\langle x,y\rangle \). A collection of nearest neighbor pairs \(\langle x,x_1\rangle ,\langle x_1,x_2\rangle , \dots , \langle x_{d-1},y\rangle \) is called a path from x to y. The distance d(x, y) on the Cayley tree is the number of edges of the shortest path from x and y.

For a fixed \(x^0 \in V\), called the root, we set

$$\begin{aligned} W_n=\{x \in V| d(x,x^0)=n\}, \quad V_n=\bigcup \limits _{m=0}^n W_m \end{aligned}$$

and denote

$$\begin{aligned} S(x)=\{y \in W_{n+1}: d(x,y)=1\}, \quad x \in W_n, \end{aligned}$$

the set of direct successors of x.

Consider models where the spin takes values in the set [0, 1], and is assigned to the vertexes of the tree. For \(A \subset V\) a configuration \(\sigma _A\) on A is an arbitrary function \(\sigma _A:A\mapsto [0,1].\) Denote \(\Omega _A=[0,1]^A\) the set of all configurations on A. A configuration \(\sigma \) on V is then defined as a function \(x \in V \mapsto \sigma (x) \in [0,1]\); the set of all configurations is \([0,1]^V\).

The (formal) Hamiltonian of the model is:

$$\begin{aligned} H(\sigma )=-J \sum \limits _{<x,y> \in L}\xi _{\sigma (x),\sigma (y)}, \end{aligned}$$
(1.1)

where \(J \in R\setminus \{0\}\) and \(\xi : (u,v) \in [0,1]^2 \mapsto \xi _{u,v} \in R\) is a given bounded, measurable function.

Let \(\lambda \) be the Lebesgue measure on [0, 1]. On the set of all configurations on A the a priori measure \(\lambda _A\) is introduced as the |A| fold product of the measure \(\lambda \). Here and further on |A| denotes the cardinality of A. We consider a standard sigma-algebra \(\mathcal {B}\) of subsets of \(\Omega =[0,1]^V\) generated by the measurable cylinder subsets. A probability measure \(\mu \) on \((\Omega , \mathcal {B})\) is called a Gibbs measure (with Hamiltonian H) if it satisfies the DLR equation, namely for any \(n=1,2,...\) and \(\sigma _n \in \Omega _{V_n}\):

$$\begin{aligned} \mu (\{\sigma \in \Omega : \sigma |_{V_n}=\sigma _n\})=\int \limits _{\Omega } \mu (d \omega )\nu _{\omega |_{W_{n+1}}}^{V_n}(\sigma _n), \end{aligned}$$

where \(\nu _{\omega |_{W_{n+1}}}^{V_n}\) is the conditional Gibbs density

$$\begin{aligned} \nu _{\omega |_{W_{n+1}}}^{V_n}(\sigma _n)=\frac{1}{Z_n(\omega |_{W_{n+1}})}\exp (\beta H(\sigma _n \mid |\omega |_{W_{n+1}})), \end{aligned}$$

and \(\beta \ge 0\) is a free parameter proportional to the inverse temperature.

2 Non-uniqueness of Gibbs measures

Let

$$\begin{aligned} C^+[0,1]=\{f\in C[0,1]: f(x)\ge 0\}. \end{aligned}$$

For every \(k\in \mathbb {N}\) we consider an integral operator \(H_{k}\) acting in the cone \(C^{+}[0,1]\) as

$$\begin{aligned} (H_{k}f)(t)=\int ^{1}_{0}K(t,u)f^{k}(u)du, \,\ k\in \mathbb {N}. \end{aligned}$$

The operator \(H_{k}\) is called Hammerstein’s integral operator of order k. This operator is well known to generate ill-posed problems. Clearly, if \(k\ge 2\) then \(H_{k}\) is a nonlinear operator.

It is known that the set of translation invariant Gibbs measures of the model (1.1) is described by the fixed points of the Hammerstein’s operator (see [6]).

In this paper we take \(k=2\) for the model (1.1) and we take concrete \(\xi \) in the following form

$$\begin{aligned} \xi _{t,u}=\xi _{t,u}(\theta ,\beta )=\frac{1}{J \beta }\ln \left( 1+\theta \root 2n+1 \of {4\left( t-\frac{1}{2}\right) \left( u-\frac{1}{2}\right) }\right) , \quad t,u \in [0,1] \end{aligned}$$
(2.1)

where \(0 \le \theta <1\). Then for the Kernel K(t, u) of the Hammerstein’s operator \(H_2\) we have

$$\begin{aligned} K(t,u)=1+\theta \root 2n+1 \of {4\left( t-\frac{1}{2}\right) \left( u-\frac{1}{2}\right) }. \end{aligned}$$

We defined the operator \(V_2:(x,y) \in R^2 \rightarrow (x', y') \in R^2\) by

$$\begin{aligned} V_2: \left\{ \begin{array}{ll} x'=x^2+\frac{2n+1}{2n+3} \root 2n+1 \of {4} \theta ^2 y^2; \\ y'= 2 \cdot \frac{2n+1}{2n+3} \theta x y. \end{array}\right. \end{aligned}$$
(2.2)

Proposition 2.1

A function \(\varphi \in C[0,1]\) is a solution of the Hammerstein’s equation

$$\begin{aligned} (H_2f)(t)=f(t) \end{aligned}$$
(2.3)

iff \(\varphi (t)\) has the following form

$$\begin{aligned} \varphi (t)=C_1 + C_2 \theta \root 2n+1 \of {4\left( t-\frac{1}{2}\right) }, \end{aligned}$$

where \((C_1, C_2) \in R^2\) is a fixed point of the operator \(V_2\) (2.2).

Proof

Necessariness Assume \(\varphi \in C[0,1]\) be a solution of the Eq. (2.3). Then we have

$$\begin{aligned} \varphi (t)=C_1 + C_2 \theta \root 2n+1 \of {4\left( t-\frac{1}{2}\right) }, \end{aligned}$$
(2.4)

where

$$\begin{aligned} C_1=\int \limits _0^1 \varphi ^2(u)\mathrm{d}u, \end{aligned}$$
(2.5)
$$\begin{aligned} C_2=\int \limits _0^1 \root 2n+1 \of {u-\frac{1}{2}}\cdot \varphi ^2(u)\mathrm{d}u. \end{aligned}$$
(2.6)

Substituting the function \(\varphi (t)\) (2.4) into (2.5) we get

$$\begin{aligned} C_1=C_1^2+\frac{2n+1}{2n+3} \root 2n+1 \of {4} \theta ^2 C_2^2, \end{aligned}$$

and substituting the function \(\varphi (t)\) into (2.6) we get

$$\begin{aligned} C_2= 2 \cdot \frac{2n+1}{2n+3} \theta C_1C_2. \end{aligned}$$

Thus, the point \((C_1,C_2) \in R^2\) is a fixed point of the operator \(V_2\) (2.2).

Sufficiency Assume that, a point \((C_1,C_2) \in R^2\) is a fixed point of the operator \(V_2\) define the function \(\varphi (t) \in C[0,1]\) by the equality

$$\begin{aligned} \varphi (t)=C_1 + C_2 \theta \root 2n+1 \of {4\left( t-\frac{1}{2}\right) }. \end{aligned}$$

Then

$$\begin{aligned}&(H_2 \varphi )(t)=\int \limits _0^1 \left( 1+\root 2n+1 \of {4} \theta \root 2n+1 \of {\left( t-\frac{1}{2}\right) \left( u-\frac{1}{2}\right) }\right) \varphi ^2(u)\mathrm{d}u=\int \limits _0^1 \varphi ^2(u)\mathrm{d}u \nonumber \\&\qquad \qquad +\root 2n+1 \of {4} \theta \root 2n+1 \of {t-\frac{1}{2}} \int \limits _0^1 \root 2n+1 \of {u-\frac{1}{2}}\varphi ^2(u) \mathrm{d}u=\int \limits _0^1 \left( C_1 + C_2\theta \root 2n+1 \of {4\left( u-\frac{1}{2}\right) }\right) ^2 \mathrm{d}u \nonumber \\&\qquad \qquad + \root 2n+1 \of {4} \theta \root 2n+1 \of {t-\frac{1}{2}} \int \limits _0^1 \root 2n+1 \of {u-\frac{1}{2}} \left( C_1+ C_2 \theta \root 2n+1 \of {4\left( u-\frac{1}{2}\right) }\right) ^2 \mathrm{d}u \nonumber \\&\quad =C_1^2 \int \limits _0^1 \mathrm{d}u + 2 C_1 C_2 \theta \int \limits _0^1 \root 2n+1 \of {4\left( u-\frac{1}{2}\right) }\mathrm{d}u +\theta ^2 C_2^2\int \limits _0^1 \left( \root 2n+1 \of {4\left( u-\frac{1}{2}\right) }\right) ^2 \mathrm{d}u \nonumber \\&\qquad + \root 2n+1 \of {4} \theta \root 2n+1 \of {t-\frac{1}{2}}\cdot (C^2_1 \int \limits _0^1 \root 2n+1 \of {u-\frac{1}{2}} \mathrm{d}u+2C_1C_2 \theta \root 2n+1 \of {4} \int \limits _0^1 \root 2n+1 \of {\left( u-\frac{1}{2}\right) ^2}\mathrm{d}u \nonumber \\&\qquad +\root 2n+1 \of {16}\theta ^2C_2^2 \int \limits _0^1 \root 2n+1 \of {\left( u-\frac{1}{2}\right) ^3}\mathrm{d}u). \end{aligned}$$
(2.7)

Now, we use the following equalities

$$\begin{aligned} \int \limits _0^1 \root 2n+1 \of {u-\frac{1}{2}} \mathrm{d}u =0; \end{aligned}$$
$$\begin{aligned} \int \limits _0^1 \root 2n+1 \of {(u-\frac{1}{2})^2} \mathrm{d}u =\frac{2n+1}{2n+3} \cdot \frac{1}{\root 2n+1 \of {4}}; \end{aligned}$$
$$\begin{aligned} \int \limits _0^1 \root 2n+1 \of {(u-\frac{1}{2})^3} \mathrm{d}u =0; \end{aligned}$$

Then from (2.7) we get

$$\begin{aligned}&=C_1^2+\frac{2n+1}{2n+3} \root 2n+1 \of {4} \theta ^2 C_2^2+2 \cdot \frac{2n+1}{2n+3} \theta C_1 C_2 \cdot \theta \root 2n+1 \of {4\left( t-\frac{1}{2}\right) }\\&\quad =C_1+C_2 \theta \root 2n+1 \of {4\left( t-\frac{1}{2}\right) }=\varphi (t), \end{aligned}$$

i.e. the function \(\varphi (t)\) is a solution of the Eq. (2.3). \(\square \)

Proposition 2.2

  1. i)

    If \( 0 \le \theta \le \frac{2n+3}{2(2n+1)}\), then the Hammerstein’s operator \(H_2\) has unique (nontrivial) positive fixed point in the C[0, 1];

  2. ii)

    If \(\frac{2n+3}{2(2n+1)}< \theta <1\), then there are exactly three positive fixed points in C[0, 1] of the Hammerstein’s operator.

Proof

It is easy to see, if \(\theta =0\) the Hammerstein’s operator \(H_2\) has unique nontrivial positive fixed points \(\varphi (t)\equiv 1\).

Let \(\theta \ne 0\). We consider the system of equations for a fixed point of the operator \(V_2\):

$$\begin{aligned} \left\{ \begin{array}{ll} x^2+\frac{2n+1}{2n+3} \root 2n+1 \of {4} \theta ^2 y^2=x,\\ 2 \cdot \frac{2n+1}{2n+3} \theta x y= y . \end{array} \right. \end{aligned}$$
(2.8)

Case \(y=0\). We get two solutions (0,0) and (1,0) in the (2.8). By Proposition 3.2 functions

$$\begin{aligned} \varphi (t)=\varphi _0(t)\equiv 0, \,\,\, \varphi (t)=\varphi _0(t)\equiv 1 \end{aligned}$$

are solutions of the equation (2.8).

Case \(y \ne 0\). Then from (2.8) we obtain \(x=\frac{2n+3}{2(2n+1) \theta }\). Hence, from the first equation of (2.8) we get

$$\begin{aligned} y^2=\frac{(2n+3)^2}{2(2n+1)^2 \root 2n+1 \of {4}\theta ^3}\cdot \left( 1-\frac{2n+3}{2(2n+1) \theta }\right) . \end{aligned}$$
(2.9)

Therefore, for \(\theta \ge \frac{2n+3}{2(2n+1)}\) from (2.9) we obtain

$$\begin{aligned} y=y_1^{\pm }= \pm \frac{2n+3}{2 (2n+1) \root 2n+1 \of {2}\theta ^2}\cdot \sqrt{\frac{2(2n+1) \theta - (2n+3)}{2n+1}}. \end{aligned}$$
(2.10)

Consequently, in the case \(0 \le \theta \le \frac{2n+3}{2(2n+1)}\) operator \(V_2\) has two fixed points: (0,0), (1,0) and in the case \(\frac{2n+3}{2(2n+1)}< \theta <1\) the operator \(V_2\) has four fixed points: (0,0), (1,0), \((x_1, y_1^{+})\) and \((x_1, y_1^{-})\), with \(x_1=\frac{2n+3}{2(2n+1) \theta }\).

Note that, there is no any other fixed point of \(V_2\). \(\square \)

Consequently,

$$\begin{aligned} \varphi _1(t)\equiv 1, \end{aligned}$$
$$\begin{aligned} \varphi _2(t)=\frac{2n+3}{2(2n+1) \cdot \theta } \left( 1+\sqrt{\frac{2(2n+1) \cdot \theta -(2n+3)}{2n+1}} \cdot \root 2n+1 \of {2(t-\frac{1}{2})}\right) , \end{aligned}$$
$$\begin{aligned} \varphi _3(t)=\frac{2n+3}{2(2n+1) \cdot \theta } \left( 1-\sqrt{\frac{2(2n+1) \cdot \theta -(2n+3)}{2n+1}} \cdot \root 2n+1 \of {2\left( t-\frac{1}{2}\right) }\right) \end{aligned}$$

are non trivial fixed points of the Hammerstein’s operator \(H_2\). Note that \(\varphi _i(t)>0\), for \(i=1, 2, 3\) and \(t \in [0,1]\). Thus we have proved the following

Theorem 2.3

  1. i)

    If \( 0 \le \theta \le \frac{2n+3}{2(2n+1)}\), then for the model (1.1) on Cayley tree \(\Gamma ^2\) there exists a unique translation-invariant Gibbs measure;

  2. ii)

    If \(\frac{2n+3}{2(2n+1)}< \theta <1\), then for the model (1.1) on Cayley tree \(\Gamma ^2\) there are three translation-invariant Gibbs measures.

Remark

Note that, in [7] the case \(n=1\) of (2.1), is considered. In the case \(n=1\) from Theorem 2.3 we get Theorem 4.2 of [7].