标题： 关于可数状态Potts模型在Cayley树上的 $p$-adic Gibbs测度 显示英文标题

标题： On $p$-adic Gibbs measures of countable state Potts model on the Cayley tree

摘要： 在本文中，我们考虑了Cayley树上的可数状态$p$-adic Potts模型。给出了依赖于权重$\l$的$p$-adic Gibbs测度的构造，并将此类测度的研究归结为考察一个无穷维递归方程。在某些关于权重的条件下研究该方程，我们证明了相变的不存在性。注意，该条件与素数$p$的取值无关，并且当自旋数量有限时，类似的结论并不成立。对于齐次模型，在上述关于权重的条件下，表明递归方程只有一个解。这意味着仅存在一个$p$-adic Gibbs测度$\m_\l$。还建立了该测度的有界性。 此外，还证明了测度 $\m_\l$ 关于 $\l$ 的连续依赖性。 最后我们给出关于 $\m_\l$ 的一个极限定理。 显示英文摘要

摘要： In the present paper we consider countable state $p$-adic Potts model on the Cayley tree. A construction of $p$-adic Gibbs measures which depends on weights $\l$ is given, and an investigation of such measures is reduced to examination of an infinite-dimensional recursion equation. Studying of the derived equation under some condition on weights, we prove absence of the phase transition. Note that the condition does not depend on values of the prime $p$, and an analogues fact is not true when the number of spins is finite. For homogeneous model it is shown that the recursive equation has only one solution under that condition on weights. This means that there is only one $p$-adic Gibbs measure $\m_\l$. The boundedness of the measure is also established. Moreover, continuous dependence the measure $\m_\l$ on $\l$ is proved. At the end we formulate one limit theorem for $\m_\l$.