A Dynamical System Approach to Phase Transitions for p-Adic Potts Model on the Cayley Tree of Order Two

In the present paper, we introduce a new kind of p-adic measures for (q + 1)-state Potts model, called generalized p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. We employ a dynamical system approach to establish phase transition phenomena for the given model. Namely, using the derived recursive relations we define a one-dimensional fractional p-adic dynamical system. We show that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. In this case, there exists a strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields the existence of a quasi phase transition.

[1]  P. Freund,et al.  Non-archimedean strings , 1987 .

[2]  David K. Arrowsmith,et al.  Some p-adic representations of the Smale horseshoe , 1993 .

[3]  On Gibbs measures of P-adic Potts model on the cayley tree , 2005, math-ph/0510025.

[4]  p-adic valued probability measures , 1996 .

[5]  V. A. Avetisov,et al.  Application of p-adic analysis to models of spontaneous breaking of the replica symmetry , 2008 .

[6]  A. C. M. van Rooij,et al.  Non-Archimedean functional analysis , 1978 .

[7]  S. V. Kozyrev,et al.  Ultrametric random field. , 2006 .

[8]  Andrei Khrennikov,et al.  Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models , 2011 .

[9]  On p-adic quasi Gibbs measures for q + 1-state Potts model on the Cayley tree , 2010 .

[10]  Giorgio Parisi,et al.  On the p-adic five-point function , 1988 .

[11]  Neal Koblitz,et al.  p-adic numbers , 1977 .

[12]  Andrei Khrennikov,et al.  The measure-theoretical approach to $p$-adic probability theory , 1999 .

[13]  D. Arrowsmith,et al.  Geometry of p -adic Siegel discs , 1994 .

[14]  Existence of a Phase Transition for the Potts p-adic Model on the Set ℤ , 2002 .

[15]  F. Mukhamedov.  On the existence of generalized gibbs measures for the one-dimensional p-adic countable state Potts model , 2009 .

[16]  Сергей Владимирович Козырев,et al.  Всплески и спектральный анализ ультраметрических псевдодифференциальных операторов@@@Wavelets and spectral analysis of ultrametric pseudodifferential operators , 2007 .

[17]  I. Volovich.  Number theory as the ultimate physical theory , 1987 .

[18]  p-adic repellers in Qp are subshifts of finite type , 2007 .

[19]  Quantum mechanics andp-adic numbers , 1972 .

[20]  ON INHOMOGENEOUS p-ADIC POTTS MODEL ON A CAYLEY TREE , 2005, math-ph/0510024.

[22]  S. V. Kozyrev.  Ultrametric pseudodifferential operators and wavelets for the case of non homogeneous measure , 2008 .

[23]  S. V. Kozyrev,et al.  Replica Symmetry Breaking Related to a General Ultrametric Space I: Replica Matrices and Functionals , 2022 .

[24]  S. V. Kozyrev,et al.  Application of p-adic analysis to models of breaking of replica symmetry , 1999 .

[25]  Jia-Yan Yao,et al.  Strict ergodicity of affine p-adic dynamical systems on Zp , 2007 .

[26]  E. Thiran,et al.  p-adic dynamics , 1989 .

[27]  A. F. Monna,et al.  Intégration non-archimédienne II , 1963 .

[28]  On $p$-adic Gibbs measures of countable state Potts model on the Cayley tree , 2007, 0705.0244.

[29]  On P-adic λ-model on the Cayley tree , 2004 .

[30]  Sergio Albeverio,et al.  A random walk on p-adics - the generator and its spectrum , 1994 .

[31]  F. Y. Wu.  The Potts model , 1982 .

[32]  WEAK SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS OVER THE FIELD OF $p$-ADIC NUMBERS , 2007, 0708.1706.

[33]  Karl-Olof Lindahl.  On Siegel's linearization theorem for fields of prime characteristic , 2004 .

[34]  Nigel P. Smart,et al.  p-adic Chaos and Random Number Generation , 1998, Exp. Math..

[35]  R. Dobrushin.  The problem of uniqueness of a gibbsian random field and the problem of phase transitions , 1968 .

[36]  Extension of measures to infinite-dimensional spaces over $p$-adic field , 2000 .

[37]  R. Dobrushin.  Prescribing a System of Random Variables by Conditional Distributions , 1970 .

[38]  Andrei Khrennikov.  Generalized probabilities taking values in non-Archimedean fields and in topological groups , 2005 .

[39]  A. Figá-Talamanca,et al.  Diffusion on locally compact ultrametric spaces , 2004 .

[40]  Igor Volovich,et al.  p-adic string , 1987 .

[41]  A. Khrennikov,et al.  Wavelets on ultrametric spaces , 2005 .

[42]  Robert L. Benedetto,et al.  Hyperbolic maps in p-adic dynamics , 2001, Ergodic Theory and Dynamical Systems.

[43]  Sergio Albeverio,et al.  Measure-valued branching processes associated with random walks on $p$-adics , 2000 .

[44]  S Albeverio,et al.  Memory retrieval as a p-adic dynamical system. , 1999, Bio Systems.

[45]  Igor Volovich,et al.  On the p-adic summability of the anharmonic oscillator , 1988 .

[46]  B. Dragovich,et al.  THE WAVE FUNCTION OF THE UNIVERSE AND p-ADIC GRAVITY , 1991 .

[47]  S. Albeverio,et al.  Theory of p -adic Distributions: Linear and Nonlinear Models: References , 2010 .