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SUMMARY:Ising-Markov chains equivalence as an inverse problem of statistic
 al physics
DTSTART;VALUE=DATE-TIME:20260922T125000Z
DTEND;VALUE=DATE-TIME:20260922T125500Z
DTSTAMP;VALUE=DATE-TIME:20260914T044159Z
UID:indico-contribution-505@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Oleg Usatenko (A.Ya. Usikov Institute for Radiophysi
 cs and Electronics Ukrainian Academy of Science)\nAn important problem in 
 the statistical description of big data random sequences is identifying ma
 croscopic characteristics that summarize the essential properties of the s
 ystem while ignoring microscopic details. These characteristics are analog
 ous to thermodynamic variables that describe the macroscopic state of a ph
 ysical system composed of many interacting degrees of freedom. Markov chai
 ns provide a natural starting point for such an analysis. Classical Nth-or
 der Markov chains allow one to describe correlations extending over long h
 istories but suffer from an exponential growth of parameters with increasi
 ng memory depth. Additive Nth-order Markov chains offer a principled reduc
 tion of this complexity by decomposing the conditional probability of the 
 next symbol into a superposition of contributions from different delay pos
 itions.\nIn Ref. [1]\, we established an exact correspondence of one- and 
 two-step-wise MC with one-dimensional Ising models with finite-range inter
 actions\, thereby introducing the macroscopic temperature to MC. We demons
 trated that information temperature could be defined through two independe
 nt approaches: (i) matching conditional probabilities\, and (ii) solving a
 n entropy-based inverse problem of statistical physics. In particular\, se
 e Ref. [2]\, it was understood that a complex system can be characterized 
 by temperature\, or in other words\, temperature is a fairly informative p
 arameter characterizing the complexity of a system.\nThe main result of th
 e work [3] is the establishment of a correspondence between an additive mu
 ltistep chain and a chain with a stepwise memory function. This correspond
 ence allows the introduction of the concept of information temperature not
  only for stepwise but also for general additive Nth-order Markov chains. 
 \nThe present work substantially extends that framework in several directi
 ons. First\, in addition to the two previous methods for introducing the i
 nformation temperature\, in this work\, we present a method based on joint
  probability. Second\, the equivalence between Markov chains and Gibbs-Isi
 ng representations is formulated within a constructive inverse statistical
  mechanics framework\, thereby making explicit the assumptions of positivi
 ty\, compatibility\, stationarity\, and finite memory. Third\, three compl
 ementary constructions of the information temperature are systematically c
 ompared and shown to be mutually consistent for the unbiased one-parameter
  step-wise Markov chain. Fourth\, the entropy-based definition is reformul
 ated as an inverse problem with an auxiliary energy scale\, clarifying its
  relation to the true Ising energy. Finally\, the analysis of the two-para
 meter inverse problem establishes the limits of this equivalence\, demonst
 rating that the coincidence of the three temperature constructions is a sp
 ecial property of the one-parameter model rather than a generic feature of
  finite-memory symbolic processes.\nThe primary objective for the future i
 s to understand the importance of introducing a macroscopic temperature pa
 rameter for characterizing Large Linguistic Models.\n\n[1] O. V. Usatenko\
 , S. S. Melnyk\, G. M. Pritula\,  and V. A. Yampol'skii\,   Phys. Rev. E  
 106\, 034127 (2022). \n[2]  O. V. Usatenko and G. M. Pritula\,  Chaos\, So
 litons and Fractals 202 (2026) 117452.\n[3] O. V. Usatenko\, G. M. Pritula
 \, and S. S. Melnyk\, Phys. Rev. E  114\, 014116 (2026).\n\nhttps://indico
 .bitp.kiev.ua/event/18/contributions/505/
LOCATION:Bogolyubov Institute for Theoretical Physics Conference Hall
URL:https://indico.bitp.kiev.ua/event/18/contributions/505/
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