Is directed percolation class for synchronization transition robust with multi-site interactions?

Manoj C. Warambhe () and Prashant M. Gade
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Manoj C. Warambhe: GH Raisoni Skill Tech University
Prashant M. Gade: RTM Nagpur University

The European Physical Journal B: Condensed Matter and Complex Systems, 2025, vol. 98, issue 4, 1-12

Abstract: Abstract Coupled map lattice with pairwise local interactions is a well-studied system. However, in several situations, such as neuronal or social networks, multi-site interactions are possible. In this work, we study the coupled Gauss map in one dimension with 2-site, 3-site, 4-site and 5-site interaction. This coupling cannot be decomposed in pairwise interactions. We coarse-grain the variable values by labeling the sites above $$x^{\star }$$ x ⋆ as up spin (+ 1) and the rest as down spin (– 1) where $$x^{\star }$$ x ⋆ is the fixed point. We define flip rate F(t) as the fraction of sites i such that $$s_{i}(t-1) \ne s_{i}(t)$$ s i ( t - 1 ) ≠ s i ( t ) and persistence P(t) as the fraction of sites i such that $$s_{i}(t')=s_{i}(0)$$ s i ( t ′ ) = s i ( 0 ) for all $$t' \le t$$ t ′ ≤ t . The dynamic phase transitions to a synchronized state is studied above quantifiers. For 3 and 5 sites interaction, we find that at the critical point, $$F(t) \sim t^{-\delta }$$ F ( t ) ∼ t - δ with $$\delta =0.159$$ δ = 0.159 and $$P(t) \sim t^{-\theta }$$ P ( t ) ∼ t - θ with $$\theta =1.5$$ θ = 1.5 . They match the directed percolation (DP) class. Finite-size and off-critical scaling is consistent with DP class. For 2 and 4 site interactions, the exponent $$\delta $$ δ and behavior of P(t) at critical point changes. Furthermore, we observe logarithmic oscillations over and above power-law decay at the critical point for 4-site coupling. Thus multi-site interactions can lead to new universality class(es). Graphical abstract

Date: 2025
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DOI: 10.1140/epjb/s10051-025-00928-z

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