Abstract
The renormalization group dynamics is studied in the four-component fermionic hierarchical model in the space of coefficients that determine the Grassmann-valued density of the free measure. This space is treated as a two-dimensional projective space. If the renormalization group parameter is greater than 1, then the only attracting fixed point of the renormalization group transformation is defined by the density of the Grassmann δ-function. Two different invariant neighborhoods of this fixed point are described, and an algorithm is constructed that allows one to classify the points on the plane according to the way they tend to the fixed point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. D. Missarov, “Renormalization group and renormalization theory in p-adic and adelic scalar models,” in Dynamical Systems and Statistical Mechanics, Ed. by Ya. G. Sinai (Am. Math. Soc., Providence, RI, 1991), Adv. Sov. Math. 3, pp. 143–164.
V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (Fizmatlit, Moscow, 1994; World Sci., Singapore, 1994).
M. D. Missarov, “Continuum limit in the fermionic hierarchical model,” Teor. Mat. Fiz. 118(1), 40–50 (1999) [Theor. Math. Phys. 118, 32–40 (1999)].
M. D. Missarov, “Renormalization group in a fermionic hierarchical model in projective coordinates,” Teor. Mat. Fiz. 173(3), 355–362 (2012) [Theor. Math. Phys. 173, 1637–1643 (2012)].
E. Yu. Lerner and M. D. Missarov, “Fixed points of renormalization group for the hierarchical fermionic model,” J. Stat. Phys. 76(3–4), 805–817 (1994).
E. Yu. Lerner and M. D. Missarov, “Global renormalization group flow and thermodynamic limit for the hierarchical fermionic model,” Teor. Mat. Fiz. 107(2), 201–212 (1996) [Theor. Math. Phys. 107, 579–588 (1996)].
M. D. Missarov, “RG-invariant curves in the fermionic hierarchical model,” Teor. Mat. Fiz. 114(3), 323–336 (1998) [Theor. Math. Phys. 114, 255–265 (1998)].
M. D. Missarov, “Critical phenomena in the fermionic hierarchical model,” Teor. Mat. Fiz. 117(3), 471–488 (1998) [Theor. Math. Phys. 117, 1483–1498 (1998)].
M. D. Missarov, “Dynamical phenomena in the hierarchical fermionic model,” Phys. Lett. A 253, 41–46 (1999).
M. D. Missarov, “Renormalization group solution of fermionic Dyson model,” in Asymptotic Combinatorics with Application to Mathematical Physics, Ed. by V. A. Malyshev and A. M. Vershik (Kluwer, Dordrecht, 2002), NATO Sci. Ser. II: Math. Phys. Chem. 77, pp. 151–166.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © M.D. Missarov, A.F. Shamsutdinov, 2014, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Vol. 285, pp. 221–231.
Rights and permissions
About this article
Cite this article
Missarov, M.D., Shamsutdinov, A.F. An algorithm for studying the renormalization group dynamics in the projective space. Proc. Steklov Inst. Math. 285, 211–221 (2014). https://doi.org/10.1134/S0081543814040142
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543814040142