Given a pair of positive integers m and d such that 2 ≤ m ≤ d, for integer n ≥ 0 the quantity bm,d(n), called the partition function is considered; this by definition is equal to the cardinality of the set (a0, a1, . . .) : n = Σkakmk, ak ∈ 0, . . . , d - 1, k ≥ 0. The properties of bm,d(n) and its asymptotic behaviour as n → ∞ are studied. A geometric approach to this problem is put forward. It is shown that C1nλ1 ≤ bm,d(n) ≤ C2nλ2 for sufficiently large n, where C1 and C2 are positive constants depending on m and d, and λ1 = lim/n→∞ log b(n)/log n and λ2 = limn→∞ log b(n)/log n are characteristics of the exponential growth of the partition function. For some pair (m, d) the exponents λ1 and λ2 are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants C1 and C2 are obtained.

Asymptotic behaviour of the partition function

Protasov, Vladimir
2000-01-01

Abstract

Given a pair of positive integers m and d such that 2 ≤ m ≤ d, for integer n ≥ 0 the quantity bm,d(n), called the partition function is considered; this by definition is equal to the cardinality of the set (a0, a1, . . .) : n = Σkakmk, ak ∈ 0, . . . , d - 1, k ≥ 0. The properties of bm,d(n) and its asymptotic behaviour as n → ∞ are studied. A geometric approach to this problem is put forward. It is shown that C1nλ1 ≤ bm,d(n) ≤ C2nλ2 for sufficiently large n, where C1 and C2 are positive constants depending on m and d, and λ1 = lim/n→∞ log b(n)/log n and λ2 = limn→∞ log b(n)/log n are characteristics of the exponential growth of the partition function. For some pair (m, d) the exponents λ1 and λ2 are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants C1 and C2 are obtained.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/123666
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 27
  • ???jsp.display-item.citation.isi??? ND
social impact