Page 172 - Special Topic Session (STS) - Volume 4

STS579 Elena Yarovaya

where

The case of weekly supercritical BRWs with a few branching sources of
various intensities is quite complicated. That is why below we consider the case
of one source of branching, that is, = 1 . If = 1 and the spectrum
(ℋ ) of the operator ℋ 1 contains for β1 ↓ βc a leading eigenvalue λ0(β1) →

1
0, the asymptotic behavior of λ0(β1), as β1 ↓ βc, see (Yarovaya, 2017c), has the
following form.
Theorem 4 Under Condition (3) the eigenvalue λ0(β) of the operator for β

↓ βc has the following asymptotic behavior:

2
(i) λ0(β) ~ c1β for d = 1,
(ii) λ0(β) ~ e −c2/β for d = 2,
2
(iii) λ0(β) ~ c3(β − βc) for d = 3,
−1
−1
(iv) λ0(β) ~ c4(β − βc) ln ((β − βc) ) for = 4,
(v) λ0(β) ~ cd(β − βc) for ≥ 5,
where , ∈ , are some positive constants.

Theorem 5 Under Condition (4) the eigenvalue λ0(β) of the operator for β

↓ βc has the following asymptotic behavior:

(i) λ0(β) ~ cd,α(Nβ) α/α−1 = 1, ∈ (1,2),
(ii) λ0(β) ~ e −cd,α/(Nβ) = 1, = 1,
(iii) λ0(β) ~ cd,α(β − βc) α/d-a = 1, ∈ (1/2, 1) = 2, ∈
(1,2) = 3, ∈ (3/2, 2),
(iv) λ0(β) ~ e W(−cd,α(β−βc)) = 1, = 21 = 2, = 1 =
3, = 3/2,
(v) λ0(β) ~ cd,α(β− βc) = 1, ∈ (0, 1/2) = 2, ∈
(0,1) = 3, ∈ (0, 3/2 ) ≥ 4, ∈ (0,2),

where , is some positive constant (for each fixed values of the parameter α
and dimension d of the lattice ℤ ), and () is the lower branch of the

Lambert W -function satisfying the condition () → −∞ for x ↑ 0.

Based on (Yarovaya, 2007), for = 1 and 0 > 0c we get that equations (5)
have the form
(6)

where

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