Iota Whitepaper: What is "k" in Section 3?

Question

The IOTA Whitepaper in Section 3 does not seem to define k. Is k the number of verifications required by a transaction as noted earlier in the Whitepaper or something else? If it's the former I do not understand. Can someone people please help me understand?

Section 3

Stability of the system, and cutsets

Let L(t) be the total number of tips in the system at time t. One expects that thestochastic process L(t) remains stable. More precisely, one expects the process to be positive recurrent, see Sections 4.4 and 6.5 of [11] for formal definitions. In particular, positive recurrence implies that the limit of P[L(t) =k] as t → ∞ should exist and be positive for all k ≥1. Intuitively, we expect that L(t) should fluctuate around a constant value, and not escape to infinity. If L(t) were to escape to infinity, many unapproved transactions would be left behind.

Section 1

This is the simplest approach. One may also study similar systems where transactions must approve k other transactions for a general k ≥ 2, or have an entirely different set of rules.

Answer 1

The k from the footnote of section 1 is the required number of transactions to be referenced by new transactions (as is noted by you in your question).

The k from section 3 is just an index/placeholder in the context of number of tips overall (not the number of tips to be referenced), and one could have used any letter here.

For example, The limit of P[L(t) = m] as t → ∞ should exist and be positive for all m ≥1 conveys the same meaning.

Answer 2

In both cases, k is a "variable" used to make the English sentence easier to understand, the variable is both defined and used within the sentence, but nowhere else. The meaning of these two k is therefore different.

So the second example could be rewritten as

One may also study similar systems where each transaction must approve a constant number of other transactions, where this number mentioned in the first half of the sentence needs to be larger than or equal to 2…