Limit Theorems in the Condensation Regime

Definitions and mathematical statement

Let $\beta$ denote the mutation probability and let $\mu$ denote the mutant-fitness distribution in the stochastic house-of-cards model. Thus $\beta\in[0,1]$; the displayed source criterion treats $\mu$ as a probability measure on $[0,1]$. Let $N(t)$ denote the population size at time $t\geq0$. The supplied record does not specify the stochastic dynamics, initial population, or additional assumptions on $\mu$, so these data require clarification before a complete probabilistic statement can be given.

Assume the stated condensation condition \[ \beta\int_{[0,1]}\frac{1}{1-x}\,\mu(\mathrm{d}x)<1. \] The integral has a singularity at $x=1$. For $\beta>0$, this condition requires a finite integral and therefore $\mu(\{1\})=0$. The source does not specify whether the endpoint mutation probabilities are allowed or how to interpret the criterion when $\beta=0$ and the integral is infinite.

The supplied statement reports exponential growth rate \[ \lambda^*=1-\beta, \] while describing the growth as not strictly exponential. The problem is to determine the correction to $e^{\lambda^*t}$ that yields a nondegenerate limit theorem for $N(t)$.

For example, a possible normalization target would be to identify a deterministic function $a(t)>0$ and a finite random variable $W$, not almost surely constant, such that \[ \frac{N(t)}{e^{(1-\beta)t}a(t)} \xrightarrow[t\to\infty]{\mathrm{d}} W. \] This display is an explicitly provisional interpretation of the requested limit theorem, not an additional assertion from the source. The source does not prescribe deterministic normalization, convergence in distribution, the meaning of nondegeneracy, or the precise meaning of "not strictly exponential." Its supplied literature assessment refers to a conjectural logarithmic correction but gives no formula or fitness-tail hypotheses that would specify such a correction.

Short English statement

In the stated condensation regime, determine how to correct the reported exponential population growth $e^{(1-\beta)t}$ to obtain a nondegenerate limit theorem. A complete formulation requires the model dynamics, fitness assumptions, normalization class, and convergence mode, which the supplied record does not specify.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002925$, source problem code catalogue entry, title "Limit Theorems in the Condensation Regime," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Interplay of Analysis and Probability in Applied Mathematics (2016), Oberwolfach Reports, pp. 1989--2064, DOI 10.4171/owr/2015/35. https://doi.org/10.4171/owr/2015/35. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Dereich--Mailler--Morters, Non-extensive condensation in reinforced branching. https://arxiv.org/abs/1601.08128.
• Supplied literature source: "Oberwolfach Report on condensation." https://doi.org/10.4171/owr/2015/35.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the reinforced-branching literature proves the exponential growth rate and non-extensive condensation, while the precise logarithmic correction and associated growth limit remain conjectural. This is supplied dated curation, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that the official report was checked to restore the stochastic growth model, condensation criterion, population variable, exponential rate, and object requiring correction. These are supplied verification claims, not source checks performed for this draft. The original recorded question asks what corrections to the non-strictly exponential growth in the condensation regime are needed to prove a limit theorem.