Gamma Profiles of Waves Entering a Condensate

Definitions and mathematical statement

Let $\mu$ denote the mutant-fitness distribution named in the supplied record, and let $\alpha>0$. The stated upper-tail hypothesis is \[ \mu\left((1-\varepsilon,1)\right)=\varepsilon^\alpha\ell(\varepsilon) \qquad (\varepsilon\downarrow0), \] where $\ell$ is positive sufficiently near zero and slowly varying at zero: for every $c>0$, \[ \lim_{\varepsilon\downarrow0}\frac{\ell(c\varepsilon)}{\ell(\varepsilon)}=1. \] The interval in this hypothesis is open at both endpoints, as in the supplied statement. The full fitness space and the treatment of possible mass at fitness $1$ are not specified.

Write $\Xi_t$ for the empirical fitness distribution at time $t$ in the model intended by the record, and write $\gamma(\beta)$ for its condensate mass. These are source-designated objects: the population dynamics, empirical normalization, parameter $\beta$, initial conditions, and conditions ensuring condensation are not supplied. For $\alpha>0$, define \[ \Gamma(\alpha)=\int_0^\infty e^{-y}y^{\alpha-1}\,{\,\mathrm{d}} y, \qquad G_\alpha(x)=\frac{1}{\Gamma(\alpha)}\int_0^x e^{-y}y^{\alpha-1}\,{\,\mathrm{d}} y \quad (x\geq0). \] Thus $G_\alpha$ is the distribution function of a gamma distribution with shape $\alpha$ and rate $1$.

The supplied conjecture says that the suitably rescaled wave entering the condensate has a gamma-distribution shape. Its supplied explicit candidate limit, for a positive scaling argument $x$, is \[ \lim_{t\to\infty}\Xi_t\!\left(\left(1-\frac{x}{t},1\right)\right) =\gamma(\beta)G_\alpha(x) =\frac{\gamma(\beta)}{\Gamma(\alpha)}\int_0^x e^{-y}y^{\alpha-1}\,{\,\mathrm{d}} y. \] Here $\Xi_t((a,b))$ denotes the empirical mass of the open interval $(a,b)$. The source does not explicitly quantify $x$, specify the mode of stochastic convergence, or state whether time is discrete or continuous. Consequently, the display records the proposed profile but does not determine a complete probabilistic assertion.

The statement names a stochastic house-of-cards condensation model, whereas the supplied literature assessment identifies the unresolved target as a reinforced branching model and distinguishes it from a simpler Kingman model. These descriptions cannot be identified precisely from the supplied information alone. In particular, no result for the simpler model is asserted here to establish the displayed conjecture for the intended model.

Short English statement

The conjecture proposes that fitness mass within distance $x/t$ below fitness $1$ tends to the condensate mass multiplied by a gamma distribution function of shape $\alpha$ and rate $1$. The precise model, condensation hypotheses, quantification of $x$, and convergence mode require clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30002926$, title "Gamma Profiles of Waves Entering a Condensate," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2015$ and supplies no named proposer.
• Supplied source 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 citation year is reproduced as supplied; its relationship to the proposed year $2015$ is not explained in the record.
• Supplied literature source: "Dereich--Mailler--Morters, reinforced branching condensation." 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 states that the gamma profile is proved for a simpler Kingman model, while the gamma-shape conjecture for the condensation wave in the reinforced branching model remains unproved. This is supplied dated curation, not an independently verified assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its verification note reports that the official report was checked and that the reconstruction restores the regular-variation hypothesis and explicit gamma-profile conjecture. These are supplied provenance claims; no independent source consultation or mathematical verification is claimed here.