> Архив сети: `01-research-reports/R4-branching-random-walks`
# R4 — Branching random walks: what we derived is textbook, and one real gap
Literature check, Aug 2026. Verdict: the probability is standard; one physics dictionary is a genuine gift; one hoped-for bridge does not exist.
## Everything we computed is already known
- **Front speed** gamma = sigma*sqrt(2 log(1+beta)) is the Biggins/Hammersley-Kingman
velocity gamma = inf_theta (1/theta) log(m E[e^{theta X}]) for Gaussian steps.
Hammersley 1974; Kingman 1975; Biggins, J. Appl. Prob. 14, 630 (1977).
- **Three lines** (two fronts + centre of mass) = Biggins' asymptotic shape theorem,
Adv. Appl. Prob. 10, 62 (1978). Rescaled occupied region -> [-gamma, +gamma], shifted by drift.
- **Centre of mass at exactly -s** is trivial: displacement mean is -s and branching is
independent of displacement. It is a martingale after removing the drift, hence an
average over exponentially many particles => small fluctuations.
- **Linear width, exponential population, exploding density** = Biggins' growth rates,
Z. Wahrsch. 48, 17 (1979): particles near position a*n grow like exp(n(log m - I(a))),
positive inside, vanishing exactly at the front.
- **GW binary criticality p = 1/2** — classical (Athreya-Ney 1972).
- **Critical GW with immigration**: mean grows linearly, extinction probability 0,
and X_n/n converges to a GAMMA law with shape 2a/b (a = immigration mean,
b = offspring variance). Seneta, JRSS B 32, 149 (1970); Pakes 1971.
- **Threshold crossing / level birth** = branching random walk with absorption at a
linear barrier. Critical slope EQUALS the front speed: Kesten 1978 (BBM dies a.s. iff
drift >= sqrt(2)); Aidekon & Jaffuel, SPA 121, 1901 (2011).
## What we MISSED, and it explains our own discrepancy
**The Bramson logarithmic correction.** R_t = gamma*t - (3/(2 lambda*)) log t + O(1).
Bramson, CPAM 31, 531 (1978); Addario-Berry & Reed, Ann. Prob. 37 (2009); Hu & Shi (2009).
Recentred maximum converges in law to a randomly-shifted Gumbel: Aidekon,
Ann. Probab. 41(3A), 1362 (2013); Lalley-Sellke (1987) for BBM.
Our measured front speeds were systematically BELOW gamma and we treated it as noise.
It is not noise. Verified:
t mean R_t R_t/t gamma std(R_t) rel. speed scatter
30 32.438 1.0813 1.1774 2.332 7.19 %
60 65.023 1.0837 1.1774 2.222 3.42 %
120 135.482 1.1290 1.1774 2.034 1.50 %
240 274.227 1.1426 1.1774 2.100 0.77 %
480 553.826 1.1538 1.1774 2.972 0.54 %
- **std(R_t) stays O(1)** (2.0-3.0, no growth with t) — exactly the Gumbel prediction.
- **Relative speed scatter falls 7.19 % -> 0.54 %.** Our earlier "13 % scatter at t=60"
was a finite-time artefact, not a property of the process.
- Speed creeps up toward gamma from below, as predicted.
- Naive fit gives c = 2.83 vs theory 3/(2 lambda*) = 1.274. The excess is our own
population CAP: keeping the top N particles is a Brunet-Derrida selection, whose
speed deficit is ~pi^2/(2 lambda*^2 (ln N)^2). For N = 120000 that is ~0.026, and the
observed residual deficit at t=480 is 0.023. Consistent. Fix: uncapped runs at small t,
or correct for the (ln N)^-2 term.
Brunet & Derrida, PRE 56, 2597 (1997); rigorous: Berard & Gouere, CMP 298 (2010);
Mueller-Mytnik-Quastel, Invent. Math. 184 (2011).
## The real gift: Derrida-Spohn
**Derrida & Spohn, "Polymers on disordered trees, spin glasses, and traveling waves",
J. Stat. Phys. 51, 817 (1988).** The BRW *is* the directed polymer on a Cayley tree.
Free energy = travelling-wave speed. The glass transition IS the front-selection
transition. This is the dictionary between our log/soph dynamics and spin glasses.
Consequences we can use:
- BRW is the canonical **log-correlated field**; exponentiating gives Gaussian
multiplicative chaos, which is genuinely **multifractal** with spectrum = Legendre
transform of the same rate function I(a) from Biggins. Rigorous: Bertacco,
arXiv:2206.06982 (EJP 2023).
- **Freezing transition** at beta_c where the Gibbs measure condenses on the extremes.
Derrida REM (PRL 45, 79, 1980) -> GREM -> BRW is a one-parameter family;
Kistler & Schmidt, arXiv:1503.04064.
- So "the spectrum of log states" has a precise meaning: a multifractal spectrum with
a freezing point, computable from the rate function.
## Critical barrier = the hierarchy mechanism
Absorption at slope -s: dies a.s. iff s >= gamma, survives iff s < gamma.
**At exactly s = gamma** the process dies a.s. but survives a long time, and the
genealogy converges to the **Bolthausen-Sznitman coalescent** — a hierarchical,
multi-scale tree. Berestycki, Berestycki & Schweinsberg, Ann. Probab. 41, 527 (2013);
survival probability arXiv:1212.3821 (PTRF 160, 489, 2014); physics predictions
Derrida & Simon, EPL 78, 60006 (2007).
This is the rigorous version of "each threshold crossing is a new level", and the
interesting regime is exactly critical.
## FKPP correspondence
McKean, CPAM 28, 323 (1975): P(max of BBM >= x) solves the FKPP equation; the front
distribution IS a travelling wave. Pulled-front speed 2 sqrt(D beta) = sigma sqrt(2 beta),
the small-beta limit of our sigma sqrt(2 log(1+beta)). Universal velocity relaxation
v(t) = v* - 3/(2 lambda* t) + ... : Ebert & van Saarloos, PRL 80, 1650 (1998);
Physica D 146, 1 (2000). Same "3/2" as Bramson, physics and probability in two languages.
## X — the Hofstadter bridge does NOT exist
We found the Hofstadter butterfly as a "wave fractal" and separately the BRW multifractal,
and hoped to link them. **There is no such link in the literature, and the structural
reason is solid:** Hofstadter/almost-Mathieu fractality is DETERMINISTIC and
number-theoretic (continued fractions of the flux alpha); BRW multifractality is
STOCHASTIC and log-correlated. There is no random field on the almost-Mathieu problem
for a BRW/GMC structure to live on.
Also our box dimension ~0.5 needs correcting:
- dim_H <= 1/2 for ALL irrational frequencies is a THEOREM (Jitomirskaya & Krasovsky,
arXiv:1909.04429) — a sharp upper bound, matching the old Thouless conjecture.
- But the true Hausdorff dimension is **frequency dependent**: exactly 0 for a dense
G_delta of frequencies (Last & Shamis, CMP 348, 729, 2016) and strictly positive for
a complementary dense set (Helffer, Liu, Qu & Zhou, CMP 368, 369, 2019).
- Cantor spectrum for all irrational alpha: Avila & Jitomirskaya, Ann. Math. 170, 303 (2009).
So "0.5" is a bound, not a universal value. Any claim pinning it at exactly 1/2 for
generic frequencies would contradict Last-Shamis.
Where the log-correlated formalism DOES apply in physics: multifractality of critical
wavefunctions at DISORDER-driven transitions — 2D Dirac fermions in a random magnetic
field (Chamon, Mudry & Wen, PRL 77, 4194, 1996, who map the exponents to a directed
polymer on a Cayley tree), the Anderson transition (Evers & Mirlin, RMP 80, 1355, 2008),
the quantum Hall transition. If the "wave fractal" can be recast as a critical
DISORDERED wavefunction rather than a quasi-periodic one, the whole machinery opens.
## Do NOT claim
**KPZ / Tracy-Widom.** The tree BRW is mean-field: its fluctuations are Gumbel and O(1)
with a log correction, NOT t^{1/3} Tracy-Widom. Directed polymers in 1+1 finite
dimensions are KPZ (Borodin-Corwin-Ferrari, arXiv:1204.1024); the tree is not.
## Novelty verdict
Blunt: the probabilistic content is entirely textbook. Potentially new is only the
FRAMING — using threshold crossings to define a discrete hierarchy of levels — and the
mechanism there is already covered by barrier-absorption theory, so expect to reproduce
gamma/spacing with Bramson corrections rather than a surprise.
The one uncharted item is the BRW <-> Hofstadter bridge, and it is uncharted for a good
structural reason.
## Next actions
1. Rerun uncapped (small t) or correct for the (ln N)^-2 Brunet-Derrida deficit, then
fit R_t = gamma t - c log t and check c ~ 3/(2 lambda*). Recovering c validates the code.
2. Recast the hierarchy as absorption at slope -s; tune s -> gamma and look for the
Bolthausen-Sznitman / Brunet-Derrida near-critical regime.
3. If a time-INHOMOGENEOUS barrier is used (thresholds spaced like sqrt(t), or a varying
variance schedule), we leave the classical regime and enter Fang-Zeitouni /
Mallein territory — t^{1/3} corrections, genuinely less solved. That is the one place
a real contribution is plausible.
## Куда дальше
- Узел: Отчёты R1–R5: https://sdominanta.net/in/works
- Ядро Алеф-теории: https://sdominanta.net/in/aleph
- Вход для моделей: https://sdominanta.net/llms.txt