Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees

By Yueyun Hu, Zhan Shi. 2007

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We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Big-gins and Kyprianou [Electron. J. Probab. 10 (2005) 609-631]. Our method applies, furthermore, to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [J. Statist. Phys. 51 (1988) 817-840]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0876d6a8f878f9a97a6228fa3b4e15dcbd53fa87

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Citations in the corpus

A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales (2009, methodology): “Similar ideas have recently been used by Kyprianou [40] to investigate the L 1-convergence of the BBM martingale (1), by Biggins and Kyprianou [4] for multi-type branching processes in discrete time, by Hu and Shi [33] for the minimal position in a branching random walk, by Geiger [16, 17] for Galto”
Gaussian multiplicative chaos and applications: A review (2013, methodology): “This approach is carried out in [58], based on the works [99] for branching Brownian motions or [1, 83] for branching random walks.”
On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, unknown): “In the case of branching Brownian motion and branching random walks, for which a tree structure is explicit, the study goes back to the seminal paper by Bramson [12], and more precise and general results have later been proven by Aı̈dékon, Hu and Shi (see [18, 2, 1]).”
Branching random walks (2015, background): “5) in [137] requires some stronger integrability conditions.”
Minima in branching random walks (2007, background): “Independently of the current work, Hu and Shi [24] have shown that assuming bounded step size, assuming that (III) holds and that E{B1+ε} < ∞ for some ε > 0, then (Mn − Λ′(n))/ log n → (−3/2t) in probability on S but that, surprisingly, (Mn − Λ′(n))/ log n does not converge almost surely.”
Convergence in Law for the Branching Random Walk Seen from Its Tip (2011, methodology): “This change of probability was also used in [16].”
The functional equation of the smoothing transform (2009, background): “Namely, under suitable conditions, the limits of the Seneta-Heyde normed version of W (α) n , see [12] and [23], are endogeneous fixed points.”
MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS (2017, background): “results of Duplantier, Rhodes, Sheﬃeld and Vargas [6] about moments of the total measure of critical Gaussian chaos (which ultimately stem from Kahane’s convexity inequality and results of Hu and Shi [17] for branching random walk), one could get another proof of Theorem 1 for q bounded away from 1.”
The Seneta–Heyde scaling for the branching random walk (2011, background): “1 leads to the following result for the lower limits of min|x|=n V (x), which was proved in [19] under stronger assumptions (namely, E[(∑|x|=1 1)1+δ]+ E[∑|x|=1 e−(1+δ)V (x)] + E[∑|x|=1 e (x)] < ∞ for some δ > 0, and (1.”
Convergence in law of the minimum of a branching random walk (2013, background): “The second order was recently found separately by Hu and Shi [17], and Addario-Berry and Reed [1], and is proved to be equal to 2 lnn in probability, though there exist almost sure fluctuations (Theorem 1.”
On Gaussian multiplicative chaos (2014, background): “inally we would like to remark there exist GMCs that are not subcritical. So far the only examples known to the author are the critical GMCs over logarithmic ﬁelds and their hierarchical counterparts [8, 6, 5]. For these critical GMCs we have for any measurable set A⊂ T EM[A] = ( 0, µ[A] = 0 ∞, µ[A]”
Convergence in law of the minimum of a branching random walk (2011, methodology): “1 ([20],[15]) (i)Under Qa, the branching random walk has the distribution of Ba.”
Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012, background): “) The literature about the analogous problem of branching random walks [2, 43] also suggests for γ = 2 a logarithmically renormalized measure obtained by multiplying by √ log(1/ε) = √ t the object [see (7) below] whose limit is taken in (1), but we expect this to converge (up to constant factor) to ”
Fixed points of the smoothing transform: two-sided solutions (2010, background): “rther, let Z := XN j=1 δS(j) (3.6) and Zn := X |v|=n: S(v)<∞ δS(v), n∈ N0 . (3.7) Then (Zn)n≥0 forms a classical BRW based on the point process Z. BRWs have been studied in many articles, see e.g. [7, 9, 8, 18] and the references therein. 8 We continue with a collection of some known facts about ”