By Francis Comets
Analyzing the section transition from diffusive to localized habit in a version of directed polymers in a random setting, this quantity areas specific emphasis at the localization phenomenon. the most questionis: What does the trail of a random stroll appear like if rewards and consequences are spatially randomly distributed?This version, which supplies a simplified model of stretched elastic chains pinned via random impurities, has attracted a lot examine task, however it (and its family) nonetheless holds many secrets and techniques, particularly in excessive dimensions. It has non-gaussian scaling limits and it belongs to the so-called KPZ universality classification whilst the distance is one-dimensional. Adopting a Gibbsian method, utilizing basic and robust instruments from likelihood thought, the discrete version is studied in complete generality. featuring the state-of-the paintings from varied views, and written within the kind of a primary direction at the topic, this monograph is geared toward researchers in likelihood or statistical physics, yet is usually obtainable to masters and Ph.D. students.
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Additional info for Directed Polymers in Random Environments: École d'Été de Probabilités de Saint-Flour XLVI – 2016
T; x/ D s/. t; x/ D s/ > d . t; x/. ˇ/ ˇ%1 ! 15) 3 The Martingale Approach and the L2 Region 38 The claim is clearly true if s D 1. ˇ/ ! 0 as ˇ ! C1. ˇ/ ! fsg/ as ˇ ! 1. eˇ! I ! eˇ! eˇ! I ! eˇ! I ! 16). fsg/ ˇh Ã : I h > 0g decreases to 1 as ˇ ! 1 Gaussian environment. 1= d /. 2 Absence of strong disorder regime. t; x/ D 1 or 0 with probability p and 1 p respectively. 12) holds for all ˇ 0 if p > d . 3 shows that, in this case, weak disorder holds for all ˇ 0. 12) holds. In this region, the natural martingale is bounded in L2 , and it allows second moment computations.
The only requirement is an upper bound on the entropy rate. 0; 1/. 14). ˇ/ for large ˇ, in terms of the marginal distribution q of !. A sufficient condition is that the environment is unbounded from above, or bounded from above with a large enough mass on its essential supremum. n; x/ 2 dh/ and s D sup suppŒq. fsg/ < 2d for ˇ > ˇ1 . 17). ˇ/ . / D eˇ! As illustrated in Fig. 1, 1. If s D C1, then 2. ˇ/ ! sup supp q when ˇ ! C1. 17) holds for large ˇ. C1 . fsg/ < 1=2d. 5 Lower bounds are less useful for our analysis.
42). 43) follows easily. e. ˇ// expf g ! expf Pˇ;! ˇ/u2 g 2 as n ! 1 for all u 2 R and ˇ 2 U2 . ˇ/ Since Wn ! s. ˇ/u2 =2g as n ! 1. t u ˇ;! ˇ/ the energy of the annealed model, but it has fluctuations of order one in this part of the weak disorder region. 1/ fluctuations. 6 Concentration Revisited 51 On the other hand, the last result shows that, due to variations from a path to another, the fluctuationspof the energy under the polymer measure are normal and of order of magnitude O. n/. 7 (At Weak Disorder but Outside L2-Region).
Directed Polymers in Random Environments: École d'Été de Probabilités de Saint-Flour XLVI – 2016 by Francis Comets
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