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Erschienen in:
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2017 | OriginalPaper | Buchkapitel

3. The Martingale Approach and the L 2 Region

verfasst von : Francis Comets

Erschienen in: Directed Polymers in Random Environments

Verlag: Springer International Publishing

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Abstract

Martingale theory is well-known as a powerful tool to study random sequences. In this section, we start to use it in our context. First of all, it is efficient for proving that equality can hold in (2.​13). The simplest sufficient condition for equality is based on a second moment computation. The parameter region where second moment computations are reliable, will be called the L 2 region.

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Vorheriges Kapitel Thermodynamics and Phase Transition
Nächstes Kapitel Lattice Versus Tree
Fußnoten
1
This means that supn n −κmax0≤ j≤n |M j| < ∞ a.s.
 
2
The main point is that, when ω(t, x) has a nearly gamma distribution, it satisfies a logarithmic Sobolev inequality w.r.t. a particular differential operator. It covers a wide class of densities, including gamma and Gaussian.
 
Literatur
8.
S. Albeverio, X.Y. Zhou, A martingale approach to directed polymers in a random environment. J. Theor. Probab. 9 (1), 171–189 (1996)MathSciNetCrossRefMATH
9.
10.
11.
K.S. Alexander, N. Zygouras, Subgaussian concentration and rates of convergence in directed polymers. Electron. J. Probab. 18 (5), 28 (2013)
15.
20.
26.
I. Ben-Ari, Large deviations for partition functions of directed polymers in an IID field. Ann. Inst. Henri Poincaré Probab. Stat. 45 (3), 770–792 (2009)MathSciNetCrossRefMATH
27.
M. Benaïm, R. Rossignol, Exponential concentration for first passage percolation through modified Poincaré inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 44 (3), 544–573 (2008)MathSciNetCrossRefMATH
28.
I. Benjamini, G. Kalai, O. Schramm, First passage percolation has sublinear distance variance. Ann. Probab. 31 (4), 1970–1978 (2003)MathSciNetCrossRefMATH
38.
M. Birkner, A condition for weak disorder for directed polymers in random environment. Electron. Commun. Probab. 9, 22–25 (electronic) (2004)
45.
C. Boldrighini, R.A. Minlos, A. Pellegrinotti, Almost-sure central limit theorem for directed polymers and random corrections. Commun. Math. Phys. 189 (2), 533–557 (1997)MathSciNetCrossRefMATH
46.
E. Bolthausen, A note on the diffusion of directed polymers in a random environment. Commun. Math. Phys. 123 (4), 529–534 (1989)MathSciNetCrossRefMATH
55.
A. Camanes, P. Carmona, The critical temperature of a directed polymer in a random environment. Markov Process. Relat. Fields 15 (1), 105–116 (2009)MathSciNetMATH
58.
P. Carmona, Y. Hu, On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Relat. Fields 124 (3), 431–457 (2002)MathSciNetCrossRefMATH
64.
S. Chatterjee, Superconcentration and Related Topics. Springer Monographs in Mathematics (Springer, Cham, 2014)
70.
F. Comets, Q. Liu, Rate of convergence for polymers in a weak disorder (2016), preprint arXiv:1605.05108
75.
F. Comets, N. Yoshida, Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34, 1746–1770 (2006)MathSciNetCrossRefMATH
79.
F. Comets, T. Shiga, N. Yoshida, Probabilistic analysis of directed polymers in a random environment: a review, in Stochastic Analysis on Large Scale Interacting Systems. Advanced Studies in Pure Mathematics, vol. 39 (Mathematical Society of Japan, Tokyo, 2004), pp. 115–142
95.
B. Derrida, M.R. Evans, E.R. Speer, Mean field theory of directed polymers with random complex weights. Commun. Math. Phys. 156 (2), 221–244 (1993)MathSciNetCrossRefMATH
99.
J. Ding, R. Eldan, A. Zhai, On multiple peaks and moderate deviations for the supremum of a Gaussian field. Ann. Probab. 43 (6), 3468–3493 (2015)MathSciNetCrossRefMATH
103.
R. Durrett, Lecture Notes on Particle Systems and Percolation. The Wadsworth and Brooks/Cole Statistics/Probability Series (Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, CA, 1988)
104.
R. Durrett, Probability: Theory and Examples, 2nd edn. (Duxbury Press, Belmont, CA, 1996)MATH
123.
127.
131.
C.H. Hall, Martingale limit theory and its applications, in Probability and Mathematical Statistics (Academic, New York/London, 1980)
141.
157.
H. Lacoin, New bounds for the free energy of directed polymers in dimension 1 + 1 and 1 + 2. Commun. Math. Phys. 294 (2), 471–503 (2010)MathSciNetCrossRefMATH
175.
G. Moreno, Convergence of the law of the environment seen by the particle for directed polymers in random media in the L 2 region. J. Theor. Probab. 23 (2), 466–477 (2010)MathSciNetCrossRefMATH
191.
P. Olsen, R. Song, Diffusion of directed polymers in a strong random environment. J. Stat. Phys. 83 (3–4), 727–738 (1996)MathSciNetCrossRefMATH
200.
A. Rényi, On stable sequences of events. Sankhyā: Indian J. Stat. Ser. A 25, 293–302 (1963)MathSciNetMATH
204.
W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill Book Co., New York, 1987)MATH
216.
Y.G. Sinai, A remark concerning random walks with random potentials. Fundam. Math. 147 (2), 173–180 (1995)MathSciNetMATH
218.
R. Song, X.Y. Zhou, A remark on diffusion of directed polymers in random environments. J. Stat. Phys. 85 (1–2), 277–289 (1996)MathSciNetCrossRefMATH
219.
F. Spitzer, Principles of Random Walk. Graduate Texts in Mathematics, vol. 34, 2nd edn. (Springer, New York-Heidelberg, 1976)
225.
226.
M. Talagrand, Mean field models for spin glasses: a first course. Lectures on probability theory and statistics (Saint-Flour, 2000). Lecture Notes in Mathematics, vol. 1816 (Springer, Berlin, 2003), pp. 181–285
234.
V. Vargas, A local limit theorem for directed polymers in random media: the continuous and the discrete case. Ann. Inst. H. Poincaré Probab. Stat. 42 (5), 521–534 (2006)MathSciNetCrossRefMATH
239.
D. Williams, Probability with Martingales. Cambridge Mathematical Textbooks (Cambridge University Press, Cambridge, 1991)CrossRefMATH
Metadaten
Titel
The Martingale Approach and the L 2 Region
verfasst von
Francis Comets
Copyright-Jahr
2017
Buch
Directed Polymers in Random Environments

Print ISBN: 978-3-319-50486-5

Electronic ISBN: 978-3-319-50487-2

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-50487-2_3