Analytic methods for obstruction to integrability in discrete dynamical systems

Abstract. We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenus [11], which can be reinterpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {Rn}n=1,2,..., which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known Logistic Map. The large-n limit of the sequence of random variables 2 −n log Rn, a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter α ∈ (0, 1), related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured in [11] that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2 < α < 1 (respectively, α < 1/2 or α = 1/2), in the sense that an arbitrarily small amount of randomness in the initial condition modifies the critical point with respect to that of the pure (i.e., non-disordered) model if α ≥ 1/2, but not if α < 1/2. Our main result is a proof of these conjectures for the case α = 1/2. We emphasize that for α > 1/2 we find the correct scaling form (for weak disorder) of the critical point shift.

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“…Altogether we have 8) and since R n is bounded from below by p(n, ∅) which tends to 1, the proof is complete.…”

Section: The Delocalized Phasementioning

confidence: 65%

Hierarchical pinning models, quadratic maps and quenched disorder

Giacomin

Lacoin

Toninelli

2009

Probab. Theory Relat. Fields

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“…To go beyond Proposition 1.2 we restrict to the d = 2 case. This restriction is made because we want to exploit directly the results in [4,5,6], that develop only the case d = 2. It is certainly possible to generalize these works, but this would not add much to the purpose of this note at the expense of rather lengthy arguments.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Oscillatory Critical Amplitudes in Hierarchical Models and the Harris Function of Branching Processes

Costin

Giacomin

2012

J Stat Phys

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Oscillatory critical amplitudes have been repeatedly observed in hierarchical models and, in the cases that have been taken into consideration, these oscillations are so small to be hardly detectable. Hierarchical models are tightly related to iteration of maps and, in fact, very similar phenomena have been repeatedly reported in many fields of mathematics, like combinatorial evaluations and discrete branching processes. It is precisely in the context of branching processes with bounded off-spring that T. Harris, in 1948, first set forth the possibility that the logarithm of the moment generating function of the rescaled population size, in the super-critical regime, does not grow near infinity as a power, but it has an oscillatory prefactor. These oscillations have been observed numerically only much later and, while the origin is clearly tied to the discrete character of the iteration, the amplitude size is not so well understood. The purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting -the pinning model -that actually just boils down to iteration of polynomial maps (and, notably, quadratic maps). In this note we show that the oscillatory critical amplitude for pinning models and the oscillating pre factor connected to the Harris random variable coincide. Moreover we make explicit the link between these oscillatory functions and the geometry of the Julia set of the map, making thus rigorous and quantitative some ideas set forth in [10].

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“…This is beyond the scope here, and will be the subject of a different paper. A less explicit expression has been obtained in [5]. We note that the constant log 2 (2π) in (1.17) of [5] should be (2π)/ log 2. where g ′ (ξ) ∼ ax θ .…”

Section: Proof Of Theoremmentioning

confidence: 93%

“…A less explicit expression has been obtained in [5]. We note that the constant log 2 (2π) in (1.17) of [5] should be (2π)/ log 2. where g ′ (ξ) ∼ ax θ . Now θ is fixed and m is arbitrary, and then the result follows.…”

Section: Proof Of Theoremmentioning

confidence: 93%

“…Natural boundaries (NBs) occur frequently in many applications of analysis, in the theory of Fourier series, in holomorphic dynamics (see [8], [7], [5] and references there), in analytic number theory, see [20], physics, see [13] and even in relatively simple ODEs such as the Chazy equation [11], an equation arising in conformal mapping theory, or the Jacobi equation.…”

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confidence: 99%

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Behavior of lacunary series at the natural boundary

Costin

Huang

2009

Advances in Mathematics

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We develop a local theory of lacunary Dirichlet series of the form ∞ P k=1 c k exp(−zg(k)), Re(z) > 0 as z approaches the boundary iR, under the assumption g ′ → ∞ and further assumptions on c k . These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with c k = 1, the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at z = 0.When sufficient analyticity information on g exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier.The singular behavior has remarkable universality and self-similarity features. If g(k) = k b , c k = 1, b = n or b = (n + 1)/n, n ∈ N, behavior near the boundary is roughly of the standard form ReThe Bötcher map at infinity of polynomial iterations of the form x n+1 = λP (xn), |λ| < λ 0 (P ), turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map P (x) = x − x 2 , λ 0 = 1, and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.

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Analytic methods for obstruction to integrability in discrete dynamical systems

Cited by 6 publications

References 27 publications

Hierarchical pinning models, quadratic maps and quenched disorder

Hierarchical pinning models, quadratic maps and quenched disorder

Oscillatory Critical Amplitudes in Hierarchical Models and the Harris Function of Branching Processes

Behavior of lacunary series at the natural boundary

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