Fluctuations of the free energy in the REM and the p-spin SK models

We consider the random fluctuations of the free energy in the $p$-spin version of the Sherrington-Kirkpatrick model in the high temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a recent paper by Talagrand on the $p$-spin version, we prove that (for $p$ even) the random corrections to the free energy are on a scale $N^{-(p-2)/4}$ only, and after proper rescaling converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, \b, smaller than a critical \b_p. We also show that \b_p\to \sqrt{2\ln 2} as $p\uparrow +\infty$. Additionally we study the formal $p\uparrow +\infty$ limit of these models, the random energy model. Here we compute the precise limit theorem for the partition function at {\it all} temperatures. For \b<\sqrt{2\ln2}, fluctuations are found at an {\it exponentially small} scale, with two distinct limit laws above and below a second critical value $\sqrt{\ln 2/2}$: For \b up to that value the rescaled fluctuations are Gaussian, while below that there are non-Gaussian fluctuations driven by the Poisson process of the extreme values of the random energies. For \b larger than the critical $\sqrt{2\ln 2}$, the fluctuations of the logarithm of the partition function are on scale one and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to 1/2.Comment: 40pp, AMSTe

tt-Martin boundary of killed random walks in the quadrant

We compute the $t$-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete $t$-harmonic functions. Our approach is uniform in $t$, and shows that there are three regimes for the Martin boundary.Comment: 18 pages, 2 figures, to appear in S\'eminaire de Probabilit\'e

Timescales of population rarity and commonness in random environments

This paper investigates the influence of environmental noise on the characteristic timescale of the dynamics of density-dependent populations. General results are obtained on the statistics of time spent in rarity (i.e.\ below a small threshold on population density) and time spent in commonness (i.e. above a large threshold). The nonlinear stochastic models under consideration form a class of Markov chains on the state space $]0, \infty[$ which are transient if the intrinsic growth rate is negative and recurrent if it is positive or null. In the recurrent case, we obtain a necessary and sufficient condition for positive recurrence and precise estimates for the distribution of times of rarity and commonness. In the null recurrent, critical case that applies to ecologically neutral species, the distribution of rarity time is a universal power law with exponent $-3/2$. This has implications for our understanding of the long-term dynamics of some natural populations, and provides a rigorous basis for the statistical description of on-off intermittency known in physical sciences

Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane

A random walk in $Z_+^2$ spatially homogeneous in the interior, absorbed at the axes, starting from an arbitrary point $(i_0,j_0)$ and with step probabilities drawn on Figure 1 is considered. The trivariate generating function of probabilities that the random walk hits a given point $(i,j)\in Z_+^2$ at a given time $k\geq 0$ is made explicit. Probabilities of absorption at a given time $k$ and at a given axis are found, and their precise asymptotic is derived as the time $k\to\infty$. The equivalence of two typical ways of conditioning this random walk to never reach the axes is established. The results are also applied to the analysis of the voter model with two candidates and initially, in the population $Z$, four connected blocks of same opinions. Then, a citizen changes his mind at a rate proportional to the number of its neighbors that disagree with him. Namely, the passage from four to two blocks of opinions is studied.Comment: 11 pages, 1 figur

On the functions counting walks with small steps in the quarter plane

Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of the numbers $q(i,j;n)$ of such walks starting at the origin and ending at $(i,j) \in {\bf Z}_+^{2}$ after $n$ steps is studied. For all non-singular models of walks, the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are continued as multi-valued functions on ${\bf C}$ having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of ${\bf C}^2$, the interval $]0,1/|\mathcal{S}|[$ of variation of $z$ splits into two dense subsets such that the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are shown to be holonomic for any $z$ from the one of them and non-holonomic for any $z$ from the other. This entails the non-holonomy of $(x,y,z)\mapsto Q(x,y;z)$, and therefore proves a conjecture of Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure

Step-wise responses in mesoscopic glassy systems: a mean field approach

We study statistical properties of peculiar responses in glassy systems at mesoscopic scales based on a class of mean-field spin-glass models which exhibit 1 step replica symmetry breaking. Under variation of a generic external field, a finite-sized sample of such a system exhibits a series of step wise responses which can be regarded as a finger print of the sample. We study in detail the statistical properties of the step structures based on a low temperature expansion approach and a replica approach. The spacings between the steps vanish in the thermodynamic limit so that arbitrary small but finite variation of the field induce infinitely many level crossings in the thermodynamic limit leading to a static chaos effect which yields a self-averaging, smooth macroscopic response. We also note that there is a strong analogy between the problem of step-wise responses in glassy systems at mesoscopic scales and intermittency in turbulent flows due to shocks.Comment: 50 pages, 18 figures, revised versio

Martin boundary of a reflected random walk on a half-space

The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial'' compactification obtained by Ney and Spitzer for the homogeneous random walks in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times\N$ to a point of on the Martin boundary does not imply convergence of the sequence $z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large deviation properties of the scaled processes and uses Pascal's method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.Comment: 42 pages, preprint, CNRS UMR 808

ПАРАМЕТРЫ ЭКОЛОГИЧЕСКОЙ ПЛАСТИЧНОСТИ СОРТОВ И СОРТООБРАЗЦОВ ЯРОВОГО ЯЧМЕНЯ АМУРСКОЙ СЕЛЕКЦИИ

The article is concerned with increasing of crop yield and explores that production quality is influenced by the variety adjusted to local conditions. This variety is most productive for plant production and important in agricultural production. New cultivar should be highly productive, highly adaptive and environmentally plastic (to form steady crop yield in different conditions). The article explores estimation of cultivars and varieties of Amur spring barley on environmental plasticity and stability. The researchers estimated environmental plasticity and stability for 3 years (2012–2014), which differed in vegetation conditions. The authors apply regression co-efficient (bi), which characterize cultivars response to agricultural changes and stability variance (s2di), which shows cultivar response to environmental changes and its stability. New Amur variety included into the State List of Selection Inventions is not stable, which is proved by estimation in 2008–2011. Earlier it was not stable but responded well to the changes; now it is not stable but more productive in favorable conditions. The authors make the idea that varieties, which belong to the group of well-responding to the changes and stable ones are the most significant varieties. The researchers define Mishka variety as a stable and well-responding. Огромную роль в повышении урожайности и улучшении качества продукции играет сорт, приспособленный к местным условиям. Он является основой производства любой растениеводческой продукции и его роль в сельскохозяйственном производстве постоянно возрастает. Новый сорт должен быть не только высокоурожайным, но обладать высокой адаптивной способностью и широкой экологической пластичностью (формировать стабильный урожай в различных условиях). Статья посвящена вопросу оценки сортов и сортообразцов ярового ячменя амурской селекции по параметрам экологической пластичности и стабильности. Расчет экологической пластичности и стабильности проводили в среднем за 3 года (2012–2014 гг.), сильно отличающиеся по условиям вегетации. Для определения данных параметров приведен расчет коэффициента регрессии (bi ), характеризующего реакцию сортов на изменение условий выращивания, и вариансы стабильности (s 2 di), которая указывает, насколько сорт отзывчив на условия среды и стабилен ли в этих условиях. Новый сорт амурской селекции Амур, внесенный в Государственный реестр селекционных достижений в 2015 г., является нестабильным, что также подтверждается и ранее проведенными расчетами (в 2008–2011 гг.). Если он ранее был нестабильным, но хорошо отзывчивым на изменение условий, то в данный момент он характеризуется как нестабильный и показывающий лучшие результаты в благоприятных условиях. Наибольшее значение имеют сорта, которые относятся к группе хорошо отзывчивых на изменение условий и являются стабильными. Из изученных нами 12 сортообразцов к этой группе можно отнести один – сортообразец Мишка

Analysis of the Karmarkar-Karp Differencing Algorithm

The Karmarkar-Karp differencing algorithm is the best known polynomial time heuristic for the number partitioning problem, fundamental in both theoretical computer science and statistical physics. We analyze the performance of the differencing algorithm on random instances by mapping it to a nonlinear rate equation. Our analysis reveals strong finite size effects that explain why the precise asymptotics of the differencing solution is hard to establish by simulations. The asymptotic series emerging from the rate equation satisfies all known bounds on the Karmarkar-Karp algorithm and projects a scaling $n^{-c\ln n}$, where $c=1/(2\ln2)=0.7213...$. Our calculations reveal subtle relations between the algorithm and Fibonacci-like sequences, and we establish an explicit identity to that effect.Comment: 9 pages, 8 figures; minor change