Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs

Chen, Joe P.; Ugurcan, Baris Evren

doi:10.1016/j.spa.2015.07.011

Cited by 5 publications

(7 citation statements)

References 36 publications

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“…As regards to the behavior of the branching random walk in presence of a hard wall, we recall similar results for other gaussian processes such as [10], [11], [2], [14], [8], [7]. The leading order term in the exponent of the probability of positivity is what is estimated, while we estimate both the leading order term and the second leading term in the exponent.…”

Section: Introductionsupporting

confidence: 74%

“…We apply a similar strategy for obtaining the lower bound as well. Let us recall (8) at this juncture along with X, and let us call the variance of X to be σ 2 d,n = 1−d −n d−1 . In (8), we condition on the value of X to obtain the following:…”

Section: Estimates On Left Tail and Positivitymentioning

confidence: 99%

“…Entropic repulsion in case of GFF on Serpinski carpet graphs has been covered in [8]. More recently entropic repulsion in |∇φ| p surfaces has been considered in [7].…”

Section: Introductionmentioning

confidence: 99%

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Branching random walk in the presence of a hard wall

Roy¹

2018

Preprint

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We consider the Branching Random Walk on d-ary tree of height n under the presence of a hard wall which restricts each value to be positive. The question of behavior of Gaussian process with long range interactions under the presence of a hard wall has been addressed, and a relation with the expected maxima of the processes has been established. We find the probability of event corresponding to the hard wall and show that the conditional expectation of the gaussian variable at a typical vertex, under positivity, is at least less than the expected maxima by order of log n.

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Section: Introductionsupporting

confidence: 74%

Section: Estimates On Left Tail and Positivitymentioning

confidence: 99%

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Branching random walk in the presence of a hard wall

Roy¹

2018

Preprint

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“…Fatou's lemmas for weakly converging measures have significant applications to various areas and fields including stochastic processes [5,7,21,26], statistics [19,31,32], control [6,12,14,17,35], game theory [22], functional analysis [20], optimization [37], and electrical engineering [28]. Our initial motivation in studying Fatou's lemma for variable probabilities was caused by its usefulness for the proof of the validity of optimality inequalities and the existence of stationary optimal policies for infinite-horizon, Borel-state, average-cost Markov decision process with noncompact action sets and unbounded costs [14].…”

Section: Introductionmentioning

confidence: 99%

Fatou's Lemma for Weakly Converging Measures under the Uniform Integrability Condition

Feinberg

Kasyanov

Liang

2020

Theory Probab. Appl.

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This note describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The note also provides new formulations of uniform Fatou's lemma, uniform Lebesgue convergence theorem, the Dunford-Pettis theorem, and the fundamental theorem for Young measures based on the equivalence of uniform integrability and the apparently weaker property of asymptotic uniform integrability for sequences of functions and finite measures.

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“…In the case of weak convergence, the lower limit of functions should be defined in a stronger sense than in the setwise convergence case and in the classic case of a single measure. Fatou's lemma for weakly converging measures is broadly used in stochastic control [7,9,19,29], game theory [15], and in other applications [6,23].…”

Section: Introductionmentioning

confidence: 99%

Uniform Fatou's lemma

Feinberg

Kasyanov

Zgurovsky

2016

Journal of Mathematical Analysis and Applications

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Fatou's lemma is a classic fact in real analysis stating that the limit inferior of integrals of functions is greater than or equal to the integral of the inferior limit. This paper introduces a stronger inequality that holds uniformly for integrals on measurable subsets of a measurable space. The necessary and sufficient condition, under which this inequality holds for a sequence of finite measures converging in total variation, is provided. This statement is called the uniform Fatou lemma, and it holds under the minor assumption that all the integrals in the inequality are well-defined. The uniform Fatou lemma improves the classic Fatou lemma in the following directions: the uniform Fatou lemma states a more precise inequality, it provides the necessary and sufficient condition, and it deals with variable measures. Various corollaries of the uniform Fatou lemma are formulated. The examples in this paper demonstrate that: (a) the uniform Fatou lemma may indeed provide a more accurate inequality than the classic Fatou lemma; (b) the uniform Fatou lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence.

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Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs

Cited by 5 publications

References 36 publications

Branching random walk in the presence of a hard wall

Branching random walk in the presence of a hard wall

Fatou's Lemma for Weakly Converging Measures under the Uniform Integrability Condition

Uniform Fatou's lemma

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