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Branching random walk in random environment with random absorption wall
Probability · Submitted 13 Sep 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that there exists a critical value $a_c>0$ such that if $a>a_c,\alpha=\frac{1}{3}$, the survival probability is positive almost surly and if $a<a_c,\alpha=\frac{1}{3}$ ,the survival probability is zero almost surely.
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The abstract, sentence by sentence
We consider the branching random walk in random environment with a random absorption wall. When we add this barrier, we discuss some topics related to the survival probability. We assume that the random environment is i.i.d., $S_i$ is a particular i.i.d. random walk depend on the random environment $\mathcal{L}$. Let the random barrier function (the random absorption wall) is $g_i(\mathcal{L}):=ai^\alpha-S_i,$ where $i$ present the generation. We show that there exists a critical value $a_c>0$ such that if $a>a_c,\alpha=\frac{1}{3}$, the survival probability is positive almost surly and if $a<a_c,\alpha=\frac{1}{3}$ ,the survival probability is zero almost surely. Moreover, if we denote $Z_n$ is the total populations in $n$-th generation in the new system (with barrier),under some conditions, we show $\ln\mathbb{P}_{\mathcal{L}}(Z_n>0)/n^{1/3}$ will converges to a negative constant almost surely if $\alpha\in[0,\frac{1}{3})$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the branching random walk in random environment with a random absorption wall. | contribution 0.58 | 0.44 | 0.03 | asserted 0.62 |
| s1 | When we add this barrier, we discuss some topics related to the survival probability. | contribution 0.85 | 0.67 | 0.03 | unclear 0.49 |
| s2 | We assume that the random environment is i.i.d., $S_i$ is a particular i.i.d. random walk depend on the random environment $\mathcal{L}$. | method 0.82 | 0.28 | 0.02 | asserted 0.98 |
| s3 | Let the random barrier function (the random absorption wall) is $g_i(\mathcal{L}):=ai^\alpha-S_i,$ where $i$ present the generation. | method 0.93 | 0.42 | 0.03 | asserted 0.97 |
| s4 | We show that there exists a critical value $a_c>0$ such that if $a>a_c,\alpha=\frac{1}{3}$, the survival probability is positive almost surly and if $a<a_c,\alpha=\frac{1}{3}$ ,the survival probability is zero almost surely. | result 1.00 | 0.94 | 0.76 | asserted 1.00 |
| s5 | Moreover, if we denote $Z_n$ is the total populations in $n$-th generation in the new system (with barrier),under some conditions, we show $\ln\mathbb{P}_{\mathcal{L}}(Z_n>0)/n^{1/3}$ will converges to a negative constant almost surely if $\alpha\in[0,\frac{1}{3})$. | result 1.00 | 0.94 | 0.19 | asserted 1.00 |
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Original abstract
We consider the branching random walk in random environment with a random absorption wall. When we add this barrier, we discuss some topics related to the survival probability. We assume that the random environment is i.i.d., $S_i$ is a particular i.i.d. random walk depend on the random environment $\mathcal{L}$. Let the random barrier function (the random absorption wall) is $g_i(\mathcal{L}):=ai^\alpha-S_i,$ where $i$ present the generation. We show that there exists a critical value $a_c>0$ such that if $a>a_c,\alpha=\frac{1}{3}$, the survival probability is positive almost surly and if $a<a_c,\alpha=\frac{1}{3}$ ,the survival probability is zero almost surely. Moreover, if we denote $Z_n$ is the total populations in $n$-th generation in the new system (with barrier),under some conditions, we show $\ln\mathbb{P}_{\mathcal{L}}(Z_n>0)/n^{1/3}$ will converges to a negative constant almost surely if $\alpha\in[0,\frac{1}{3})$.
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2 arXiv papers cite this in 2 sentences
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Cites as background 2 papers
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The model named branching random walk on $$ with random environment in time (BRWre) has been introduced in [ref] and [ref] .
Branching random walk with a random environment in time · Introduction · 0.99
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The model named branching random walk on $$ with random environment in time (BRWre) has been introduced in [ref] and [ref] .
Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment · Introduction · 0.99
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