Parabolic Anderson model on critical Galton-Watson trees in a Pareto environment

Anne Pein; Eleanor Archer

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Parabolic Anderson model on critical Galton-Watson trees in a Pareto environment

Eleanor Archer, Anne Pein

Probability · Submitted 17 Feb 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,893 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.43 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.02 · states quantitative result 0.54 · compares with prior results 0.94 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$. The main difference with earlier results on $\mathbb{Z}^d$ is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The parabolic Anderson model is the heat equation with some extra spatial randomness.	background 1.00	0.04	0.01	asserted 1.00
s1	In this paper we consider the parabolic Anderson model with i.i.d.	contribution 0.80	0.54	0.02	asserted 0.57
s2	Pareto potential on a critical Galton-Watson tree conditioned to survive.	contribution 0.64	0.29	0.04	asserted 0.53
s3	We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$.	result 1.00	0.96	0.81	asserted 1.00
s4	Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$.	result 0.93	0.96	0.21	asserted 1.00
s5	The main difference with earlier results on $\mathbb{Z}^d$ is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.	contribution 0.44	0.48	0.05	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$. The main difference with earlier results on $\mathbb{Z}^d$ is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Firstly, it follows from the Rayleigh-Ritz formula (e.g. see [ref] ) for the principal eigenvalue of the Anderson Hamiltonian that [equation] Then, taking $y=Ẑ_t^(2)$ in [ref] we have for $x∈Λ_t∖{Ẑ_t^(1),Ẑ_t^(2)}$, [equation] Now let $(T_v)_v ∈γ_x,t^(2)$ denote a set of independent exponential random variables with respective parameters $(v)_v ∈γ_x,t^(2)$.

The parabolic Anderson model on a Galton-Watson tree · Bounds on eigenfunctions · 0.88

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