Inverting the operation of conditioning a branching process on extinction

Ewain Gwynne; Jiaqi Liu

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Inverting the operation of conditioning a branching process on extinction

Ewain Gwynne, Jiaqi Liu

Probability · Submitted 9 Nov 2024 · Published version

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

The authors' main claim We show that such a supercritical branching process (which we call a conjugate branching process) exists under mild hypotheses on the original subcritical branching process.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.79 · human participants 0.03 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.31 · mentions code or data availability 0.02.

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It is well-known that conditioning a supercritical (multi-type) branching process on the event that it eventually becomes extinct yields a subcritical branching process. We study the corresponding inverse problem: given a subcritical branching process, does there exist a supercritical branching process with the property that when we condition it on extinction, we get back the original subcritical branching process? We show that such a supercritical branching process (which we call a conjugate branching process) exists under mild hypotheses on the original subcritical branching process. We also show by example that if there are at least two types, then the conjugate branching process is not necessarily unique. Our results are relevant to the problem of constructing natural random planar maps whose scaling limit is given by supercritical Liouville quantum gravity. Moreover, conjugate branching processes can also be used to give alternative evolutionary hypotheses in cancer modeling.

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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	It is well-known that conditioning a supercritical (multi-type) branching process on the event that it eventually becomes extinct yields a subcritical branching process.	background 1.00	0.03	0.04	asserted 1.00
s1	We study the corresponding inverse problem: given a subcritical branching process, does there exist a supercritical branching process with the property that when we condition it on extinction, we get back the original subcritical branching process?	contribution 0.64	0.28	0.04	unclear 0.70
s2	We show that such a supercritical branching process (which we call a conjugate branching process) exists under mild hypotheses on the original subcritical branching process.	result 0.98	0.96	0.82	asserted 1.00
s3	We also show by example that if there are at least two types, then the conjugate branching process is not necessarily unique.	result 0.97	0.94	0.08	asserted 0.85
s4	Our results are relevant to the problem of constructing natural random planar maps whose scaling limit is given by supercritical Liouville quantum gravity.	interpretation 0.97	0.22	0.03	asserted 0.67
s5	Moreover, conjugate branching processes can also be used to give alternative evolutionary hypotheses in cancer modeling.	interpretation 0.95	0.31	0.03	hedged possibility 0.76

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

It is well-known that conditioning a supercritical (multi-type) branching process on the event that it eventually becomes extinct yields a subcritical branching process. We study the corresponding inverse problem: given a subcritical branching process, does there exist a supercritical branching process with the property that when we condition it on extinction, we get back the original subcritical branching process? We show that such a supercritical branching process (which we call a conjugate branching process) exists under mild hypotheses on the original subcritical branching process. We also show by example that if there are at least two types, then the conjugate branching process is not necessarily unique. Our results are relevant to the problem of constructing natural random planar maps whose scaling limit is given by supercritical Liouville quantum gravity. Moreover, conjugate branching processes can also be used to give alternative evolutionary hypotheses in cancer modeling.

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