Timeline for Decay of eigenfunctions for Laplacian
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| Jul 31, 2019 at 6:57 | answer | added | RaphaelB4 | timeline score: 0 | |
| Jul 30, 2019 at 8:32 | answer | added | Mateusz Kwaśnicki | timeline score: 0 | |
| Jul 29, 2019 at 22:29 | comment | added | Yannis Pimalis | @MateuszKwaśnicki Thank you, yet although I found endless papers discussing the lowest eigenvalue, I could not find any paper discussing the connection with the asymptotics of eigenvectors. If you could point me to relevant literature or elaborate eventually, I would be very grateful. | |
| Jul 29, 2019 at 21:04 | comment | added | Mateusz Kwaśnicki | I am afraid I have no time now to elaborate. However, you may like to search for "Dirichlet heat kernel bounds on graphs", there are dozens of papers dealing with these subjects. | |
| Jul 29, 2019 at 20:51 | comment | added | Yannis Pimalis | @MateuszKwaśnicki I deliberately asked the question a bit open ended cause I was not sure how much can be said. Thanks for looking into this. I think it would be interesting for me to understand first based on a soft argument why the "soft estimate" holds. So if you say this is a general thing for positive matrices, I am curious to hear about it. Then, if you can actually prove the asymptotics using some kind of robust argument, I would be very curious to hear about it. I am grateful for any insights you would like to share about this estimate. | |
| Jul 28, 2019 at 20:46 | comment | added | Mateusz Kwaśnicki | OK, I see now what you mean by decay. Still, I am not sure if you like to get an exact bound $v_{1,1} \sim C n^{-3/2}$ (note that in fact $v_{i,j} \sim C_{i,j} n^{-3/2}$ for any fixed $i$ and $j$), or a softer estimate of the form $|v_{1,1}| \le |v_{i,1}|$ for any $i$. The latter would work for any totally positive matrix, I think. The former requires more assumptions, and I can think of various proofs (for example, if your matrix is a generator of a symmetric nearest-neighbour Markov chain, then a probabilistic argument could work). | |
| Jul 26, 2019 at 22:36 | history | edited | Yannis Pimalis | CC BY-SA 4.0 |
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| Jul 26, 2019 at 22:30 | comment | added | Yannis Pimalis | I elaborated a bit on the question. It is about proving that we have these fixed decay rates for certain entries. But why? | |
| Jul 26, 2019 at 22:30 | history | edited | Yannis Pimalis | CC BY-SA 4.0 |
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| Jul 26, 2019 at 20:40 | comment | added | Mateusz Kwaśnicki | Do I understood correctly that the question is about the eigenvectors and eigenvalues of a tri-diagonal matrix $L$ with $-2$ on the main diagonal and $1$ on the neighbouring two diagonals? In this case $-L$ is a totally positive matrix, which automatically tells a lot about the eigenvalues and eigenvectors (e.g. the interlacing property). However, all eigenvectors are just discretised sine waves, so they do not really decay anywhere. Thus, I must be getting something wrong. Can you elaborate a bit? | |
| Jul 26, 2019 at 15:40 | history | edited | Yannis Pimalis | CC BY-SA 4.0 |
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| Jul 26, 2019 at 13:50 | history | edited | Yannis Pimalis | CC BY-SA 4.0 |
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| Jul 26, 2019 at 13:10 | review | First posts | |||
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| Jul 26, 2019 at 13:05 | history | asked | Yannis Pimalis | CC BY-SA 4.0 |