$a_{n+m+2} leq a_m+a_n+g(n)$ with $g(n) = o(n)$. Show that $a_n geq (n+2)lambda-g(n)$ where $lambda = lim...
up vote
1
down vote
favorite
I was working on the proof of the exponential decay on supercritical percolation as shown in Grimmett's Percolation (1999, 2 ed. pg 206 - 210) and he uses as a lemma a form of the subadditive theorem (fekete) in which he states (he says it is easy to find) that
if you have a sequence $(a_n)_n$ such that $a_{n+m+2} leq a_m+a_n+g(n)$ and $lim_{n to infty} frac{g(n)}{n}=0$ then
the limit $lim_{n to infty} frac{a_n}{n} = lambda$ exists and it's equal to the $inf left{frac{a_n}{n}right}$.
$a_n geq (n+2)lambda-g(n);forall n$.
Now, although I coundn't prove the existence of the limit using only what was stated on the theorem, I manage to prove it adding some restrictions which are valid on Grimmet's theorem (I asked it in here: Subadditive lemma for $a_{m+n+2}leq a_n+a_m+g(m)$ and I found that, in my application, the last condition was satisfied).
Now assuming the limit exists, I couldn't manage to show that $$a_ngeq (n+2) lambda-g(n)$$
Any tips on how to proceed? Thanks in advance.
real-analysis analysis convergence percolation
asked Nov 21 at 21:20
Matheus barros castro
375110
add a comment |
up vote
1
down vote
favorite
I was working on the proof of the exponential decay on supercritical percolation as shown in Grimmett's Percolation (1999, 2 ed. pg 206 - 210) and he uses as a lemma a form of the subadditive theorem (fekete) in which he states (he says it is easy to find) that
if you have a sequence $(a_n)_n$ such that $a_{n+m+2} leq a_m+a_n+g(n)$ and $lim_{n to infty} frac{g(n)}{n}=0$ then
the limit $lim_{n to infty} frac{a_n}{n} = lambda$ exists and it's equal to the $inf left{frac{a_n}{n}right}$.
$a_n geq (n+2)lambda-g(n);forall n$.
Now, although I coundn't prove the existence of the limit using only what was stated on the theorem, I manage to prove it adding some restrictions which are valid on Grimmet's theorem (I asked it in here: Subadditive lemma for $a_{m+n+2}leq a_n+a_m+g(m)$ and I found that, in my application, the last condition was satisfied).
Now assuming the limit exists, I couldn't manage to show that $$a_ngeq (n+2) lambda-g(n)$$
Any tips on how to proceed? Thanks in advance.
real-analysis analysis convergence percolation
asked Nov 21 at 21:20
Matheus barros castro
375110
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I was working on the proof of the exponential decay on supercritical percolation as shown in Grimmett's Percolation (1999, 2 ed. pg 206 - 210) and he uses as a lemma a form of the subadditive theorem (fekete) in which he states (he says it is easy to find) that
if you have a sequence $(a_n)_n$ such that $a_{n+m+2} leq a_m+a_n+g(n)$ and $lim_{n to infty} frac{g(n)}{n}=0$ then
the limit $lim_{n to infty} frac{a_n}{n} = lambda$ exists and it's equal to the $inf left{frac{a_n}{n}right}$.
$a_n geq (n+2)lambda-g(n);forall n$.
Now, although I coundn't prove the existence of the limit using only what was stated on the theorem, I manage to prove it adding some restrictions which are valid on Grimmet's theorem (I asked it in here: Subadditive lemma for $a_{m+n+2}leq a_n+a_m+g(m)$ and I found that, in my application, the last condition was satisfied).
Now assuming the limit exists, I couldn't manage to show that $$a_ngeq (n+2) lambda-g(n)$$
Any tips on how to proceed? Thanks in advance.
real-analysis analysis convergence percolation
asked Nov 21 at 21:20
Matheus barros castro
375110
I was working on the proof of the exponential decay on supercritical percolation as shown in Grimmett's Percolation (1999, 2 ed. pg 206 - 210) and he uses as a lemma a form of the subadditive theorem (fekete) in which he states (he says it is easy to find) that
if you have a sequence $(a_n)_n$ such that $a_{n+m+2} leq a_m+a_n+g(n)$ and $lim_{n to infty} frac{g(n)}{n}=0$ then
the limit $lim_{n to infty} frac{a_n}{n} = lambda$ exists and it's equal to the $inf left{frac{a_n}{n}right}$.
$a_n geq (n+2)lambda-g(n);forall n$.
Now, although I coundn't prove the existence of the limit using only what was stated on the theorem, I manage to prove it adding some restrictions which are valid on Grimmet's theorem (I asked it in here: Subadditive lemma for $a_{m+n+2}leq a_n+a_m+g(m)$ and I found that, in my application, the last condition was satisfied).
Now assuming the limit exists, I couldn't manage to show that $$a_ngeq (n+2) lambda-g(n)$$
Any tips on how to proceed? Thanks in advance.
real-analysis analysis convergence percolation
real-analysis analysis convergence percolation
asked Nov 21 at 21:20
Matheus barros castro
375110
asked Nov 21 at 21:20
Matheus barros castro
375110
asked Nov 21 at 21:20
Matheus barros castro
375110
asked Nov 21 at 21:20
Matheus barros castro
375110
asked Nov 21 at 21:20
Matheus barros castro
375110
375110
add a comment |
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008384%2fa-nm2-leq-a-ma-ngn-with-gn-on-show-that-a-n-geq-n2-lamb%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008384%2fa-nm2-leq-a-ma-ngn-with-gn-on-show-that-a-n-geq-n2-lamb%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
