JustWannaKnow
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Member for 4 years, 11 months
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Strong topology on a topological vector space
@JochenWengenroth this is because, in this topology, a net of operators $T_{\alpha}$ converges to $T$ iff $||T_{\alpha}x-Tx||\to 0$ for every $x \in X$, right?
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Strong topology on a topological vector space
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Strong topology on a topological vector space
Oh, right. My bad. Gonna fix it right now! Thanks
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Strong topology on a topological vector space
@NateEldredge thanks for the comment! Simon's terminology 'strong operator topology' for $\mathcal{L}(X,\mathbb{C}) = X^{*}$ is defined when $X$ is Banach. Here, you are defining this topology as I proposed?
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Reformulation - Construction of thermodynamic limit for GFF
Amazing! Thank you so much!
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Reformulation - Construction of thermodynamic limit for GFF
PS: you should definitely write a book or some lecture notes on these topics if you have the chance. You have a very nice approach to all this and we lack introductory books dealing with these topics in so rigorous and careful way, in my opinion.
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Reformulation - Construction of thermodynamic limit for GFF
Right! I'll keep this in mind. I prefer to work with topological vector spaces too. But I'm giving my first steps here so I feel that is important to know more than one point of view.
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Reformulation - Construction of thermodynamic limit for GFF
Thanks! I'm always studying your previous answers and they always help me a lot. Out of curiosity: why do some references prefer to address these problems as Gaussian processes? Is it just a matter of taste? I think this is where my confusion begun. If you check, say, Velenik's book, this very same problem is addresses in terms of random variables, Gaussian processes and Kolmogorov. I prefer to deal with it as you did, but these other references treat these problems in a way that they seem to be completely different.
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Reformulation - Construction of thermodynamic limit for GFF
It couldn't be more clear! Perfect answer again! Thank you so much for always saving me!
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Reformulation - Construction of thermodynamic limit for GFF
You said I can define $\tau_{L}$ as my favorite extension. If I put outise the box, then I would get the point but as I'm using the torus, I should extend it using periodization, don't I? If I understood it correctly, you said I should take $\tau_{L}$ as $\phi \mapsto \mathcal{F}_{per}$ but it is not clear to me that $\mathcal{F}_{per} \subset s'$.
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Reformulation - Construction of thermodynamic limit for GFF
Ok, I think I'm getting it. So, basically I must define $\tau_{L}: \mathbb{R}^{\Lambda_{L}}\to s'$. I have a finite volume measure $\mu_{\Lambda_{L}}$ on $\mathbb{R}^{\Lambda_{L}}$ on one side and $\tilde{\mu}_{K}$ on $s'$ on the other. This is where Levy's continuity should take place, right? Because I should prove $\mathbb{E}_{\mu_{\Lambda_{L}}}[E^{it\cdot \phi}] = \mathbb{E}_{\tilde{\mu}_{K}}[e^{i(t,\phi)}]$ for each $\tau_{L}$, and this would lead to weak convergence by Levy's continuity, right?
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Reformulation - Construction of thermodynamic limit for GFF
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Reformulation - Construction of thermodynamic limit for GFF
Just to clarify something here: each $\mu_{L}$ must be defined as a measure on $\mathbb{R}^{\mathbb{Z}^{d}}$ so that its restriction to $s'$ must converge weakly to $\tilde{\mu}_{K}$ right? And the construction of such measures on $\mathbb{R}^{\mathbb{Z}^{d}}$ must be done by Kolmogorov's Extension Theorem right? Then, the weak convergence should be proved using the characteristic function of these measures, if I understood it correctly.
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Reformulation - Construction of thermodynamic limit for GFF
no problem! I will keep trying to figure it out anyway! Thanks so much!
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Reformulation - Construction of thermodynamic limit for GFF
Also, he writes down the Green function as a limit of Riemann sums, not the Kernel of $-\Delta+m^{2}$ itself. Well, as you may notice I'm completely lost here.
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Reformulation - Construction of thermodynamic limit for GFF
@AbdelmalekAbdesselam yes, this is what I'm trying to prove. I'm following Kupiainen's lecture notes on renormalization group (2014). As far as I understand, he difines the Gaussian measure on $s'$ and he treats the infinite volume Gibbs measure as this measure. But this measure is defined by means of the function $W$ I defined in my post, and this could be done directly in the infinite volume limit in my opinion. So I don't get how this construction connects with the thermodynamic limit and the finite volume measures.