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16h
comment 2 D random walk first return to origin
Let us continue this discussion in chat.
19h
comment 2 D random walk first return to origin
@CuriousOne : Ok, I give up, you're a lost cause for mathematical physics ;) . That of course does not make you a bad physicist, but it's a pity: you're missing on so many fascinating things. To each his own, I guess. Well, have a nice day too!
20h
comment 2 D random walk first return to origin
@CuriousOne : [...] As a different example, the Mermin-Wagner theorem for ferromagnetic classical XY model with interactions of the form $-J(x-y)\vec S_x\cdot\vec S_y$ holds if and only if the random walk with transition probabilities $J_{x}/\sum_y J_y$ is recurrent. Of course, there are zillions other examples of their importance.
20h
comment 2 D random walk first return to origin
@CuriousOne : but maybe what you want are non-trivial applications of random walks in stat. mechanics (not this specific estimate)? Of course, they have obvious use to model diffusions or polymers. But they play also an important role in many other places: there are, for example, many types of random walk representations for correlation functions of various models in stat. mech. and field theory (e.g., those originally derived by Symanzik in physics). [...]
20h
comment 2 D random walk first return to origin
@CuriousOne : Well, and concerning the technical content of my papers, I suppose that you haven't published many maths papers. Mine are often long, technical, and I certainly do not remember all the (classical) technical tools I made use of in papers published years ago. There is no doubt that I have been using similar estimates in several papers (though maybe not explicitly this one). The fact that I cannot even assert this shows how much random walks are relevant to stat. mechanics: I rely on their properties all the time.
20h
comment 2 D random walk first return to origin
@CuriousOne : Concerning the distribution of first return to the origin for the random walk on $\mathbb{Z}^d$, you can find early result in the classical paper by Dvoretzky and Erdös (Some problems on random walks in space), published in 1951. There, the authors are mostly interested in the range of the range (number of distinct visited vertices) of the simple random walk on $\mathbb{Z}^d$, but along the way they derive asymptotics for the probability that the random has not yet returned at its starting point after time $n$, which might be what the OP is asking about (who knows?).
1d
comment 2 D random walk first return to origin
@CuriousOne : I am not saying that this particular question is good: I haven't upvoted it, nor made any effort to answer it. It is even far from clear what is being asked. That being said, estimates on first hitting times do play an important role in many probabilistic arguments (also in a statistical mechanical context; I don't have an immediate example, although I am pretty sure I could even find some in my own papers if I tried) . Anyway, what I find really annoying is your systematic criticism of mathematical topics. Like it or not, maths is the proper language to discuss physics.
1d
comment 2 D random walk first return to origin
@CuriousOne : it doesn't matter. You are once more confusing real systems (in the lab) with our modelization of them. Physics without an underlying theoretical understanding (and, yes, this means mathematical understanding) is just completely empty. Now, these theoretical constructs on which we base our current understanding of many, completely different, phenomena often depend on properties of random walks (and closely related objects, such as harmonic functions, etc.). Of course, these mathematical objects might not exist in the lab, but this is completely irrelevant...
1d
comment 2 D random walk first return to origin
@CuriousOne : Again, Nature might not, but properties of random walks on lattice do play an important role in many aspects of our current understanding of physics. You really have much too narrow a vision of Physics.
Feb
7
answered Why do we work in thermodynamic limit in statistical physics?
Feb
1
comment Relation between the $N$ particle partition function and probability?
I agree with the answer, but disagree with the characterization of the partition function as the most important quantity in statistical physics (even though it is often presented as such in textbooks). This central role should be reserved to the probability measure itself. Indeed, the latter gives you access to much more information (even about macroscopic quantities) than the partition function, which only gives you access to the statistical properties of the macroscopic quantities appearing in the Boltzmann weight: energy, magnetization, etc.).
Jan
28
awarded  Custodian
Jan
28
reviewed Approve What exactly “triggers” a matter/antimatter detonation?
Jan
22
comment Reference request: 2D conformal field theory and functions on the triangular lattice
You should google "discrete complex analysis". There are many papers introducing these ideas, for example those by Chelkak, Smirnov, etc.
Jan
22
comment Cluster Expansion
Done. I tried to make a readable answer out of the comments, but had only limited time, so there might be typos.
Jan
22
answered Cluster Expansion
Jan
22
comment Cluster Expansion
Yes you sum over all possible values of $n_\ell$ for each possible values of $\ell$ (so $\{n_\ell\}$ specifies the values of $n_1, n_2, n_3, \ldots$). Then, once these are fixed, you take the product of all the functions $f_\ell(n_\ell)$ (for these specific values of $n_1,n_2,n_3,\ldots$).
Jan
22
comment Cluster Expansion
Sure. Imagine that the only allowed cluster sizes are $\ell=1$ and $\ell=2$ (so that I can write things down explicitely). Let us use the notation $f_\ell(n_\ell) = \frac{1}{n_\ell!}\bigl( \frac{e^{\ell\beta\mu}b_\ell}{\lambda^{3\ell}\ell!} \bigr)^{n_\ell}$. Then your identity becomes $\sum_{\{n_1,n_2\} \in\{0,1,2,\ldots\}^2} \prod_{\ell=1}^2 f_\ell(n_\ell) = \sum_{n_1\geq 0} \sum_{n_2\geq 0} f_1(n_1) f_2(n_2)= \sum_{n_1\geq 0} f_1(n_1) \sum_{n_2\geq 0} f_2(n_2)= \prod_{\ell=1}^2 \sum_{n_\ell\geq 0} f_\ell(n_\ell)$
Jan
21
comment Cluster Expansion
This is just linearity of the sum. Maybe it will become obvious if you look at the following particular case: $\sum_{i,j} a_i b_j = \sum_i \sum_j a_i b_j = \Bigl(\sum_i a_i\Bigr)\Bigl(\sum_j b_j\Bigr)$, which follows by pulling out $a_i$ from the inner sum and then pulling out $\sum_j b_j$ from the outer sum. Now, in your formula $\{n_\ell\}$ denotes the number of clusters of each possible length (so $\sum_{\{n_\ell\}} = \sum_{n_1\geq 0}\sum_{n_2\geq 0}\cdots = \prod_{\ell\geq 1} \sum_{n_\ell\geq 0}$).
Jan
19
comment What are alternative ways to think about transfer matrix as used in Ising model?
@Nathaniel : yes, that's exactly the point. If you denote by $\varphi^{*,1}$ the left Perron-Frobenius eigenvector (associated, of course, to the same eigenvalue $\lambda_1$), then the only thing that changes is the expression for the invariant measure, which becomes $\mu( \sigma) = \varphi^{*,1}_\sigma \varphi^1_\sigma$. (Note also that you could define another Markov chain with transition probabilities $\pi^*(\sigma,\sigma') = \frac{\varphi^{*,1}_{\sigma'}}{\lambda_1\varphi^{*,1}_\sigma} T_{\sigma',\sigma}$; this corresponds to the time-reversed chain (which has the same invariant measure).