Search Results

Since the product of real numbers is commutative, for any permutation $\sigma$, $$ \int_{S'}\phi(f_{\sigma(1)})\cdots\phi(f_{\sigma(n)})\ d\mu(\phi)= \int_{S'}\phi(f_{1})\cdots\phi(f_{n})\ d\mu(\phi) …

I am not aware of a theorem which has UV stability as a hypothesis and delivers the construction of a continuum limit, even modulo taking subsequences. UV stability is just a combination of suitable u …

My formula in the comment for $\|z\|=1$, was obtained as follows. First note that the integral is well defined in $[0,\infty]$, and by monotone convergence, is given by $$ \int_{S^{2n-1}}\frac{1}{|1-\ …

There is a choice to be made here: working with Banach spaces or with spaces of distributions like $\mathscr{S}'(\mathbb{R}^d)$. There are pros and cons for each of these two settings. Let me stick to …

This is easy to do using a graphical calculus for contractions of old-fashioned tensors. See this recent article for an example of application of such techniques and hopefully useful references. Here …

This is easy to compute using techniques from perturbative quantum field theory, i.e., Wick's Theorem (due to Isserlis) for moments of Gaussian measures. Consider $$ (2\pi)^{-\frac{p}{2}}\int_{\mathbb …

I could add a couple things to the already existing good answers. Continuous time processes are not that bad/that much more difficult to define or to simulate numerically in practice. For a Markov ch …

Cumulants have many other names depending on the context (statistics, quantum field theory, statistical mechanics,...): seminvariants, truncated correlation functions, connected correlation functions, …

The result is $$ I(k)=\frac{2\pi^4\ \Gamma\left(\frac{k}{2}+1\right)\Gamma\left(\frac{k}{2}+2\right)}{\Gamma(k+4)}\ , $$ which can be simplified a bit more using the Legendre Duplication Formula. More …

While waiting for more context around the question, one can already mention the main definitions and tools for this topic. I will assume the space on which these probability measures live is the space …

The axioms don't tell you what theory you constructed. For that you need to go beyond the construction of correlation functions of the elementary field $\phi$ (the basic chapter on renormalization in …

Essentially, what is asked is the continuation of my previous MO answer Reformulation - Construction of thermodynamic limit for GFF and the solution of the exercise I mentioned at the end of that answ …

For $x\in\mathbb{Z}^d$ I will denote by $\bar{x}$ the corresponding equivalence class in the discrete finite torus $\Lambda_{L}=\mathbb{Z}^d/L\mathbb{Z}^d$. I will view a field $\phi\in\mathbb{R}^{\La …

The source of the confusion is not saying explicitly what are the sets and $\sigma$-algebras the measures are supposed to be on. For example, a sentence like ''By Kolmogorov's Extension Theorem, there …

Let me comment on points 4) and 5). The problem with infinities in QFT or traditional equilibrium statistical field theory is related to the one addressed by Martin's theory but there are some differe …