All Questions
12 questions
2
votes
1
answer
158
views
Definition of average $\langle \langle \cdot \rangle \rangle$
I started reading the paper Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model and I would like to clarify the notation $\langle \langle \cdot \rangle\rangle$ the authors use in ...
- 3,515
2
votes
1
answer
238
views
Thermodynamic limit and Gaussian measures
Let $\Lambda \subset \mathbb{Z}^{d}$ be finite and fixed and consider $\mathbb{R}^{|\Lambda|}$ be the vector space of all sequences $\varphi = (\varphi_{x})_{x\in \Lambda}$. We equip $\mathbb{R}^{|\...
- 3,515
1
vote
1
answer
184
views
Measure, volume and cardinality on Minlos' book on statistical physics
The following content was based on Minlos' book on statistical physics. Let $\Lambda \subset \mathbb{R}^{d}$ be fixed (Minlos takes $d=3$ but I think the ideas follow without change to $d \ge 1$). We ...
- 1,305
4
votes
2
answers
267
views
Grand-canonical Gibbs measure for continuous systems
Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+...
- 3,515
2
votes
2
answers
294
views
Imprecise Definition of a $\sigma$-algebra
I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...
- 3,515
1
vote
1
answer
176
views
Gaussian Property of the Renormalization Group
Let $\Lambda \subset \mathbb{Z}^{d}$ be a finite set and $\varphi = (\varphi_{x})_{x\in \Lambda} \in \mathbb{R}^{|\Lambda|}$. Let $F^{\Lambda}=F^{\Lambda}(\varphi)$ be a real-valued global function, ...
- 3,515
3
votes
0
answers
126
views
Other than Brownian motion, when else is it possible to define "normalized weighted infinite dimensional Lebesgue measure"?
In this article Sourav Chatterjee poses the question, how do we define the measure:
$$d\mu(A)=\frac{1}{Z}\exp\left(-\frac{1}{4g^2}S_{YM}(A)\right)dA$$
The $Z$ here is an infinite normalizing ...
user69208
5
votes
1
answer
945
views
Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?
In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement ...
- 1,026
2
votes
1
answer
245
views
Probability measures on $L^p$
Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
- 8,512
10
votes
2
answers
2k
views
When is a space of measures a measurable space?
Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
- 8,512
5
votes
1
answer
437
views
Stationary, ergodic measures from the structuralist point of view
Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
- 8,512
10
votes
1
answer
3k
views
Applications of Banach-Tarski Paradox to Probability Theory?
I was just curious, since the B-T paradox is a measure theoretic result, if there are any consequences of this paradox in probability theory? Also, is there is a way of stating the B-T paradox in the ...
- 137