All Questions
Tagged with pr.probability mp.mathematical-physics
29 questions
35
votes
5
answers
11k
views
What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
- 921
63
votes
3
answers
7k
views
A roadmap to Hairer's theory for taming infinities
Background
Martin Hairer gave recently some beautiful lectures in Israel on "taming infinities," namely on finding a mathematical theory that supports the highly successful computations from quantum ...
- 24.7k
7
votes
0
answers
579
views
Guises of the noncrossing partitions (NCPs)
From "Noncrossing partitions in surprising locations" by Jon McCammond:
Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
- 10.5k
74
votes
16
answers
8k
views
Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?
What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...
6
votes
1
answer
999
views
A set of questions on continuous Gaussian Free Fields (GFF)
As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
- 3,515
6
votes
1
answer
707
views
Reformulation - Construction of thermodynamic limit for GFF
I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...
- 3,515
11
votes
1
answer
626
views
Formula for $U(N)$ integration wanted
Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group.
What I would like is a formula ...
6
votes
1
answer
274
views
Spectrum asymptotics for a product of $k$ random matrices?
How does the spectrum of a product of $k$ random matrices behave around 0?
In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=...
- 2,252
6
votes
2
answers
3k
views
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
- 1,026
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
94
votes
1
answer
11k
views
The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 23.5k
54
votes
4
answers
9k
views
Why is Quantum Field Theory so topological?
I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always ...
15
votes
1
answer
660
views
Which limit to take as a key applied math decision
The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...
11
votes
2
answers
353
views
Exponential decay of voltage potential difference
Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
- 1,495
11
votes
2
answers
2k
views
Free Boson Correlator $ \langle X(z)X(w) \rangle =- \ln |z - w| $
In physics papers, the massless free boson has a definition involving an action:
$$ S(X) = \frac{1}{8\pi} \int d\sigma^2\, \partial X \overline{\partial X}$$
The random functions $X(z)$ are ...
- 22.8k
10
votes
2
answers
2k
views
Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
- 2,071
8
votes
1
answer
527
views
A q,t-extension of Plancherel Measure thru Yang-Mills Theory ?
Buried in the physics paper by Nekrasov and Okounkov, a strange identity is proven:
$$ \prod_{n > 0} (1 - q^n)^{\mu^2-1} = \sum_{\mathbf{k}} q^{|\mathbf{k}|} \prod_{\square \in k} \left( 1 - \frac{\...
- 22.8k
8
votes
3
answers
2k
views
References request: constructive quantum field theory
I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman ...
- 663
8
votes
3
answers
2k
views
randomness in nature [closed]
What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...
- 307
7
votes
0
answers
497
views
Extreme unitary minimal models of conformal field theory
Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge
$$
c=1-\frac{6}{m(m+1)}\ .
$$
I ...
6
votes
2
answers
5k
views
Can I relate the L1 norm of a function to its Fourier expansion?
I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
- 185
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
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
4
votes
2
answers
564
views
A relation between the second moment of a distribution and one of its particular probability
I had recently posted a question here: To prove a relation involving a probability distribution
The relations quoted in the above question are used extensively in fluid mechanics and many other fields,...
user318428
2
votes
0
answers
1k
views
Can we pass to the limit in Poincaré-Jaynes-Bretthorst interpolation and deconvolution?
In Science and Hypothesis, chapter XI, The calculus of probabilities, Henri Poincaré deals informally with the fundamental problem of interpolation. He concludes (see http://ia600308.us.archive.org/21/...
2
votes
1
answer
161
views
Expected value of global functions in renormalization group
This is related to my previous question. I'm having some problems understanding the local to global program discussed in Brydge's lecture notes. We are assuming $C=C_{1}+\cdots+C_{N}$ is a covariance ...
- 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
2
votes
0
answers
491
views
Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question
This is a prequel to my question:
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Clearly my ...
- 1,026
1
vote
1
answer
505
views
Gaussian measures on non-separable spaces
Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero ...
- 8,512