1,517 reputation
516
bio website unige.ch/math/folks/velenik
location Geneva, Switzerland
age 44
visits member for 2 years, 2 months
seen 1 hour ago

Professor at the Mathematics department of the University of Geneva. Mostly working at the intersection of probability theory and statistical physics.

With a colleague, we are in the process of writing an introductory book on the (classical) equilibrium statistical physics of lattice spin systems, from a mathematically rigorous point of view. A more detailed description of the project, as well as early drafts of several chapters can be found here.


Jul
7
comment Numerical Ising Model - Wolff algorithm and correlations
@Learningisamess: Let me reiterate what I said as a comment to the other answer. You can completely avoid the issue of determining the number of steps required to thermalize: just use a perfect simulation algorithm; when the latter stops, you are guaranteed that the system has thermalized! One such (cluster) algorithm is described in this paper: M. Huber, a bounding chain for Swendsen-Wang, Random Structures & Algorithms, 22(1) :43–59, 2003. It is trickier to implement than the standard algorithm, but it is very efficient and you get all the benefits of perfect simulation.
Jul
6
comment Numerical Ising Model: Swendsen–Wang algorithm, Percolationtheory?
If you want to understand the relation between the Swendsen-Wang algorithm and percolation, relevant keywords are Fortuin-Kasteleyn's random cluster representation and Edwards-Sokal coupling. There exist several mathematical analyses of relaxation properties of this dynamics. See for example Mario Ullrich's PhD thesis and references therein; the latter can be found on his homepage.
Jul
5
comment Numerical Ising Model - Wolff algorithm and correlations
Why not implement a perfect simulation algorithm? Then you'll be guaranteed that each sample is drawn without any statistical error from the Gibbs distribution. It is possible to do that also with cluster algorithm. The one I implemented (years ago, to make illustrations for a course I was giving) was based on this paper: M. Huber. A bounding chain for Swendsen-Wang. Random Structures Algorithms, 22(1) :43–59, 2003. It's a bit tricky to implement properly, but very efficient, and you get all the benefits of perfect simulation.
Jul
2
comment Critical temperature difference between Ising and XY model
No it's just that for the Ising model, the loop and spin models coincides. This is really due to the fact that the spins take values $\pm 1$, since this implies that $e^{\beta \sigma_i\sigma_j} = \cosh(\beta)(1+\tanh(\beta)\sigma_i\sigma_j)$. (Of course the prefactor $\cosh(\beta)$ in the right-hand side has no impact on the probability distribution.)
Jul
1
comment Critical temperature difference between Ising and XY model
I have no idea what the value of the coupling constant of the spin O(2) model on the triangular lattice is. There are very efficient cluster methods for this model (as far as I know, I am not a specialist in the numerical aspects) which should provide precise estimates.
Jul
1
comment Critical temperature difference between Ising and XY model
The value of the critical points of the spin and loop models should not be related at all. However, the critical behaviors are expected to be the same (they should belong to the same universality class).
Jul
1
comment Critical temperature difference between Ising and XY model
No, the loop model is an approximation of the O(n) spin model. You can look, for example, at the explanations in Nienhuis' Les Houches lectures on Loop models, which can be downloaded from his homepage.
Jun
28
comment Critical temperature difference between Ising and XY model
Your intuition is correct. It is not difficult to prove that the 2-point function of the Ising model is always an upper bound on the corresponding quantity for the XY model. In particular, if $\beta_c^{XY}$ denotes the inverse temperature at which the Kosterlitz-Thouless phase transition occurs, then $\beta_c^{XY}\geq 2\beta_c^{I}$, where $\beta_c^I$ is the inverse critical temperature for the Ising model. You can find the proof here. Note that the formula you give above is not for the $O(n)$ spin model, but for the $O(n)$ loop model.
Jun
24
comment Is there a reasonable lower bound for free energy per site of the 2D Ising model in the presence of an external field?
I am not sure that I understand your question. There are many possible explicit lower bounds on the free energy. What additional properties do you want? For example, a well-known lower bound on the partition function (and thus on $\psi$) is given by the mean-field partition function.
Jun
5
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
@NikosM.: he says "there are no known examples of classical probabilities that don't have a quantum mechanical origin", and wants to reduce it to QM. He's not limiting his discussion to application of PT to QM, but speaking of PT in general. From this point of view, this is plain wrong.
Jun
4
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
@CountIblis: You're missing my point. There is no need to have any randomness to apply probability theory. So there is absolutely no need of quantum mechanics to derive or justify classical probability theory. To say that it requires quantum mechanics because the universe is fundamentally quantum, would be the same as saying that, say, literature is reducible to quantum mechanics because the brains of writers are quantum objects... Maybe true, but utterly useless.
Jun
4
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
@CountIblis: Even if in a fully deterministic universe, probability theory would be indispensable. Including in its application to physics. You (and the authors of this paper) have an extremely narrow view on this field.
Jun
3
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
Real randomness (inasmuch as such a thing exist) might indeed have quantum origin, but one does not need anything truly random to be able to apply probability theory successfully, just situations of sufficient complexity that renouncing a precise description is the only way to proceed.
Jun
3
comment Why is the application of probability in QM fundamentally different from application of probability in other areas?
Absolutely not. Probability theory has a much broader scope than the "physical" probability theory you seem to have in mind. Probability theory can really be seen as an extension of logic, and as such applies to all situations in which only partial information is available, or in which the probability represents subjective assessments of likelihood. You should have a look, for example, at Jaynes' book "Probability Theory: The Logic of Science".
May
26
comment 2d Ising model in CFT and statistical mechanics
Maybe you could look at Cardy's lecture notes, for example (both the 1988 and the 2008 ones). They can be found on his homepage.
May
26
comment Fluctuations in energy for macro and micro canonical ensembles
Consider a super simple example: you have $N$ objects each of which may be either in its "ground state" of energy $0$ or in an "excited state" of energy $\epsilon$. There is no interaction between these objects. The partition function is $\sum_{k=0}^N\binom{N}{k}e^{-\beta\epsilon k}=(1+e^{-\beta\epsilon})^N$, and thus the probability that the energy is equal to $E=M\epsilon$ is $\binom{N}{M}e^{-\beta\epsilon M}/(1+e^{-\beta\epsilon})^N$, which you can check develops a sharp peak (of width of order $\sqrt{N}$) around its expected value $N\epsilon/(1+e^{\beta\epsilon})$.
May
25
comment Fluctuations in energy for macro and micro canonical ensembles
The Boltzmann distribution is not an exponential distribution: the Boltzmann weight has certainly an exponential form, but you're forgetting the entropy (that is, how many microstates of a given energy there are). The distribution of the energy density in the canonical ensemble concentrates sharply on its average (with fluctuations of the order of $1/\sqrt{N}$). This is why you get equivalent results for both microcanonical (fixed energy) and canonical (fluctuating energy) ensembles (well, at least in the absence pf phase transition).
May
23
answered Minimum connectivity required for mean field to be a good approximation?
May
23
comment Minimum connectivity required for mean field to be a good approximation?
If you're interested in critical exponents, then you can, in principle, extract (probably bad) bounds from this paper: arXiv:0712.0312.
May
23
comment Minimum connectivity required for mean field to be a good approximation?
Well, a good approximation in what sense? If you want, say, close free energy and magnetization, or the ability to say that the system has a first order phase transition if its mean-field approximation has, then you might be able to extract estimates (although rather pessimistic, I suspect) from this paper: arXiv:math/0207242. There are also two subsequent papers, if I remember correctly.