2D Ising model by Dmitry Chelkak : this is a topic which has been studies longtime, but there are still many questions open and the most tremendous progress is found in recent year. We may know the scaling limit of the interface of hexagon is $SLE_6$, but how about the others ? In fact, the scaling limit of the configuration is related to a similar random process - CLE conformal loop ensemble, another varied version of SLE. Moreover, a complete system of discrete complex analysis and discrete holomorphic tools are developed. There are a lot of open questions like Ising model on random maps.
Recurrence relation and dimer model by Paul Melotti : We can find a direct correspondent relation between recurrence series and the partition function of dimer model, which helps us express it explicitly, I ask the question if we could also find the recurrence for given partition function ? The answer is negative since the recurrence is always polynomial and this is not the case for any partition function - the integrable problem.
Flips of the triangulation on the sphere by Thomas Budzinski : The uniform triangulation is now a very popular topics. However, how we construct them ? The natural way is Monte-Carlo. We start from a configuration and we try to flip the configuration like MCMC metropolis method. Using the $n^4$ growth and the bottleneck property, Thomas gives a inferior bound for the mixing time as $n^{\frac{5}{4}}$. However, the upper bound is missing (but he says numerical result is like this.) And we want if we could do better ?
Expander by Simon Coste : OK, this is the topic of my PSC at Polytechnique. This talk still gives me some new idea. From the point of random graph, the distribution of random value has demi-circle law and this can be generalized to some more general case - graph oriented, We ask if random map could be a expander ? It seems not but we need to add the weight ... Anyway, the spectral analysis also is an important technique in random model.
A nonlinear SPDE by Perla EI Kettani : WOW, I find something that I look forward longtime. Once, I ask if Gaussian free field has some application in analysis and this is one example. If we consider the coefficient of each element of orthogonal base is not only a gaussian but also a Brownian motion, we generally use the stochastic analysis frame to treat PDE, Therefore, the GFF is one type of noise as one of my friend comments on my simulation. So, probability and analysis interacts and I believe that there will be more applications.
Deformation of random field by Julie Fournier : I didn't understand much but it seems like the deformation of random field.
Bismut-Elworthy-Li formula by Henri Elad Altman : Henri fires ! This talk is about the strong Feller property. Just like the transport equation can only keep the regularity but the diffusion equation can improve the regularity, we ask the same question for the semi-group. This formula applies for a general Ito process given that the drift isn't so degenerated. This tool is essential for the study of some kinds of equation like equation of KPZ.
Cost functional for large random trees by Marion Sciauveau : We would like to generalize the cost functional from a discrete random tree to a continuous tree coded by Brownian excursion. Before we do the convergence and in fact, we can also embed the discrete tree into the continuous tree. The cost functional helps study the DC complexity.
Hypercube percolation by Remco van der Hofstad : A exhaustive study about the percolation on the hypercube, It is hard to imagine that the work is done without simulation. The critical point is given by
$$
\mathbb{E}_{p_c}[C_0] = 2^{n/3}
$$
and the window is about $2^{-n/3}$, which means in this period we see a drastic transition of phase.
I think a seminar like this gives us a quick understand of different direction of probability, since the subject is always very various in this domain. On the other hand, to make new friends during the seminar is also interesting. However, to see those who grow up together from prepa, master and become collaborator is an envy.
Recurrence relation and dimer model by Paul Melotti : We can find a direct correspondent relation between recurrence series and the partition function of dimer model, which helps us express it explicitly, I ask the question if we could also find the recurrence for given partition function ? The answer is negative since the recurrence is always polynomial and this is not the case for any partition function - the integrable problem.
Flips of the triangulation on the sphere by Thomas Budzinski : The uniform triangulation is now a very popular topics. However, how we construct them ? The natural way is Monte-Carlo. We start from a configuration and we try to flip the configuration like MCMC metropolis method. Using the $n^4$ growth and the bottleneck property, Thomas gives a inferior bound for the mixing time as $n^{\frac{5}{4}}$. However, the upper bound is missing (but he says numerical result is like this.) And we want if we could do better ?
Expander by Simon Coste : OK, this is the topic of my PSC at Polytechnique. This talk still gives me some new idea. From the point of random graph, the distribution of random value has demi-circle law and this can be generalized to some more general case - graph oriented, We ask if random map could be a expander ? It seems not but we need to add the weight ... Anyway, the spectral analysis also is an important technique in random model.
A nonlinear SPDE by Perla EI Kettani : WOW, I find something that I look forward longtime. Once, I ask if Gaussian free field has some application in analysis and this is one example. If we consider the coefficient of each element of orthogonal base is not only a gaussian but also a Brownian motion, we generally use the stochastic analysis frame to treat PDE, Therefore, the GFF is one type of noise as one of my friend comments on my simulation. So, probability and analysis interacts and I believe that there will be more applications.
Deformation of random field by Julie Fournier : I didn't understand much but it seems like the deformation of random field.
Bismut-Elworthy-Li formula by Henri Elad Altman : Henri fires ! This talk is about the strong Feller property. Just like the transport equation can only keep the regularity but the diffusion equation can improve the regularity, we ask the same question for the semi-group. This formula applies for a general Ito process given that the drift isn't so degenerated. This tool is essential for the study of some kinds of equation like equation of KPZ.
Cost functional for large random trees by Marion Sciauveau : We would like to generalize the cost functional from a discrete random tree to a continuous tree coded by Brownian excursion. Before we do the convergence and in fact, we can also embed the discrete tree into the continuous tree. The cost functional helps study the DC complexity.
Hypercube percolation by Remco van der Hofstad : A exhaustive study about the percolation on the hypercube, It is hard to imagine that the work is done without simulation. The critical point is given by
$$
\mathbb{E}_{p_c}[C_0] = 2^{n/3}
$$
and the window is about $2^{-n/3}$, which means in this period we see a drastic transition of phase.
I think a seminar like this gives us a quick understand of different direction of probability, since the subject is always very various in this domain. On the other hand, to make new friends during the seminar is also interesting. However, to see those who grow up together from prepa, master and become collaborator is an envy.
没有评论:
发表评论