Abstract: We construct a natural coupling between the continuum Gaussian free field (GFF) and the critical Ising magnetisation field (IMF) in a planar domain. In fact, we show that two independent IMFs with + boundary conditions and two independent IMFs with free boundary conditions are a deterministic function of a single instance of the GFF together with a sequence of independent coin flips. This construction should be seen as an extension of the bosonisation phenomenon, and to the best of our knowledge its existence has not been predicted before.
We arrive at our main result in the continuum by studying novel discrete structures. Our starting point is a coupling resembling the Edwards-Sokal coupling between the Ising model and the Fortuin-Kasteleyn random cluster model, though with role of the latter played by a different percolation model obtained from the double random current model. By taking a scaling limit of the coupling at criticality, we obtain a continuum Edwards-Sokal-like representation of the IMFs in terms of certain two-valued sets of the GFF introduced by Aru, Sepúlveda and Werner.
In the first talk we will review the known results on geometric representations of the IMF, and we will introduce the background necessary to state our main theorem. In the second talk we will discuss the new discrete Edwards-Sokal coupling, its fundamental properties like (partial) positive association, and some of the ingredients of the proof of the main result.
Joint work with Tomás Alcalde López.
Join the Zoom meeting with the link below: https://northeastern.zoom.us/j/94673771688?pwd=HhuTbvRVVM8OKxAzkgAVf8gV352KfM.1.
Note that you have to be signed in to a Zoom account to join. Please turn off your personal videos to reduce the bandwidth used!
You can access a recording of this seminar, as well as those of past seminars, in the Youtube channel: https://www.youtube.com/@oneworldprobability3373/videos
If you are having troubles with zoom, or if the capacity of the zoom room gets exceeded, you can also access to the Youtube live stream at the channel of the seminar: https://www.youtube.com/channel/UCiLiEQGTp6bZEhuHDM-WNWQ
We use zoom, a simple webinar tool. Zoom is not free, but only the hosts (that's us) need a license. All others can log-in via a link. Before the talk you can find the link on this website. Just click on the link and you will be sent to a webinar. There will be a host with control over the microphones.
Please subscribe to our mailing list for updates about the upcoming seminars and other One World events.