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We know that two dimensional discrete GFF(2d-DGFF) on a box $V_N=N[0,1]^2\cap\mathbb{Z}$ with Dirichlet boundary condition will converge in distribution to the 2d continuum GFF with Dirichlet boundary condition, in the sense of convergence of the field acting on any suitable test functions. And we know 2d-DGFF can be seen as an analogue of branching random walks, especially 4-ary branching random walks with i.i.d. Gaussian increments.

So we embed the family tree of a 4-ary tree on the interval $[0,1]$, in the following sense. We code the root as 0, and code its each offspring with 0 and add another integer in the order of 0,1,2, and 3: 00,01,02,03. Continue doing this we get a code for the whole family tree. For a tree of depth $n$, we get $4^n$ leaves each labelled by $(n+1)$ integers from $\{0,1,2,3\}$. And we map these nodes to the interval $[0,1]$ in the 4-adic way. For $u=(0,u_1,\cdots,u_n)$ we map it to $\pi(u):=\sum_{i=1}^n \frac{u_i}{4^i}$. Finally, we assign i.i.d. standard Gaussian increments $Y_{e}$ on the edges and let $X_{\pi(u)}=X_u:=\sum_{e\in 0\leftrightarrow u}Y_e$ as the Gaussian branching random walk embedded on $[0,1]$.

Question is: does this field converge in distribution(in the sense against any test functions) to some random generalized function on $[0,1]$ like continuum GFF? I think the answer should be yes, then how to describe it?

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    $\begingroup$ That's basically a Mandelbrot cascade. See section 2 of the review by Duplantier et al on log-correlated fields arxiv.org/abs/1407.5605 and the article by Benjamini and Schramm arxiv.org/abs/0806.1347 Note that it is a bit artificial to construct the continuum object on $\mathbb{R}^n$. A more natural limit object is the GFF over $p$-adics. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Aug 18, 2023 at 14:43
  • $\begingroup$ @AbdelmalekAbdesselam Thanks for your comment! Is there any reference for the convergence to GFF over p-adics? $\endgroup$
    – MikeG
    Commented Aug 19, 2023 at 17:23
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    $\begingroup$ I don't know of good references for your question. Convergence here means weak convergence of probability measures on the Schwartz-Bruhat space of distributions $S'(\mathbb{Q}_p^d)$. The needed result is the Levy Continuity Theorem. Via the isomorphism to $\mathbb{R}^{N}$, this reduces to the theorem discussed in mathoverflow.net/questions/277678/… Then, because these are Gaussian measures, you only need to show pointwise convergence of the covariance... $\endgroup$
    – Abdelmalek Abdesselam
    Commented Aug 21, 2023 at 14:06
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    $\begingroup$ ...bilinear form applied to fixed test functions when removing the UV/short distance truncation. This can be done by hand through an explicit computation. I gave a talk at the Leipzig AQFP seminar on related matters, maybe that can help, see youtube.com/watch?v=CZLGJPRXyq0 $\endgroup$
    – Abdelmalek Abdesselam
    Commented Aug 21, 2023 at 14:10
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    $\begingroup$ In my talk the quantity $[\phi]$ would have to be set equal to zero, for the log-correlated case. Also, the measure and Levy continuity limit should be on $S'_0(\mathbb{Q}_p^d)$ instead of $S'(\mathbb{Q}_p^d)$. Namely, it lives on the strong dual of the space of test functions with zero spatial integral. This is similar to the real GFF, which can also be obtained as a $\epsilon\rightarrow 0$ limit of a fractional field, see mathoverflow.net/questions/172916/… $\endgroup$
    – Abdelmalek Abdesselam
    Commented Aug 21, 2023 at 15:25

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