I'm looking for references for the rate of escape and return probability for a group of intermediate growth.


Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then (via isoperimetry, the only reference I have for this is Woess' book) one gets that the return probability $P\_{2n}(e,e) \preceq \mathrm{exp}(-n^{\frac{\alpha}{\alpha+2}})$.


Let $0< \beta < 1$. Using entropy and Amir-Virag's paper on the rate of escape (proposition 8), if the volume growth is $\preceq \mathrm{exp}(n^\beta)$, then the rate of escape ($\mathbb{E}(|W\_n|)$ where $W\_n$ is the time $n$ distribution and $|\cdot|$ is word length) is $\mathbb{E}(|W\_n|) \preceq n^{\frac{1+\beta}{2}}$


$\mathbf{Question:}$ Assuming the volume growth is $\succeq \mathrm{exp}(n^\alpha)$ and $\preceq \mathrm{exp}(n^\beta)$ for $0<\alpha < \beta < 1$, is there any other known bound (lower or upper) known for these quantities? 


P.S.: It's not possible to improve these upper bounds for exponential growth (i.e. $\alpha = \beta = 1$), and the lower bounds (in this generality) are $\mathrm{exp}(-n)$ (e.g. for a non-amenable group) and $n^{1/2}$ (e.g. for a polycylic group, see above reference).



Estimates for simple random walks in groups of intermediate growth
It was pointed out to me that Lemma 5.1 in [this](https://eudml.org/doc/116198) paper of Erschler (Critical constants for recurrence of random walks on $G$-spaces. Ann. Inst. Fourier (Grenoble), 55(2):493--509, 2005) gives bounds on speed and entropy in terms of volume (the bound on entropy is implicit in the proof). Namely, volume growth $\preceq \mathrm{exp}(n^\beta)$ implies $\mathbb{E}|W\_n| \preceq n^{1/(2-\beta)}$.


Some of these estimates are done for measures which are not finitely supported in section 4 of 


[arXiv:1403.1195](http://arXiv.org/abs/1403.1195) -- A. Gournay. The Liouville property and Hilbertian compression.


And an amazing relation between return probability and speed [through entropy] is descirbed in


[arXiv:1510.08830](http://arXiv.org/abs/1510.08830) -- L. Saloff-Coste and T. Zheng. Isoperimetric profiles and random walks on some permutation wreath products.


For groups of exponential growth, here are references on the possible behaviour of speed and entropy:


[arXiv:1505.03294](http://arxiv.org/abs/1505.03294) -- J. Brieussel. About the speed of random walks on solvable groups.


[arXiv:1509.00256](http://arXiv.org/abs/1509.00256) -- G. Amir. On the joint behaviour of speed and entropy of random walks on groups.


[arXiv:1510.08040](http://arxiv.org/abs/1510.08040) -- J. Brieussel, T. Zheng. Speed of random walks, isoperimetry and compression of finitely generated groups


