# Reference request: log Sobolev inequality for uniform measure (uniform distribution over discrete set)

Suppose that $$N \in \mathbb N_+$$ is fixed and denote by $$\mu = (\mu_0,\ldots,\mu_N)$$ the uniform distribution on the set $$\{0,1,\ldots,N\}$$ (i.e., $$\mu_n = \frac{1}{N+1}$$ for each $$0\leq n\leq N$$). I am wondering if there exists a log-Sobolev inequality for the uniform distribution $$\mu$$, in the sense that $$\sum_{n=0}^N \mu_n f^2_n \log f^2_n \leq C\sum_{n=0}^{N-1} \mu_n(f_{n+1}-f_n)^2 \label{1}\tag{1}$$ for some universal constant $$C$$, where $$f = (f_0,\ldots,f_N) \in \mathbb{R}^{N+1}_+$$ satisfies $$\sum_{n=0}^N \mu_n f^2_n =1$$. Of course, \eqref{1} is merely a discrete analog of the log-Sobolev inequality for the uniform measure on a one dimensional compact interval, which takes the following form: $$\int_0^N f^2 \log f^2 \mathrm{d}\mu \leq C \int_0^N |\nabla f|^2 \mathrm{d}\mu$$ where $$\mu$$ is the uniform distribution on the interval $$[0,N]$$ and $$f \colon [0,N] \to \mathbb{R}_+$$ satisfies the constraint that $$\int_0^N f^2 \mathrm{d}\mu = 1$$. I am aware of some literatures on log Sobolev inequalities for uniform measures on compact intervals (or finite Lebesgue measure) such as this reference, but it seems very hard to locate the precise reference for the corresponding result (for discrete state space) stated as in the form of \eqref{1}. Any help or pointer to related references are greatly appreciated (and I am not interested in the optimal/sharp log-Sobolev constant) !

Remark: when $$N=1$$, the log-Sobolev inequality \eqref{1} boils down to the well-known log-Sobolev inequality for (symmetric) Bernoulli distribution due to Leonard Gross (1975) that can be found in here for instance.

Edit: Here $$N$$ is fixed and for sure $$C$$ can depend on $$N$$, but $$C$$ cannot depend on the specific choice of $$f$$

• No. This fails when $f_1=\sqrt{N}$, $f_j=0$ otherwise (or very small if you want to avoid the value zero). Commented Jan 31 at 21:18
• That is, there is no constant that is independent of $N$. Existence of a $C(N)$ should follow from compactness of the set of $f$'s considered. Commented Jan 31 at 21:20
• Hello Christian: $N$ is fixed and of course $C$ is allowed to depend on $N$, but $C$ cannot depend on the choice of $f$ (as long as $f$ satisfies the stated constraint) Commented Jan 31 at 21:32
• @FeiCao By scaling, the continuous version has $C \propto N^2$, so that's what one would expect here (one can see that one cannot do better, by considering $f_n = F(n/N)$ for $F$ smooth). Commented Feb 4 at 10:12
• @AndréSchlichting Thank you very much for this new reference preprint (which is very recent)! I think that's all I need Commented Feb 7 at 19:09

One can use the results by Diaconis & Saloff-Coste "Logarithmic Sobolev inequalities for finite Markov chains" in Section 4.2, where the proof the result for the symmetric random walk on the circle with $$N$$ nodes is given and takes the form $$\frac{1}{N}\sum_{n=0}^N f^2_n \log f^2_n \leq C^\circ_N \frac{1}{N}\sum_{n=1}^{N} (f_{n+1}-f_n)^2 \tag{LSI^\circ_N} \,$$ where $$f_{N+1}=f_1$$ (note that I slightly changed the numbering from the OP to the set {1,\dots,N}, which is more consistent with the reference). Then the inequality (1) of the OP can be recovered by noting that $$(f_N-f_1)^2 = \left( \sum_{n=1}^{N-1} (f_{n+1}-f_n) \right)^2 \leq N^2 \sum_{n=1}^N (f_{n+1}-f_n)^2 .$$ Hence, we get that $$C$$ in (1) can be estimated by $$C^{\circ}_N + N^2$$.
To apply the results from Diaconis & Saloff-Coste, we translate the more functional analytic form of LSI$$^\circ_N$$ to Markov chains, see also Appendix A in arXiv:2312.13284 from where I took the material below.
The right-hand side of LSI$$^\circ_N$$ is related to the Dirichlet form of the symmetric random walk on the circle $$\{1,\dots,N\}$$ with jump kernel $$K(\kappa,\kappa\pm 1)=\frac{1}{2}$$ and zero else. Indeed, we find $$\begin{equation*} \mathcal{D}(f,f) = \frac{1}{2N} \sum_{\kappa,\lambda=1}^N \bigl(f_\kappa-f_\lambda\bigr)^2 K(\kappa,\lambda)= \frac{1}{2N} \sum_{\kappa=1}^N \bigl( f_\kappa - f_{\kappa+1})^2 %= \frac{\delta^2}{2} \delta \sum\nolimits_\kappa \abs*{ \partial^{\delta}_+ f}^2 \end{equation*}$$ The left-hand side of LSI$$^\circ_N$$ is the relative entropy of the measure $$f^2/N$$ with respect the uniform measure on $$\{1,\dots,N\}$$ defined by $$\operatorname{Ent}(f^2) = \frac{1}{N} \sum_{\kappa=1}^N f_\kappa^2 \log \frac{f_\kappa^2}{\frac{1}{N}\sum_{\lambda=1}^N f_\lambda^2} = \frac{1}{N} \sum\nolimits_\kappa f_\kappa^2 \log f_\kappa^2 ,$$ provided $$f^2$$ is normalized such that $$\frac{1}{N}\sum_{\lambda=1}^N f_\lambda^2=1$$. Then, the logarithmic Sobolev constant is defined by $$\alpha=\min\bigl\{ \mathcal{D}(f,f)/\operatorname{Ent}(f^2): \operatorname{Ent}(f^2)\neq 0\bigr\}$$. The result in Diaconis & Saloff-Coste §4.2 implies that $$\alpha \geq 8\pi^2/(25 N^2)$$, which translates to $$C^{\circ}_N \leq \frac{25}{16\pi}N^2 \qquad\text{hence}\qquad C \leq \left(\frac{25}{16\pi}+1\right) N^2 \approx 1.5 N^2 .$$ all exact numerical values and factors of two without guarantees