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Entropic curvature on graphs along Schrödinger bridges at zero temperature

Abstract

Lott–Sturm–Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. Léonard by replacing \(W_2\)-Wasserstein geodesics by Schrödinger bridges in the definition of entropic curvature (Léonard in Discrete Contin Dyn Syst A 34(4):1533–1574, 2014; Ann Probab 44(3):1864–1915, 2016; in: Gigli N (ed) Measure theory in non-smooth spaces. Sciendo Migration,Warsaw, pp 194–242, 2017). As a remarkable fact, as a temperature parameter goes to zero, these Schrödinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice \(\mathbb {Z}^n\) endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli–Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar–Maas results on graphs (Erbar and Maas in Arch Ration Mech Anal 206(3):997–1038, 2012; Discrete Contin Dyn Syst A 34(4):1355–1374, 2014; Maas in J Funct Anal 261(8):2250–2292, 2011), entropic curvature results of this paper imply new Prékopa–Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube \(\{0,1\}^n\) and for the Bernoulli Laplace model, a new \(W_2-W_1\) transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see Gozlan et al. in J Funct Anal 273(11):3327–3405, 2017) associated to the so-called convex-hull method by Talagrand (Publ Math l’Inst Hautes Etudes Sci 81(1):73–205, 1995).

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Correspondence to Paul-Marie Samson.

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This research is partly funded by the Bézout Labex, funded by ANR, reference ANR-10-LABX-58. The author is supported by a grant of the Simone and Cino Del Duca Foundation.

Appendices

Appendix A: Basic lemmas

Lemma 4.1

The transport-entropy inequality (29) implies the \(W_2\) transport-entropy inequality (30) for the standard Gaussian measure \(\gamma \).

Proof

The result follows from the transport-entropy inequality (29) for the uniform probability measure \(\mu \) on the hypercube (\(\alpha _i=1/2\) for all \(i\in [n]\)), and by using the central limit Theorem with the projection map

$$\begin{aligned} T_n(x):=\frac{2}{\sqrt{n}} \left( \sum _{i=1}^n x_i-\frac{n}{2}\right) , \quad x,y\in \{0,1\}^n. \end{aligned}$$

By density, it is sufficient to prove (30) for any probability measure \(\nu \) on \(\mathbb {R}\) with continuous density f and compact support K. Let \(\nu ^n\) denotes the probability measure on \(\{0,1\}^n\) with density \(f_n\) with respect to \(\mu \) given by

$$\begin{aligned} f_n(x):= \frac{f(T_n(x))}{\int f\circ T_n \,d\mu },\qquad x\in \{0,1\}^n. \end{aligned}$$

Applying (29) with \(\nu _0:=\mu \) and \(\nu _1:= \nu ^n\), one gets

$$\begin{aligned} \frac{2}{n} {T_{c_2} }(\mu ,\nu ^n)\le H(\nu ^n|\mu ). \end{aligned}$$
(77)

By the weak convergence of \(T_n \#\mu \) to the standard Gaussian law \(\gamma \), one has

$$\begin{aligned} \lim _{n\rightarrow \infty } H(\nu ^n |\mu )= H(\nu |\gamma ), \end{aligned}$$
(78)

and for \(k=1\) or \(k=2\),

$$\begin{aligned} \lim _{n\rightarrow \infty } \int |w|^k \, d(T_n\#\nu ^n)(w)=\lim _{n\rightarrow \infty } \int |T_n(x)|^k \, f_n(T_n(x)) \,d\mu (x)= \int |w|^k \,d\nu (w).\nonumber \\ \end{aligned}$$
(79)

Since \( d(x,y)\ge \frac{\sqrt{n}}{2} \left| T_n(x)-T_n(y)\right| \) and the monotonicity property of the function \(c_2:\mathbb {R}\rightarrow \mathbb {R}^+\) on \([2,+\infty )\) implies

$$\begin{aligned} \frac{2}{n}\, c_2(d(x,y))\ge \frac{2}{n} c_2\left( \frac{\sqrt{n}}{2} \left| T_n(x)-T_n(y)\right| \right) \mathbbm {1}_{\frac{\sqrt{n}}{2} \left| T_n(x)-T_n(y)\right| \ge 2}, \end{aligned}$$

and therefore

$$\begin{aligned} \frac{2}{n}\,{ T_{c_2}}(\mu ,\nu ^n)^2\ge \frac{1}{2} \inf _{\pi _n \in \Pi (T_n\#\mu ,T_n\#\nu ^n)}\iint c_n(z,w) \,d\pi _n(z,w), \end{aligned}$$

where for any \(z,w\in \mathbb {R}\)

$$\begin{aligned} c_n(z,w)&:= 4n c_2\left( \frac{\sqrt{n}\,|z-w|}{2}\right) \mathbbm {1}_{ |z-w|\ge 4/\!\sqrt{n}}\\&=\left[ |z-w|^2-\frac{4(1+\log (\sqrt{n} /2))}{\sqrt{n}}\,|z-w| - \frac{4}{\sqrt{n}} \, |z-w|\log |z-w|\right] \mathbbm {1}_{|z-w|\ge 4/\!\sqrt{n}} . \end{aligned}$$

Let \(c(z,w):=|z-w|^2\), \(z,w\in \mathbb {R}\). One has, for any \(z,w\in \mathbb {R}\), \(c(z,w)\ge c_n(z,w)\) and

$$\begin{aligned} c(z,w)-c_n(z,w)&= |z-w|^2\mathbbm {1}_{|z-w|< 4/\!\sqrt{n}} +\left[ \frac{4(1+\log (\sqrt{n} /2))}{\sqrt{n}}\,|z-w|\right. \\&\quad \left. + \frac{4}{\sqrt{n}} \, |z-w|\log |z-w|\right] \,1_{|z-w|\ge 4/\!\sqrt{n}}\\&\le \frac{16}{n} + \frac{4(1+\log n)}{\sqrt{n}} \left[ |z|+|w| +1+2|z|^2+2|w|^2\right] , \end{aligned}$$

where the last inequality follows from \(|u\log u|\le 1+u^2, u>0\). Since

$$\begin{aligned} \int |z|\,d(T_n \#\mu )(z)\le \left( \int |z|^2 d(T_n \#\mu )(z)\right) ^{1/2}= \left( \int T_n^2 d\mu \right) ^{1/2}=1, \end{aligned}$$

it follows that for any \(\pi _n \in \Pi (T_n\#\mu ,T_n\#\nu ^n)\),

$$\begin{aligned} \iint c_n \,d\pi _n&\ge \iint c\, d\pi _n-\frac{16}{n} - \frac{4(1+\log n)}{\sqrt{n}} \iint \left[ |z|+|w| +1+2|z|^2+2|w|^2\right] \,d\pi _n(z,w)\\&\ge \iint c\, d\pi _n - \frac{32(1+\log n)}{\sqrt{n}} \left[ 1+ \int |w| \, d(T_n\#\nu ^n)(w) +\int |w|^2 \, d(T_n\#\nu ^n)(w)\right] , \end{aligned}$$

and therefore

$$\begin{aligned} \frac{2}{n}\,{T_{c_2} }(\mu ,\nu ^n)^2&\ge \frac{1}{2}\,W^2_2(T_n\#\mu ,T_n\#\nu ^n) \\&\quad - \frac{16(1+\log n)}{\sqrt{n}}\left[ 1+ \int |w| \, d(T_n\#\nu ^n)(w) +\int |w|^2 \, d(T_n\#\nu ^n)(w)\right] . \end{aligned}$$

From the weak convergence in \(\mathcal {P}_2(\mathbb {R})\) of the sequences \((T_n\#\mu )\) and \((T_n\#\nu ^n)\) and using (79), the last inequality implies as n goes to infinity

$$\begin{aligned} \liminf _{n\rightarrow +\infty } \frac{2}{n}\,{T_{c_2} }(\mu ,\nu ^n)\ge \frac{1}{2} \,W_2^2(\nu ,\gamma ). \end{aligned}$$

Finally, Talagrand’s inequality \(W_2^2(\nu ,\gamma )\le 2 H(\nu |\gamma )\), follows from (77) and (78). \(\square \)

Lemma 4.2

If the convexity property (21) holds, then for any \(\nu _0,\nu _1\in \mathbb {P}_b(\mathcal {X})\),

$$\begin{aligned} H(\nu _0|\mu )\le H(\nu _1|\mu ) +\sum _{x\in \mathcal {X}}\sum _{x'\in \mathcal {X}, x'\sim x} \left( \log (f(x)-\log f(x')\right) \, \Pi ^{x'}_\rightarrow (x) \,\nu _0(x) -\frac{1}{2}\liminf _{t\rightarrow 0} C_t(\widehat{\pi }). \end{aligned}$$

where \(\Pi ^{x'}_\rightarrow (x):=\int \mathbbm {1}_{x'\in [x,y]} d(x,y) r(x,x',x',y) \, d\widehat{\pi }_\rightarrow (y|x)\).

Proof

The convexity property (21) implies, for any \(\nu _0,\nu _1\in \mathbb {P}_b(\mathcal {X})\) and for any \(t\in (0,1)\)

$$\begin{aligned} H(\nu _0|\mu )\le H(\nu _1|\mu ) -\frac{H({\widehat{Q}}_t|\mu )-H(\nu _0|\mu )}{t} - \frac{(1-t)}{2} C_t(\widehat{\pi }). \end{aligned}$$
(80)

The first step is to compute the left-hand side of this inequality as t goes to zero. According to the expression (28) of \({Q_t}\!^{ x,y}\), for any \(x,y,z\in \{0,1\}^n\),

$$\begin{aligned} \partial _t {Q_t}\!^{ x,y}(z)&=r(x,z,z,y)\,\left( {\begin{array}{c}d(x,y)\\ d(x,z)\end{array}}\right) \,\mathbbm {1}_{[x,y]}(z) \left( d(x,z)t^{d(x,z)-1}(1-t)^{d(z,y)}\right. \\&\quad \left. -d(z,y) t^{d(x,z)}(1-t)^{d(z,y)-1}\right) , \end{aligned}$$

and therefore

$$\begin{aligned} \partial _t Q_t^{x,y}(z)_{|t=0}&=r(x,z,z,y)\,\left( {\begin{array}{c}d(x,y)\\ d(x,z)\end{array}}\right) \left( \mathbbm {1}_{[x,y]}(z)\mathbbm {1}_{z\sim x}-d(x,y)\mathbbm {1}_{x=z}\right) \\&=\sum _{x'\in [x,y], x'\sim x} d(x,y) r(x,x',x',y)\left( \delta _{x'}(z)-\delta _x(z)\right) . \end{aligned}$$

Since \(\partial _t {\widehat{Q}}^\gamma _t(z)_{|t=0} =\sum _{x,y\in \mathcal {X}} \partial _t Q_t^{x,y}(z)_{|t=0} \, \widehat{\pi }(x,y)\), it follows that

$$\begin{aligned}&\lim _{t\rightarrow 0} \frac{ H({\widehat{Q}}_t|\mu )-H(\nu _0|\mu )}{t}=\partial _t H({\widehat{Q}}^\gamma _t|\mu )_{|t=0}=\sum _{z\in \mathcal {X}} \partial _t {\widehat{Q}}_t(z)_{|t=0} \log f(z)\,\mu (z)\\&\quad =\sum _{x,y\in \mathcal {X}} \sum _{x'\in [x,y], x'\sim x} d(x,y) \left( \log f(x')-\log f(x)\right) \, d(x,y)\,r(x,x',x',y)\, \widehat{\pi }(x,y)\\&\quad =\sum _{x\in \mathcal {X}}\sum _{x'\in \mathcal {X}, x'\sim x} \left( \log (f(x')-\log f(x)\right) \, \left( \sum _{y\in \mathcal {X}, x'\in [x,y]} d(x,y) r(x,x',x',y) \, \widehat{\pi }_{_\rightarrow }(y|x) \right) \,\nu _0(x) \end{aligned}$$

The proof of Lemma 4.2 ends from (80) as t goes to 0. \(\square \)

Lemma 4.3

Let \(\mathcal {X}\) be a graph with graph distance d. Let \(\nu _0,\nu _1\in \mathcal {P}(\mathcal {X})\) and assume that \(\widehat{\pi }\in \mathcal {P}(\mathcal {X}\times \mathcal {X})\) is a \(W_1\)-optimal coupling of \(\nu _0\) and \(\nu _1\), namely

$$\begin{aligned} W_1(\nu _0,\nu _1)=\iint d(x,y) \,d\widehat{\pi }(x,y). \end{aligned}$$
  1. (i)

    Let

    $$\begin{aligned} C_{_\rightarrow }:=\Big \{(z,w)\in \mathcal {X}\times \mathcal {X}\,\Big |\,z\ne w, \exists (x,y)\in \mathrm{supp}(\widehat{\pi }), (z,w)\in [x,y]\Big \}. \end{aligned}$$

    If \((z_1,w)\in C_{_\rightarrow }\) and \((w,z_2)\in C_{_\rightarrow }\) then \(d(z_1,z_2)\ge 2\) and \(w\in [z_1,z_2]\).

  2. (ii)

    Let

    $$\begin{aligned} C_{_\leftarrow }:=\Big \{(z,w)\in \mathcal {X}\times \mathcal {X}\,\Big |\, (w,z) \in C_{_\rightarrow } \Big \}. \end{aligned}$$

    The sets \(C_{_\rightarrow }\) and \(C_{_\leftarrow }\) are disjoint.

  3. (iii)

    If d is the Hamming distance then the following sets \(D_{_\rightarrow }\) and \(D_{_\leftarrow }\) are disjoint,

    $$\begin{aligned} D_{_\leftarrow }:=\Big \{ w\in \mathrm{supp}(\nu _1)\,\Big |\, \exists x\in \mathcal {X}, w\ne x, (x,w)\in \mathrm{supp}(\widehat{\pi }) \Big \}, \end{aligned}$$

    and

    $$\begin{aligned} D_{_\rightarrow }:=\Big \{ w\in \mathrm{supp}(\nu _0)\,\Big |\, \exists y\in \mathcal {X}, w\ne y, (w,y)\in \mathrm{supp}(\widehat{\pi }) \Big \}. \end{aligned}$$

Proof

  1. (i)

    Let \((z_1,w)\in C_{_\rightarrow }\) and \((w,z_2)\in C_{_\rightarrow }\). There exists \((x,y)\in \mathrm{supp}(\widehat{\pi })\) such that \((z_1,w)\in [x,y]\) and there exists \((x',y')\in \mathrm{supp}(\widehat{\pi })\) such that \((w,z_2)\in [x',y']\). One has

    $$\begin{aligned} d(z_1,w)+d(w,z_2)&=\big ((d(x,y)-d(x,z_1)-d(w,y)\big )\\&\quad +\big (d(x',y')-d(x',w)-d(z_2,y')\big ). \end{aligned}$$

    It is well known that the support of any optimizer of \(W_1(\nu _0,\nu _1)\) is d-cyclically monotone (see [43, Theorem 5.10]. By definition, it means that for any family \((x_1,y_1),\ldots ,(x_N,y_N)\) of points in the support of \(\widehat{\pi }\)

    $$\begin{aligned} \sum _{i=1}^N d(x_i,y_i) \le \sum _{i=1}^N d(x_i,y_{i+1}), \end{aligned}$$

    with the convention \(y_{N+1}=y_1\). It follows that

    $$\begin{aligned} d(x,y)+d(x',y')\le d(x,y')+d(x',y), \end{aligned}$$

    and therefore, from the above identity,

    $$\begin{aligned} d(z_1,w)+d(w,z_2)\le d(x,y')+d(x',y) -d(x,z_1)-d(w,y)-d(x',w)-d(z_2,y'). \end{aligned}$$

    By the triangular inequality, it follows that

    $$\begin{aligned} 2&\le d(z_1,w)+d(w,z_2)\le \big (d(x,z_1)+d(z_1,z_2)+d(z_2,y')\big )\\&\quad +\big (d(x',w)+d(w,y)\big ) -d(x,z_1)-d(w,y)-d(x',w)-d(z_2,y')=d(z_1,z_2) . \end{aligned}$$

    This implies that \(d(z_1,z_2)\ge 2\) and \(w\in [z_1,z_2]\).

  2. (ii)

    Assume there exists \((z,w)\in C_{_\rightarrow }\cap C_{_\leftarrow }\). Then \((w,z)\in C_{_\rightarrow }\) and therefore, according to (i), \(z\in [w,w]=\{w\}\). This is impossible since \(z\ne w\).

  3. (iii)

    We assume that \(d(x,y)=\mathbbm {1}_{x\ne y}\) for any \(x,y\in \mathcal {X}\). If the two sets \(D_{_\rightarrow }\) and \(D_{_\leftarrow }\) intersect, then there exists \((x,w)\in C_{_\rightarrow }\) and \((w,y)\in C_{_\rightarrow }\). Point (i) implies \(w\in [x,y]\), and since \(d(x,y)=1\), we get either \(w=x\) or \(w=y\), which is impossible.

\(\square \)

Lemma 4.4

Let \(\nu _0\) and \(\nu _1\) some probability measures in \({{\mathcal {P}}}(\mathcal {X})\) with bounded support.

  1. (i)

    If (13) holds (\(\exists S\ge 1,\sup _{x\in \mathcal {X}}|L(x,x)|\le S\)), then for any \(x,y\in \mathcal {X}\) and any integer k,

    $$\begin{aligned} L^k(x,y)\le (2S)^k. \end{aligned}$$
  2. (ii)

    If (14) holds (\(\exists I\in (0,1],\inf _{x,y\in \mathcal {X},x\sim y} L(x,y)\ge I\)), then for any \(x,y\in \mathcal {X}\), \(L^{d(x,y)}(x,y)\ge I^{d(x,y)}.\)

  3. (iii)

    If (13) and (14) hold, then for any \(x,y\in \mathcal {X}\), any \(t\in [0,1]\), and any \(\gamma \in (0,1)\), one has

    $$\begin{aligned} {P}_t^{\gamma }(x,y) = \frac{ L^{d(x,y)}(x,y)}{d(x,y)!} \, (\gamma t)^{d(x,y)}\left( 1+ \gamma K^{d(x,y)}O(1)\right) , \end{aligned}$$

    where \(K:=2S/I\) and O(1) denotes a quantity uniformly bounded in xyt and \(\gamma \).

  4. (iv)

    If (13) holds then for any \(x,y,z\in \mathcal {X}\) and for any \(t\in [0,1]\)

    $$\begin{aligned} \lim _{\gamma \rightarrow 0}{Q_t^\gamma }^{x,y}(z)={Q_t}\!^{ x,y}(z):=\mathbbm {1}_{[x,y]}(z)\,r(x,z,z,y) \,{\rho }_t^{d(x,y)}(d(x,z)). \end{aligned}$$
  5. (v)

    If (13) holds then for any \(x,y\in \mathcal {X}\),

    $$\begin{aligned} {P}_t^{\gamma }(x,y) \ge \frac{ L^{d(x,y)}(x,y) }{d(x,y)!} \,(t\gamma )^{d(x,y)}e^{-\gamma t S}. \end{aligned}$$

    For a fixed \(x_0\in \mathcal {X}\), let \(D:=\max _{x\in \mathrm{supp}(\nu _0),y\in \mathrm{supp}(\nu _1)}(d(x_0,x), d(x_0,y))\). It follows that if (13) and (14) hold then for any \(\gamma \in (0,1)\) and \(t\in (0,1)\),

    $$\begin{aligned} 0<e^{-S} \left( \frac{t\gamma I}{d(x_0,z)+1+D}\right) ^{d(x_0,z)+1+D} \min _{w\in \mathrm{supp}(\nu _0)} f^\gamma (w) \le {P}_t^\gamma f^\gamma (z)\le \max _{w\in \mathrm{supp}(\nu _0)} f^\gamma (w). \end{aligned}$$
    (81)
  6. (vi)

    If (13) holds then \({{\mathbb {E}}}_{R^\gamma }[\ell |X_0=x,X_1=y]\le \frac{\gamma S}{{P}^\gamma _1(x,y)}\).

  7. (vii)

    Assume (13) and (14) hold. For a fixed \(x_0\in \mathcal {X}\), let \( D:=\max _{x\in \mathrm{supp}(\nu _0),y\in \mathrm{supp}(\nu _1)}(d(x_0,x), d(x_0,y))\). For any \(x\in \mathrm{supp}(\nu _0) \) and \(y\in \mathrm{supp}(\nu _1) \), one has for any \(t \in (0,1)\) and any \(\gamma \in (0,1)\)

    $$\begin{aligned} {Q_t^\gamma }^{x,y}(z)\le O(1) \left( \mathbbm {1}_{[x,y]}(z)+ \left( 1-\mathbbm {1}_{[x,y]}(z)\right) \gamma \left( \gamma K^2\right) ^{[2d(x_0,z)-4D-1]_+}\right) , \end{aligned}$$

    where \(K:=2S/I\) and O(1) denotes a constant that only depends on SID and \(K:=2S/I\).

    As a consequence, setting

    $$\begin{aligned} B:=\bigcup _{x\in \mathrm{supp}(\nu _0),y\in \mathrm{supp}(\nu _1)} [x,y], \end{aligned}$$

    one has

    $$\begin{aligned} {\widehat{Q}}_t^\gamma (z)\le O(1) \,\gamma \left( \gamma K^2\right) ^{[2d(x_0,z)-4D-1]_+},\qquad \forall z\in \mathcal {X}\setminus B. \end{aligned}$$
    (82)
  8. (viii)

    Assume (13) and (14) hold. Let \(x_0\in \mathcal {X}\), \(t\in (0,1)\) and \(\gamma \in (0,1)\). For any \(w,z,z'\in \mathcal {X}\) with \(d(z,z')\le 2\) and \(w\in \mathrm{supp}(\nu _0)\) one has

    $$\begin{aligned} \frac{P_t^\gamma (z',w)}{P_t^\gamma (z,w)}\le \frac{\max \left( 1, d(x_0,z)^{d(z,z')}\right) K^{d(x_0,z)}\,O(1)}{(\gamma t)^{d(z,z')}}, \end{aligned}$$

    where \(K:=2S/I\) and O(1) is a positive constant that does not depend on \(z,z',\gamma ,t\). It follows that

    $$\begin{aligned}&\frac{(\gamma t)^{d(z,z')}}{\max \left( 1, d(x_0,z)^{d(z,z')}\right) K^{d(x_0,z)}\,O(1)}\le \frac{{P}_t^\gamma f^\gamma (z')}{{P}_t^\gamma f^\gamma (z)} \nonumber \\&\quad \le \frac{\max \left( 1, d(x_0,z)^{d(z,z')}\right) K^{d(x_0,z)}\,O(1)}{(\gamma t)^{d(z,z')}}. \end{aligned}$$
    (83)
  9. (ix)

    Let \((\gamma _\ell )_{\ell \in \mathbb {N}}\) be a sequence of positive numbers converging to zero. If (12), (13), (14) and (15) hold, then for any \(t\in [0,1]\)

    $$\begin{aligned} \lim _{\gamma _\ell \rightarrow 0} H({\widehat{Q}}_t^{\gamma _\ell }|m) =H({\widehat{Q}}_t^{0}|m). \end{aligned}$$

Proof

  1. (i)

    Given (13), we want to show that for any \(x\in \mathcal {X}\), \(S_k(y):= \sup _{x\in \mathcal {X}} |L^k(x,y)|\le (2S)^k\). It follows by induction on k from the inequality

    $$\begin{aligned} S_{k+1}(y)=\sup _{x\in \mathcal {X}} \Big |L(x,x) L^k(x,y)+\sum _{z, z\sim x} L(x,z) L^k(z,y) \Big |\le 2\sup _{x\in \mathcal {X}}|L(x,x)| \,\, S_{k}(y). \end{aligned}$$
  2. (ii)

    For \(x=y\), one has \(L^{d(x,y)}(x,y)=1\) and by definition for \(x\ne y\),

    $$\begin{aligned} L^{d(x,y)}(x,y):= \sum _{\alpha }L_\alpha , \end{aligned}$$

    where the sum is over all path \(\alpha \) from x to y of length d(xy), \(\alpha =(z_0,\ldots , z_{d(x,y)})\) with \(z_0=x\) and \(z_{d(x,y)}=y\), and

    $$\begin{aligned} L_\alpha :=L(z_0,z_1)L(z_1,z_2)\ldots L(z_{d(x,y)-1},z_{d(x,y)}). \end{aligned}$$

    Such a path \(\alpha \) is a geodesic. Since we assume in this paper that \(L(x,y)>0\) if and only if x and y are neighbour, one has \(L_\alpha >0\). By irreducibility it always exists at most one geodesic path from x to y, and from assumption (13), for such a path \(\alpha \), \(L_\alpha \ge I^{d(x,y)}\). As a consequence we get \(L^{d(x,y)}(x,y)\ge I^{d(x,y)}\).

  3. (iii)

    According to (16), for any \(x,y\in \mathcal {X}\),

    $$\begin{aligned}&{P}_t^{\gamma }(x,y) = \frac{ L^{d(x,y)}(x,y)}{d(x,y)!} \\&\quad (\gamma t)^{d(x,y)}\left( 1+ \gamma \sum _{k,k\ge d(x,y)+1} \frac{ L^k(x,y)}{L^{d(x,y)}(x,y)}\,\frac{d(x,y)!}{k!}\,t^{k-d(x,y)}\gamma ^{k-d(x,y)-1}\right) . \end{aligned}$$

    Applying Lemma 4.4 (i) and  (ii), we get

    $$\begin{aligned}&\Big |{P}_t^{\gamma }(x,y) - \frac{ L^{d(x,y)}(x,y)}{d(x,y)!} \, (\gamma t)^{d(x,y)}\Big |\\&\quad \le \gamma \, \frac{ L^{d(x,y)}(x,y)}{d(x,y)!} \, (\gamma t)^{d(x,y)} \sum _{k,k\ge d(x,y)+1} K^{d(x,y)} \frac{(2S)^{k-d(x,y)}}{(k-d(x,y))!}\\&\quad \le \gamma \, \frac{ L^{d(x,y)}(x,y)}{d(x,y)!} \, (\gamma t)^{d(x,y)} K^{d(x,y)} e^{2S}, \end{aligned}$$

    from which the expected result follows.

  4. (iv)

    Let \(x,y,z\in \mathcal {X}\) and \(t\in [0,1]\). If (13) holds, according to (16), the Taylor expansion of \({P}_t^{\gamma }(x,y)\) as \(\gamma \) goes to zero is given by

    $$\begin{aligned} {P}_t^{\gamma }(x,y) = \frac{ L^{d(x,y)}(x,y)}{d(x,y)!} \, (\gamma t)^{d(x,y)} + o(\gamma ^{d(x,y)}), \end{aligned}$$

    As a consequence, the Taylor expansion of \({Q_t^\gamma }^{x,y}(z)\), defined by (9), is

    $$\begin{aligned} {Q_t^\gamma }^{x,y}(z)&=\gamma ^{d(x,z)+d(z,y)-d(x,y)} \frac{L^{d(x,z)}(x,z) L^{d(z,y)}(z,y)}{L^{d(x,y)}(x,y)} \,\frac{d(x,y)!}{d(x,z)!d(z,y)!} \,t^{d(x,z)}(1-t)^{d(z,y)} \\&\quad +o(\gamma ^{d(x,z)+d(z,y)-d(x,y)}). \end{aligned}$$

    The expected result follows since one has \(\gamma ^{d(x,z)+d(z,y)-d(x,y)}=1\) if \(z\in [x,y]\), and \(\lim _{\gamma \rightarrow 0} \gamma ^{d(x,z)+d(z,y)-d(x,y)}=0 \) otherwise.

  5. (v)

    On some probability space \((\Omega ',{\mathcal {A}}, \mathbb { P})\), let \((N_s)_{s\ge 0}\) be a Poisson process with parameter \(\gamma S\) and \((Y_n)_{n\in \mathbb {N}}\) be a Markov chain on \(\mathcal {X}\) with transition matrix K given by

    $$\begin{aligned} \mathrm{K}(z,w): =\frac{L^\gamma (x,w)}{\gamma S}, \quad \text{ for }\, w\ne z\in \mathcal {X}, \text{ and } \quad \mathrm{K}(z,z):=\frac{\gamma S+L^\gamma (z,z)}{\gamma S}. \end{aligned}$$

    We assume that \((Y_n)_{n\in \mathbb {N}}\) and \((N_s)_{s\ge 0}\) are independent. It is well known that the law of the process \((X_t)_{t\ge 0}\) under \(R^{\gamma }\) given \(X_0=x\) is the same as the law of the process \(({\widetilde{X}}_t)_{t\ge 0}\) under \({\mathbb {P}}\) given \({\widetilde{X}}_0=x\) defined by \({\widetilde{X}}_t:=Y_{N_t}\). As a consequence, one has for any \(y\in \mathcal {X}\),

    $$\begin{aligned} {P}^\gamma _t(x,y)=R^{\gamma }\left( X_t=y\,|\,X_0=x\right) ={{\mathbb {P}}}\left( {\widetilde{X}}_t=y\,|\,{\widetilde{X}}_0=x\right) . \end{aligned}$$

    Let \(n=d(x,y)\) and \({\widetilde{N}}_t\) denotes the number of jumps of the process \({\widetilde{X}}_t\), one has

    $$\begin{aligned} {P}_t^\gamma (x,y)&\ge {{\mathbb {P}}}\left( {\widetilde{X}}_t=y, {\widetilde{N}}_t =n \,|\,{\widetilde{X}}_0=x\right) \\&= {{\mathbb {P}}}\left( Y_1,\ldots , Y_{n} \text{ are } \text{ all } \text{ different }, Y_{n}=y, N_t =n\, |\,{\widetilde{X}}_0=x\right) \\&={{\mathbb {P}}}\left( N_t =n)\, {{\mathbb {P}}}(Y_1,\ldots , Y_{n} \text{ are } \text{ all } \text{ different }, Y_{n}=y \,|\,{\widetilde{X}}_0=x\right) \\&=\frac{(\gamma t S)^{n}}{n!} \,e^{-\gamma t S} \sum _{\alpha =(x_0,\ldots ,x_{n}),\,\alpha \, \mathrm{geodesic} \, \mathrm{from}\, x\, \mathrm{to} \, y} \mathrm{K}(x_0,x_1)\cdots \mathrm{K}(x_{n-1},x_n)\\&=\frac{(\gamma t )^{n}}{n!} \,e^{-\gamma t S} L^{d(x,y)}(x,y). \end{aligned}$$

    This ends the proof of the first part of (v). Observe that from the Schrödinger system (7), \(f^{\gamma }(w)>0\) if and only if \(w\in \mathrm{supp}(\nu _0)\). Since \(\nu _0\) has bounded support, it follows that for any \(w\in \mathrm{supp}(\nu _0)\),

    $$\begin{aligned} 0<\min _{u\in \mathrm{supp}(\nu _0)} f^{\gamma }(u)\le f^{\gamma }(w)\le \max _{u\in \mathrm{supp}(\nu _0)} f(u), \end{aligned}$$

    and therefore for any \(z\in \mathcal {X}\),

    $$\begin{aligned} \min _{u\in \mathrm{supp}(\nu _0)} f(u) \min _{w\in \mathrm{supp}(\nu _0)} {P}^\gamma _t(z,w)&\le \sum _{w\in \mathrm{supp}(\nu _0)} f^{\gamma }(w) {P}^\gamma _t(z,w)\\&= {P}^\gamma _t f^{\gamma }(z)\le \max _{u\in \mathrm{supp}(\nu _0)} f(u). \end{aligned}$$

    From (14) and (ii) and since \(d(z,w)\le d(z,x_0)+1+D\) for any \(w\in \mathrm{supp}(\nu _0)\), one gets

    $$\begin{aligned} \min _{w\in \mathrm{supp}(\nu _0)} {P}^\gamma _t(z,w)\ge e^{-S} \left( \frac{t\gamma I}{d(x_0,z)+1+D}\right) ^{d(x_0,z)+1+D}, \end{aligned}$$

    from which the second part of (v) follows.

  6. (vi)

    The length \(\ell (\omega )\) of a path \(\omega \in \Omega \) represents the number of jumps of the process \(X_t\) between times 0 and 1. Therefore according to the definition of the process \(({\widetilde{X}}_t)_{t\ge 0}\) above,

    $$\begin{aligned} {{\mathbb {E}}}_{R^\gamma }&[\ell \,|\,X_0=x,X_1=y]={{\mathbb {E}}}_{{\mathbb {P}}}\left[ {\widetilde{N}}_1\,| \,{\widetilde{X}}_0=x, {\widetilde{X}}_1=y\right] \\&\le {{\mathbb {E}}}_{{\mathbb {P}}}\left[ N_1\,|\, {\widetilde{X}}_0=x, {\widetilde{X}}_1=y\right] = \frac{ {{\mathbb {E}}}_{{\mathbb {P}}}\left[ N_1\mathbbm {1}_{{\widetilde{X}}_1=y} \,| \,{\widetilde{X}}_0=x\right] }{{{\mathbb {P}}}\left( {\widetilde{X}}_1=y\,|\, {\widetilde{X}}_0=x\right) }\le \frac{ {{\mathbb {E}}}_{{\mathbb {P}}}\left[ N_1\right] }{{P}^\gamma _1(x,y)}, \end{aligned}$$

    which ends the proof since \( {{\mathbb {E}}}_{{\mathbb {P}}}\left[ N_1\right] =\gamma S\).

  7. (vii)

    From (iii) and (v), one gets for any \(x,z,y\in \mathcal {X}\),

    $$\begin{aligned} {Q_t^\gamma }^{x,y}(z)&= \frac{{P}_t^\gamma (x,z){P}_{1-t}^\gamma (z,y)}{P_1^\gamma (x,y)}\nonumber \\&\le \gamma ^{d(x,z)+d(z,y)-d(x,y)} r(x,z,z,y)\, \frac{d(x,y)!}{d(x,z)!d(z,y)!} t^{d(x,z)}(1-t)^{d(z,y)}\,e^{\gamma S}\nonumber \\&\quad \left( 1+\gamma K^{d(x,z)}O(1)\right) \left( 1+\gamma K^{d(z,y)}O(1)\right) . \end{aligned}$$
    (84)

    If \(z\in [x,y]\) then thanks to (i) and  (ii), the right-hand side of this inequality is bounded from above by

    $$\begin{aligned} \left( \frac{2S}{I}\right) ^{d(x,y)}e^{d(x,y)}e^{\gamma S}4 K^{2d(x,y)}O(1), \end{aligned}$$

    and the maximum of this quantity over all \(x\in \mathrm{supp}(\nu _0)\) and \(y\in \mathrm{supp}(\nu _1)\) is a constant O(1), independent of xzy and \(\gamma \).

    If \(z\not \in [x,y]\), then \(d(x,z)+d(z,y)-d(x,y)\ge \max \{1, 2d(x_0,z)-4D\}\), and the right-hand side of (84) is bounded by

    $$\begin{aligned}&\gamma ^{d(x,z)+d(z,y)-d(x,y)}\,\frac{(2S)^{d(x,z)+d(z,y)}}{I^{d(x,y)}}\,d(x,y)!\,e^{\gamma S}4K^{d(x,z)+d(z,y)}O(1)\\&\quad \le \gamma ^{1+[2d(x_0,z)-4D-1]_+} \frac{(2S)^{2d(x_0,z)+ 2D}}{I^{d(x,y)}}\,d(x,y)!\,e^{\gamma S}4K^{2d(x_0,z)+2D} O(1). \end{aligned}$$

    The maximum over all \(x\in \mathrm{supp}(\nu _0)\) and \(y\in \mathrm{supp}(\nu _1)\) of the right-hand side quantity is bounded by \(O(1)\,\gamma ^{1+[2d(x_0,z)-4D-1]_+} K^{4d(x_0,z)}\). This ends the proof of the first inequality of (vii). The second inequality easily follows since

    $$\begin{aligned} {\widehat{Q}}_t^\gamma (z)= \sum _{x\in \mathrm{supp}(\nu _0),y\in \mathrm{supp}(\nu _1) } {Q_t^\gamma }^{x,y}(z) \,\,\widehat{\pi }^\gamma (x,y). \end{aligned}$$
  8. (viii)

    Using (iii) and (v), one gets for any \(z,z'\in \mathcal {X}\) and any \(w\in \mathrm{supp}(\nu _0)\),

    $$\begin{aligned}&\frac{{P}_t^\gamma (z',w)}{{P}_t^\gamma (z,w)}\le \frac{L^{d(z',w)}(z',w)}{L^{d(z,w)}(z,w)}\, \frac{d(z,w)!}{d(z',w)!}\,\left( \frac{1}{\gamma t}\right) ^{d(z,w)-d(z',w)} e^{\gamma t S}\left( 1+\gamma K^{d(z',w)} O(1)\right) \\&\quad \le K^{d(z,z')+ d(z,x_0)+d(x_0,w)}\max \left( 1,d(z,w)^2\right) \left( \frac{1}{\gamma t}\right) ^{d(z,z')}2e^{ S}K^{d(z,z')+ d(z,x_0)+d(x_0,w)}O(1) \\&\quad \le \frac{K^{2d(z,x_0)}\max \left( 1,d(z,x_0)^2\right) O(1)}{(\gamma t)^{d(z,z')}}, \end{aligned}$$

    where one maximizes over all \(w\in \mathrm{supp}(\nu _0)\) to get the last inequality. Inequality (83) follows since

    $$\begin{aligned} \frac{{P}_t^\gamma f^\gamma (z')}{{P}_t^\gamma f^\gamma (z)}= \sum _{w\in \mathrm{supp}(\nu _0) }\frac{{P}_t^\gamma (z',w)}{{P}_t^\gamma (z,w)} \frac{f^\gamma (w){P}_t^\gamma (z,w)}{{P}_t^\gamma f^\gamma (z)}, \end{aligned}$$

    with \(\sum _{w\in \mathrm{supp}(\nu _0) } \frac{f^\gamma (w){P}_t^\gamma (z,w)}{{P}_t^\gamma f^\gamma (z)}=1\).

  9. (ix)

    Recall that

    $$\begin{aligned} H({\widehat{Q}}^{\gamma _\ell }_t|m)= \sum _{z\in \mathcal {X}}\log \frac{{\widehat{Q}}^{\gamma _\ell }_t(z)}{m(z)}\, {\widehat{Q}}^{\gamma _\ell }_t(z). \end{aligned}$$

    Let us consider the finite set B defined in Lemma 4.4 (vii). From the weak convergence of the sequence \(({\widehat{Q}}^{\gamma _\ell }_t)\) to \({\widehat{Q}}^{0}_t\) and since \(\mathrm{supp}({\widehat{Q}}^{0}_t)\subset B\), one has

    $$\begin{aligned} \lim _{\gamma _\ell \rightarrow 0} \sum _{z\in B}\log \frac{{\widehat{Q}}^{\gamma _\ell }_t(z)}{m(z)}\, {\widehat{Q}}^{\gamma _\ell }_t(z)= H({\widehat{Q}}^{0}_t|m). \end{aligned}$$

    Therefore it remains to prove that

    $$\begin{aligned} \lim _{\gamma _\ell \rightarrow 0} \sum _{z\in \mathcal {X}\setminus B}\log \frac{{\widehat{Q}}^{\gamma _\ell }_t(z)}{m(z)}\, {\widehat{Q}}^{\gamma _\ell }_t(z)=0. \end{aligned}$$

    From Lemma 4.4 (vii) and hypothesis (12) one has, for any \(z\in \mathcal {X}\setminus B\),

    $$\begin{aligned} \frac{{\widehat{Q}}^{\gamma _\ell }_t(z)}{m(z)}\le \frac{O(1)\,\gamma _\ell \, \left( \gamma _\ell K^2\right) ^{[2d(x_0,z)-4D-1]_+}}{\inf _{z\in \mathcal {X}} m(z)}. \end{aligned}$$

    Using the inequality \(|v\log v|\le \sqrt{v}\) for \(v\in (0,1]\), we get for \(0<\gamma _\ell \le \min \left( \frac{\inf _{z\in X}m(z)}{O(1)},\frac{1}{K^2}\right) \),

    $$\begin{aligned} \sum _{z\in \mathcal {X}\setminus B}\log \frac{{\widehat{Q}}^{\gamma _\ell }_t(z)}{m(z)}\, {\widehat{Q}}^{\gamma _\ell }_t(z) \le O(1)\sup _{z\in \mathcal {X}} m(z) \sqrt{\gamma _\ell } \sum _{z\in \mathcal {X}} \left( \gamma _\ell K^2\right) ^{[2d(x_0,z)-4D-1]_+/2}. \end{aligned}$$

    Hypothesis (15) then implies that there exists \(\tilde{\gamma }>0\) such that for any \(0<\gamma _\ell <\tilde{\gamma }\)

    $$\begin{aligned} \sum _{z\in \mathcal {X}\setminus B} \log \frac{{\widehat{Q}}^{\gamma _\ell }_t(z)}{m(z)}\, {\widehat{Q}}^{\gamma _\ell }_t(z) \le O(1) \sqrt{\gamma }_\ell , \end{aligned}$$

    and the expected result follows.

\(\square \)

Appendix B: Proofs of Lemmas 3.13.23.3, and 3.4

Proof of Lemmas 3.2 and 3.3

Let \(\gamma \) denotes a fixed parameter of temperature that can be chosen as small as we want. To simplify the notations, the dependence in the temperature parameter \(\gamma \) is sometimes omitted. For \(t\in (0,1)\), let us note \(f_t:= P_t^\gamma f^\gamma \) and \(g_t:=P_{1-t}^\gamma g^\gamma \) and recall that \(F_t :=\log f_t\), \(G_t:=\log g_t\) and

$$\begin{aligned} \varphi (t)=\int F_t f_t \, g_t \,dm, \qquad \psi (t)=\int G_t f_t \,g_t \,dm. \end{aligned}$$

Observe that for \(\gamma \) sufficiently small, these two functions are well defined on (0, 1) since (81) and (82) implies

$$\begin{aligned} \int |F_t| f_t \, g_t \,dm&=\sum _{z\in \mathcal {X}} \left| \log (P_t^\gamma f^\gamma (z))\right| {\widehat{Q}}_t^\gamma (z)\\&\le O(1) +O(1)\sum _{z\in \mathcal {X}\setminus B} (d(x_0,z)+1+D) \\&\quad \left( \log \frac{1}{t\gamma I}+\log \left( {d(x_0,z)+1+D}\right) \right) \,\gamma \left( \gamma K^2\right) ^{[2d(x_0,z)-4D-1]_+}. \end{aligned}$$

According to hypothesis (15), the right-hand side of this inequality is finite if \((\gamma K^2)^2<\gamma _o\). Identically, one could check that \(\int |G_t| f_t \,g_t \,dm\) is finite for \(\gamma \) sufficiently small.

The proof is based on \(\Gamma _2\)-calculus by using backward equations, \(\partial _t f_t =L f_t\), \(\partial _t g_t=-L g_t\). We only present the proof of the expression of \(\varphi '(t)\) and \(\varphi ''(t)\). Same arguments provide the expression of \(\psi '(t)\) and \(\psi ''(t)\). We start with a general statement that we will apply twice. Let \((t,z)\in (0,1)\times \mathcal {X}\rightarrow V_t(z)\in \mathbb {R}\) denotes some differentiable function in t (that also depends of the parameter \(\gamma \)) satisfying for any \(\varepsilon \in (0,1/2)\), and any \(x_0\in \mathcal {X}\),

$$\begin{aligned} \sup _{t\in (\varepsilon ,1-\varepsilon )} |V_t(z)|\le O(1) \frac{A^{d(x_0,z)}}{\gamma ^{10}}, \end{aligned}$$
(85)

and

$$\begin{aligned} \sup _{t\in (\varepsilon ,1-\varepsilon )} |\partial _tV_t(z)|\le O(1) \frac{B^{d(x_0,z)}}{\gamma ^{10}}, \end{aligned}$$
(86)

for all \(z\in \mathcal {X}\) where O(1), AB denote constants that do not depend on \(t,\gamma \) and z. Then the following identity holds: for any \(t\in (0,1)\),

$$\begin{aligned} \partial _t \left( \int V_t f_t \,g_t \,dm\right)&=\int \partial _t(V_t f_t \,g_t)\,dm \nonumber \\&=\int (\partial _t V_t) \,f_t \, g_t+V_t \,(Lf_t)\, g_t - V_t \,f_t \,(Lg_t)\,dm \nonumber \\&=\int (\partial _t V_t) \,f_t \, g_t+V_t \,(Lf_t)\, g_t - L(V_t f_t ) g_t \,dm\nonumber \\&= \int \Big [\partial _t V_t(z) - \sum _{z', \,z'\sim z} e^{\nabla F_t(z,z') }\nabla V_t(z,z')\,L(z,z')\Big ]f_t(z) g_t(z)\,dm(z). \end{aligned}$$
(87)

It suffises to justify this identity for any \(\varepsilon \in (0,1/2)\) and any \(t\in (\varepsilon ,1-\varepsilon )\). The second equality of (87) is due to the backward equations. The first equality of (87) is justified by applying Lebesgue’s theorem with hypothesis (15), provided that for \(\gamma \) sufficiently small, one has

$$\begin{aligned} \sup _{t\in (\varepsilon ,1-\varepsilon )}|\partial _t(V_t f_t \,g_t)(z) \,m(z)|\le O(1) \,\gamma _o^{d(x_0,z)}. \end{aligned}$$

This is indeed the case, since for any \(z\in \mathcal {X}\),

$$\begin{aligned} \partial _t(V_t f_t \,g_t)(z) \,m(z)= \left[ (\partial _t V_t)(z)+V_t(z) \frac{L{P}_t^\gamma f^\gamma (z)}{{P}_t^\gamma f^\gamma (z)}- V_t(z) \frac{L{P}_{1-t}^\gamma g^\gamma (z)}{{P}_{1-t}^\gamma g^\gamma (z)}\right] {\widehat{Q}}_t^\gamma (z), \end{aligned}$$

with according to (83), for any \(t\in (\varepsilon ,1)\),

$$\begin{aligned} \left| \frac{L{P}_t^\gamma f^\gamma (z)}{{P}_t^\gamma f^\gamma (z)}\right|&\le S d_{\mathrm{max}} \left( 1+\max _{z', z'\sim z}\left| \frac{{P}_t^\gamma f^\gamma (z')}{{P}_t^\gamma f^\gamma (z)}\right| \right) \\&\le Sd_{\mathrm{max}} \frac{\max (1, d(x_0,z)) K^{d(x_0,z)}\,O(1)}{\gamma \varepsilon }\le O(1) \,\frac{K^{d(x_0,z)}}{\gamma }. \end{aligned}$$

One identically shows that \(\left| \frac{L{P}_{1-t}^\gamma g^\gamma (z)}{{P}_{1-t}^\gamma g^\gamma (z)}\right| \le O(1) \frac{K^{d(x_0,z)}}{\gamma },\) for any \(t\in (0,1-\varepsilon )\) and \(z\in \mathcal {X}\). Together with (82), we get the bound, for any \(z\in \mathcal {X}\) and \(t\in (\varepsilon ,1-\varepsilon )\),

$$\begin{aligned} |\partial _t(V_t f_t \,g_t)(z) m(z)|\le O(1)\left( B^{d(x_0,z)}+ (AK)^{d(x_0,z)}\right) \frac{\left( \gamma K^2\right) ^{2d(x_0,z)}}{\gamma ^{11}}\le O(1) \,\gamma _o^{d(x_0,z)}, \end{aligned}$$

for any \(\gamma >0\) with \(\gamma ^2(B+AK) K^4\le \gamma _o\). The third equality of (87) is due to Fubini’s theorem together with the reversibility property of m with respect to L. The last equality of (87) is a simple rearrangement of the terms.

At first, one applies (87) with \(V_t=F_t\), since according to (81), for any \(t\in (\varepsilon , 1-\varepsilon )\), for any \(z\in \mathcal {X}\),

$$\begin{aligned}&|F_t(z)| \le O(1) \left( {d(x_0,z)+1+D}\right) \left( \log \frac{1}{\varepsilon \gamma I}+\log \left( {d(x_0,z)+1+D}\right) \right) \\&\quad \le O(1) \,\frac{2^{d(x_0,z)}}{\gamma }, \end{aligned}$$

and

$$\begin{aligned} |\partial _t F_t(z)|= \left| \frac{L{P}_t^\gamma f^\gamma (z)}{{P}_t^\gamma f^\gamma (z)}\right| \le O(1) \, \frac{K^{d(x_0,z)}}{\gamma }. \end{aligned}$$
$$\begin{aligned} \partial _t F_t (z)=\sum _{z'\in \mathcal {X}} e^{\nabla F_t(z,z') }L(z,z')= \sum _{z', \,z'\sim z} \left( e^{\nabla F_t(z,z') }-1\right) L(z,z'),\qquad z\in \mathcal {X}, \end{aligned}$$

one gets the expected result

$$\begin{aligned} \varphi '(t)&=\int \sum _{z',\, z'\sim z} \left( e^{\nabla F_t(z,z') }-1- \nabla F_t(z,z') e^{\nabla F_t(z,z') }\right) L(z,z') f_t(z) g_t(z)\,dm(z)\\&=-\int \sum _{z',\, z'\sim z}\zeta \left( e^{\nabla F_t(z,z') }\right) L(z,z') \,d{\widehat{Q}}^\gamma _t(z). \end{aligned}$$

We want now to apply again (87) with \(V_t(z)=\sum _{z', z'\sim z}\zeta \left( e^{\nabla F_t(z,z') }\right) L(z,z') \), \(z\in \mathcal {X}\). From the inequality, \(|\zeta (a)|\le 2 +a^2,a>0\) and using (83), one may check as above that (85) holds. The backward equations ensure that

$$\begin{aligned} \partial _t V_t(z)&= \sum _{z',\, z'\sim z} \left( \frac{Lf_t(z')}{f_t(z)} -\frac{f_t(z')Lf_t(z)}{f_t^2(z)}\right) \zeta '\left( e^{\nabla F_t(z,z') }\right) L(z,z')\\&= \sum _{z',\, z'\sim z} e^{\nabla F_t(z,z') }\left( \frac{Lf_t(z')}{f_t(z')} -\frac{Lf_t(z)}{f_t(z)}\right) \nabla F_t(z,z') \,L(z,z')\\&=\sum _{z',\,{z''},\, z\sim z'\sim {z''}} \nabla F_t(z,z') \,e^{\nabla F_t(z,z') }\left( e^{\nabla F_t(z',{z''})}-1\right) L(z,z')\,L(z',{z''}) \\&\qquad -\sum _{z',\,w', \,z'\sim z, \,w'\sim z} \nabla F_t(z,z')\, e^{\nabla F_t(z,z') }\left( e^{\nabla F_t(z,w')} -1\right) L(z,z')\,L(z,w'). \end{aligned}$$

Simple computations together with (83) show that (86) holds too.

Applying the identity (87), since

$$\begin{aligned}&\sum _{z',\, z'\sim z} e^{\nabla F_t(z,z') }\nabla V_t(z,z')\,L(z,z')\\&\quad = \sum _{z',\,{z''},\, z\sim z'\sim {z''}}e^{\nabla F_t(z,z') } \zeta \left( e^{\nabla F_t(z',{z''}) }\right) L(z,z')\,L(z',{z''}) \\&\qquad - \sum _{z',\,w', \,z'\sim z, \,w'\sim z} e^{\nabla F_t(z,z')} \zeta \left( e^{\nabla F_t(z,w') }\right) L(z,z')\,L(z,w'), \end{aligned}$$

one gets for any \(t\in (0,1)\),

$$\begin{aligned} \varphi ''(t)&=-\int \bigg [\sum _{z',\,w', \,z'\sim z, \,w'\sim z} \!\!\!\! \!\!\!\!\Big [ \zeta \left( e^{\nabla F_t(z,w') }\right) - \nabla F_t(z,z') \left( e^{\nabla F_t(z,w')} -1\right) \Big ]\\&\quad \times e^{\nabla F_t(z,z')}L(z,z')\,L(z,w')\\&\quad + \!\!\!\! \!\!\!\!\sum _{z',\,{z''},\, z\sim z'\sim {z''}} \!\!\!\! \!\!\!\!\Big [\nabla F_t(z,z') \left( e^{\nabla F_t(z',{z''})}-1\right) - \zeta \left( e^{\nabla F_t(z',{z''}) }\right) \Big ]\\&\quad +e^{\nabla F_t(z,z') } L(z,z')\,L(z',{z''})\bigg ] d{\widehat{Q}}^\gamma _t(z)\\&=-\int \Big [\sum _{z',\,w',\, z'\sim z,\, w'\sim z} \Big ( \left( \nabla F_t(z,w') - \nabla F_t(z,z')\right) -1\Big )\\&\quad \times e^{\nabla F_t(z,w')+\nabla F_t(z,z')} L(z,z')\,L(z,w')\\&\quad \left. +\sum _{z',\,w',\, z'\sim z,\, w'\sim z} \left( \nabla F_t(z,z')+1\right) e^{\nabla F_t(z,z')}L(z,z')\,L(z,w')\right. \\&\quad \left. -\sum _{z',\,{z''}, \,z\sim z'\sim {z''}}\left( \nabla F_t(z,z')+1\right) e^{\nabla F_t(z,z')}L(z,z')\,L(z',{z''}) \right. \\&\quad -\sum _{z',\,{z''}, \,z\sim z'\sim {z''}} \rho \left( e^{\nabla F_t(z,z') },e^{\nabla F_t(z,{z''}) }\right) L(z,z')\,L(z',{z''})\Big ] d{\widehat{Q}}^\gamma _t(z), \end{aligned}$$

where the last equality holds since \(\nabla F_t(z,z')+\nabla F_t(z',{z''})=\nabla F_t(z,{z''})\). The expected expression of \(\varphi ''(t)\) follows by symmetrization of the first sum in \(z'\) and \(w'\), and since \(\sum _{w',\, w'\sim z} L(z,w')=-L(z,z)\). \(\square \)

Proof of Lemma 3.1

Let \(\varepsilon \in (0,1/2)\). We first prove that if (13), (14) and (15) hold then \(\varphi ''_\gamma (t)\) is uniformly lower bounded over all \(t\in [\varepsilon , 1]\) and \(\gamma \in (0,\bar{\gamma }]\) for some \(\bar{\gamma }\in (0,1)\). According to (48) and inequality (50) and (51), for any \(t\in [\varepsilon , 1]\) and \(\gamma >0\),

$$\begin{aligned} \varphi ''_\gamma (t)&\ge -O(1)\left[ \frac{|\gamma \log \gamma |}{\varepsilon }\int d^2(x_0,z)K^{d(x_0,z)} d{\widehat{Q}}_{t}^\gamma (z)\right. \\&\quad \left. +\frac{1}{\varepsilon ^2}\int \Big (d^2(x_0,z)+1\Big ) K^{2 d(x_0,z)}d{\widehat{Q}}_{t}^\gamma (z)\right] \\&\ge -O(1) \int d^2(x_0,z) K^{2 d(x_0,z)}d{\widehat{Q}}_{t}^\gamma (z), \end{aligned}$$

where O(1) denotes a positive constant that only depends on \(\bar{\gamma }\) and \(\varepsilon \). Using Lemma 4.4 (vii) and the fact that \(\nu _0\) and \(\nu _1\) have bounded support, it follows that

$$\begin{aligned} \varphi ''_\gamma (t)&\ge - O(1) \!\! \!\! \!\! \!\!\sum _{x\in \mathrm{supp}(\nu _0), y\in \mathrm{supp}(\nu _1)}\!\! \max _{ z\in [x,y]}\left( d^2(x_0,z) K^{2 d(x_0,z)}\right) \\&\quad - O(1) \sum _{z\in \mathcal {X}}d^2(x_0,z)\left( \gamma K^3\right) ^{[2 d(x_0,z)-4D-1]_+} \\&=-O(1) -O(1)\sum _{z\in \mathcal {X}}d^2(x_0,z)\left( \gamma K^3\right) ^{[2 d(x_0,z)-4D-1]_+} \end{aligned}$$

From hypothesis (15), choosing \(\bar{\gamma }\) so that \((\bar{\gamma }K^3)^2<\gamma _o\), one gets

$$\begin{aligned} \inf _{\gamma \in (0,\bar{\gamma }), t\in [\varepsilon , 1]} \varphi ''_\gamma (t) \ge -O(1) . \end{aligned}$$

One may similarly proved by symmetry that if (13), (14) and (15) hold, then \(-\psi ''_\gamma (t)\) is also uniformly lower bounded, namely

$$\begin{aligned} \inf _{\gamma \in (0,\bar{\gamma }), t\in [0, 1-\varepsilon ]} \psi ''_\gamma (t)\ge - O(1) . \end{aligned}$$

Let \(\varepsilon \in (0,1/2)\), and for \(\gamma \in [0,1)\), let

$$\begin{aligned} F_\gamma ^\varepsilon (t)= H({\widehat{Q}}^\gamma _{(1-\varepsilon )t+\varepsilon (1-t)}|m), \qquad t\in [0,1]. \end{aligned}$$

We will first prove a convexity property for the function \(F_0^\varepsilon \) from a convexity property of \(F^{\gamma _\ell }_\varepsilon \) as the sequence \((\gamma _\ell )\) goes to zero. We use the identity, for any \(t\in (0,1)\)

$$\begin{aligned} (1-t) F_{\gamma _\ell }^\varepsilon (0)+tF_{\gamma _\ell }^\varepsilon (1)- F_{\gamma _\ell }^\varepsilon (t)=\frac{t(1-t)}{2}\int _0^1 K_t(s){(F_{\gamma _\ell }^\varepsilon )}''(s)\,ds, \end{aligned}$$
(88)

where the kernel \(K_t\) is defined by (38). Observe that

$$\begin{aligned} \int _0^1 K_t(s){(F_{\gamma _\ell }^\varepsilon )}''(s)\,ds=(1-2\varepsilon ) \int _\varepsilon ^{1-\varepsilon } K_t\left( \frac{u-\varepsilon }{1-2\varepsilon }\right) \left( \varphi _{\gamma _\ell }''(u) +\psi _{\gamma _\ell }''(u)\right) du. \end{aligned}$$

The above uniform bounds on \(\varphi _{\gamma }''\) and \(\psi _{\gamma }''\) for \(\gamma \in (0,\bar{\gamma })\) allow to apply Fatou’s Lemma. Together with Lemma 4.4 (ix) it implies, for any \(\varepsilon \in (0,1/2)\)

$$\begin{aligned}&(1-t) F_{0}^\varepsilon (0)+tF_{0}^\varepsilon (1)- F_{0}^\varepsilon (t) \nonumber \\&\quad \ge \frac{t(1-t)}{2} (1-2\varepsilon )\int _\varepsilon ^{1-\varepsilon } K_t\left( \frac{u-\varepsilon }{1-2\varepsilon }\right) \liminf _{\gamma _\ell \rightarrow 0}\left( \varphi _{\gamma _\ell }''(u) +\psi _{\gamma _\ell }''(u)\right) du. \end{aligned}$$
(89)

For any \(t\in [0,1] \) the support of the measure \({\widehat{Q}}_t\) is finite, included in the set B defined Lemma 4.4 (vii). As a consequence, the function \(t\in [0,1]\rightarrow H({\widehat{Q}}_t|m)\) is continuous as a finite sum of continuous functions. It follows that for any \(t\in [0,1]\),

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} F_{0}^\varepsilon (t)= H({\widehat{Q}}_t|m). \end{aligned}$$

Consequently, using hypothesis (39) and applying Fatou’s Lemma as \(\varepsilon \) goes to zero, equality (89) provides

$$\begin{aligned}&(1-t) H(\nu _0|m)+tH(\nu _1|m)- H({\widehat{Q}}_t|m)\\&\quad \ge \frac{t(1-t)}{2}\int _0^1 K_t\left( u\right) \Big (\liminf _{\gamma _\ell \rightarrow 0}\varphi _{\gamma _\ell }''(u) +\liminf _{\gamma _\ell \rightarrow 0}\psi _{\gamma _\ell }''(u)\Big ) \,du\\&\quad \ge \frac{t(1-t)}{2}\int _0^1 K_t\left( u\right) \xi ''(u) \,du\\&\quad = (1-t)\xi (0)+t\xi (1)-\xi (t) \end{aligned}$$

were the last equality is a consequence of identity (88) applied with \(\xi \). \(\square \)

Proof of Lemma 3.4

Let \(z\in {\widehat{Z}}\) and \(z'\in V(z)\). One will only compute the expression of \(\lim _{\gamma _\ell \rightarrow 0} \left( \gamma _\ell A_t^{\gamma _\ell }(z,z')\right) \) and similar calculations provide \(\lim _{\gamma _\ell \rightarrow 0} \left( \gamma _\ell B_t^{\gamma _\ell }(z,z')\right) \). For any \(\gamma >0\), let

$$\begin{aligned} a_t^\gamma (z,y) :={\widehat{Q}}^\gamma (X_t=z|X_1=y)= \int {Q_t^{\gamma }}^{w,y}(z) \,d\widehat{\pi }^{\gamma }_{_\leftarrow }(w|y), \end{aligned}$$

and

$$\begin{aligned}&{\mathrm a}_t^\gamma (z,z',y):=\int \alpha _t^\gamma (y,z,z',w) \,d\widehat{\pi }^\gamma _{_\leftarrow }(w|y),\quad \\&\quad \text{ with }\quad \alpha _t^\gamma (y,z,z',w)=\, \frac{P_{1-t}^\gamma (y,z)P^\gamma _t(z',w)}{P_1^\gamma (y,w)}. \end{aligned}$$

Using equality (11) and since \(P^\gamma _1 f^\gamma (y)>0\) for any \(\gamma >0\), one easily check that for any \(\gamma >0\),

$$\begin{aligned} A_t^\gamma (z,z') = \,\frac{P_t^\gamma f^\gamma (z')}{P_t^\gamma f^\gamma (z)}=\frac{{\mathrm a}_t^\gamma (z,z',y)}{a_t^\gamma (z,y)}. \end{aligned}$$

From the expression (41) of \(a_t(z,y)\) and since \(\mathrm{supp}(\widehat{\pi }^{\gamma _\ell }_{_\leftarrow }(\cdot |y))\subset \mathrm{supp}(\nu _0)\), one has

$$\begin{aligned} \left| \, a^{\gamma _\ell }_t(z,y)-a_t(z,y)\,\right|&\le \sup _{w\in \mathrm{supp}(\nu _0)} \left| \,{Q_t^{\gamma _\ell }}^{w,y}(z)-{Q_t}^{w,y}(z)\,\right| \\&\quad + \sum _{w\in \mathrm{supp}(\nu _0)} \left| \,\widehat{\pi }^{\gamma _\ell }_{_\leftarrow }(w|y)-\widehat{\pi }_{_\leftarrow }(w|y)\,\right| . \end{aligned}$$

Therefore, the weak convergence of \((\widehat{\pi }^{\gamma _\ell })_{k\in \mathbb {N}}\) to \(\widehat{\pi }\) and Lemma 4.4 (iv) imply

$$\begin{aligned} \lim _{\gamma _\ell \rightarrow 0} a_t^{\gamma _\ell }(z,y)=a_t(z,y). \end{aligned}$$
(90)

Let us now consider the behaviour of \(\gamma _\ell {\mathrm a}_t^{\gamma _\ell }(z,z',y)\) as \(\gamma _\ell \) goes to zero. Lemma 4.4 (iii) provides the following Taylor expansion,

$$\begin{aligned} \gamma \alpha _t^\gamma (y,z,z',w)&= \gamma ^{d(y,z)+1+d(z',w)-d(y,w)}r(y,z,z',w)\\&\quad \times \frac{d(y,w)!}{d(y,z)! d(z',w)!}\,(1-t)^{d(y,z)}t^{d(z',w)}\\&\quad \cdot \left( 1+\gamma \left( K^{d(y,z)}+K^{d(z',w)}+K^{d(y,w)}\right) O(1) \right) , \end{aligned}$$

where O(1) is a quantity uniformly bounded in \(t,\gamma ,z,z',x,y\). By the triangular inequality and since \(z\sim z'\), one has \(d(y,w)\le d(y,z)+1+d(z',w),\) with equality if and only if \((z,z')\in [y,w]\). Therefore, one gets

$$\begin{aligned} \lim _{\gamma \rightarrow 0} \gamma \alpha _t^{\gamma }(y,z,z',w)= \alpha _t(y,z,z',w), \end{aligned}$$

with

$$\begin{aligned} \alpha _t(y,z,z',w) :=\mathbbm {1}_{(z,z')\in [y,w]}\,r(y,z,z',w)d(y,w){\rho }_t^{d(y,w)-1}(d(z,w)-1). \end{aligned}$$

Moreover, Lemma 4.4 (i), (ii) and (iii) ensures that for any \(w\in \mathrm{supp}(\nu _0)\) and \(y\in \mathrm{supp}(\nu _1)\),

$$\begin{aligned} \gamma \alpha _t^\gamma (y,z,z',w)&\le O(1) \,\gamma ^{d(y,z)+1+d(z',w)-d(y,w)}\;{(2S)^{d(y,z)+d(z',w)-d(y,w)}}\;K^{d(y,z)+d(z',w)}\\&\qquad \cdot \,\max _{w\in \mathrm{supp}(\nu _0),y\in \mathrm{supp}(\nu _1)}\frac{(2S)^{d(y,w)} d(y,w)! K^{d(y,w)}}{I^{d(y,w)}}\\&\le O(1) \,(\gamma 2SK)^{d(y,z)+d(z',w)+1-d(y,w)}, \end{aligned}$$

where O(1) is a constant independent of \(t,y,z,z',w\). Therefore \(\gamma \alpha _t^\gamma (y,z,z',w)\le O(1)\) as soon as \(\gamma <1/(2SK)\). As a consequence, for any \(\gamma _\ell <1/(2SK)\), it holds

$$\begin{aligned} \left| \, \gamma _\ell {\mathrm a}^{\gamma _\ell }_t(z,z', y)-{\mathrm a}_t(z,z', y)\,\right|&\le \sup _{w\in \mathrm{supp}(\nu _0)} \left| \,\gamma _\ell \alpha _t^{\gamma _\ell }(y,z,z',w)-\alpha _t(y,z,z',w)\,\right| \\&\quad + O(1) \sum _{w\in \mathrm{supp}(\nu _0)} \left| \,\widehat{\pi }^{\gamma _\ell }_{_\leftarrow }(w|y)-\widehat{\pi }_{_\leftarrow }(w|y)\,\right| , \end{aligned}$$

As \(\gamma _\ell \) goes to 0, this inequality with the weak convergence of \(\widehat{\pi }^{\gamma _\ell }\) to \(\widehat{\pi }^{0}\) implies

$$\begin{aligned} \lim _{\gamma _\ell \rightarrow 0} \gamma _\ell \,{\mathrm a}^{\gamma _\ell }_t(z,z', y)={\mathrm a}_t(z,z', y), \end{aligned}$$

The set \({\widehat{Y}}_z\) is not empty since \(z \in {\widehat{Z}}\). Since for any \(y\in {\widehat{Y}}_z\), \(a_t(z,y)\ne 0\), it follows from (90) that \(\gamma _\ell A_t^{\gamma _\ell }(z,z')\) converges as \(\gamma _\ell \) goes to zero with for any \(y\in {\widehat{Y}}_z\),

$$\begin{aligned} \lim _{\gamma _\ell \rightarrow 0} \gamma _\ell A_t^{\gamma _\ell }(z,z')=\frac{{\mathrm a}_t(z,z', y)}{a_t(z,y)}. \end{aligned}$$

The proof of the first part of Lemma 3.4 is completed.

We now turn to the proof of the second part of Lemma 3.4. One will only compute \(\lim _{\gamma _\ell \rightarrow 0} \left( \gamma _\ell ^2 A_t^{\gamma _\ell }(z,{z''})\right) \) for \(z\in {\widehat{Z}},{z''}\in \mathbb {V}(z)\) and the expression of \(\lim _{\gamma _\ell \rightarrow 0} \left( \gamma _\ell ^2 B_t^{\gamma _\ell }(z,{z''})\right) \) follows from similar calculations. For any \(y\in \mathcal {X}\) and any \(t>0\), one has

$$\begin{aligned} A_t^{\gamma }(z,{z''}) =\frac{ {\mathrm a}_t^\gamma (z,{z''},y)}{a^\gamma _t(z,y)}, \end{aligned}$$

with

$$\begin{aligned} {\mathrm a}_t^\gamma (z,{z''},y):=\int \, \alpha _t^{\gamma }(y,z,{z''},w)\,d\widehat{\pi }^\gamma _{_\leftarrow }(w|y). \end{aligned}$$

It remains to compute \(\lim _{\gamma _\ell \rightarrow 0} \gamma _\ell ^2 {\mathrm a}_t^{\gamma _\ell }(z,{z''},y)\) to prove (45). As above, Lemma 4.4 (iii) provides

$$\begin{aligned} \gamma \alpha _t^\gamma (y,z,{z''},w)&= \gamma ^{d(y,z)+2+d({z''},w)-d(y,w)}r(y,z,{z''},w)\,\frac{d(y,w)!}{d(y,z)! d({z''},w)!}\,(1-t)^{d(y,z)}t^{d({z''},w)}\\&\quad \cdot \left( 1+\gamma \left( K^{d(y,z)}+K^{d({z''},w)}+K^{d(y,w)}\right) O(1) \right) , \end{aligned}$$

where O(1) is a quantity uniformly bounded in \(t,\gamma ,z,{z''},x,y\). Since \(d(y,w)\le d(y,z)+2+d({z''},w)\) with equality if and only if \((z,{z''})\in [y,w]\), it follows that

$$\begin{aligned}&\lim _{\gamma \rightarrow 0}\gamma ^2 \alpha _t^\gamma (y,z,{z''},w) =\alpha _t(y,z,{z''},w)\\&\quad :={\mathbbm {1}_{(z,{z''})\in [y,w]}} \,r(y,z,{z''},w)\, d(y,w) (d(y,w)-1){\rho }_t^{d(y,w)-2}(d(z,w)-2). \end{aligned}$$

Moreover, Lemma 4.4 (i), (ii) and (iii) gives that for any \(w\in \mathrm{supp}(\nu _0)\) and \(y\in \mathrm{supp}(\nu _1)\),

$$\begin{aligned} \gamma ^2\alpha _t^\gamma (y,z,{z''},w) \le O(1)\, (\gamma 2SK)^{d(y,z)+d(z',w)+2-d(y,w)}, \end{aligned}$$

where O(1) is a constant independent of \(t,y,z,{z''},w\). As above, the proof ends as \(\gamma _\ell \) goes to 0 from the inequality

$$\begin{aligned}&\left| \, \gamma _\ell ^2 {\mathrm a}^{\gamma _\ell }_t(z,{z''}, y)-{\mathbbm {a}}_t(z,{z''}, y)\,\right| \\&\quad \le \sup _{w\in \mathrm{supp}(\nu _0)} \left| \,\gamma _\ell ^2\alpha _t^{\gamma _\ell }(y,z,{z''},w)-\alpha _t(y,z,{z''},w)\,\right| \\&\qquad + O(1) \sum _{w\in \mathrm{supp}(\nu _0)} \left| \,\widehat{\pi }^{\gamma _\ell }_{_\leftarrow }(w|y)-\widehat{\pi }_{_\leftarrow }(w|y)\,\right| , \end{aligned}$$

for all \(\gamma _\ell <1/(2SK)\). The end of the proof of the second part of Lemma 3.4 is identical to the one the first part. \(\square \)

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Samson, PM. Entropic curvature on graphs along Schrödinger bridges at zero temperature. Probab. Theory Relat. Fields 184, 859–937 (2022). https://doi.org/10.1007/s00440-022-01167-4

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Keywords

  • Displacement convexity property
  • Ricci curvature
  • Graphs
  • Bernoulli Laplace model
  • Discrete hypercube
  • Schrödinger bridges
  • Transport-entropy inequalities
  • Concentration of measure
  • Prékopa–Leindler inequalities

Mathematics Subject Classification

  • 60E15
  • 32F32
  • 39A12